**MATHEMATICS AND 5D**

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**Introduction.**

**The Fractal Universe**: its bio-topo-logic properties, its 5 Dimotions & s=T symmetries.

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**Book I. The spatial view. Geometry. Its S=T symmetry. Numbers**

**∞ Mind Spaces.**Its different p.o.v.s Artistic Geometries.**Philosophy of Mathematics:**5 sub-disciplines as mirrors of ¬∆@st. 3 Ages. S-point=∆-numbers symmetry.

**III.** **∆-scales: Number theory.** S=T Dualities: Space Points = Scale numbers. Polygons. Primes. Closure: N,Z,R,Q,C.

**Algebra:**Its operands as Dimotions. The Fractal Generator. Existential Algebra.

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**Fractal points:**¬E Points. ¡logic Geometry. Its 5 Non-E Postulates as Mirrors of scalar growth.**Vital Topology.**The 3±¡ varieties of reality.

**VII.** **Time Geometry:3 ages:** 2D Geometry->Analytic & Differential Geometry ->5D hyperbolic geometry & Topology.

**VIII. Space Geometry: Entanglement. **Trinity and Pentalogic. Its 3 Classes: S@: Mental; S=T; Topologic & 5D Spaces.

**Scalar Geometry of the 5**Network spaces.^{th}Dimension.

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**X**.**1**Greek, lineal Geometry in the bidimensional plane.^{st}

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**2**Analytic geometry: S=T symmetries. Conics & Curves.^{nd}Age.

**XII.** **2 ^{nd} Age. **Differential Geometry: Motions of a point.

**XIII.** **2 ^{nd} Age. **Vectorial Spaces

**XIV.** **3 ^{rd} Age.** ∆±¡: Hyperbolic geometry

**3**Metric spaces: Riemann’s r=evolution. From phase spaces to Hilbert Spaces.^{rd}Age:

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**XVI. ∆±¡ Age:** The Future of Geometry. Pangeometry of Existential actions. Ethonomics. Its vital frame of reference.

**XVII. ∆±¡ Age: **Evolution of vital geometric space-time: how mental spaces and vital topologies construct the world.**
Abstract**. This article introduces mathematics as anexperimental mirror language of the 5D Universe, and its 3 ∆ST essential elements, studied by its 3 main sub-disciplines. Spatial fractal points studied by geometry; Scalar numbers studied by Algebra, and time=change dimotions studied by analysis. As the 3 elements, ∆ST are entangled into ‘planes of space-time’, so are the 3 sub-disciplines of science. As

*scales of space-time include the other two elements, so Algebra includes all other elements and numbers become the main substance of mathematics. As entanglement means a huge number of different possible,*¡logic mirror symmetries between all those elements, when a ‘mind’ creates a language to observe it, mathematics has multiple forms to perceive ∆st.

How to put a sequential order to all this? We don’t beyond using the simplest ‘age scheme’ of growing complexity that provides the best sequential progression that mimics the worldcycle of any existence, mental or physical. But the true ‘jump’ of understanding is to force your mind into parallel, synchronous, simultaneous thinking and perceive the ‘symmetries’, dualities, trinities, pentalogic structures that cross between disciplines.

Consider the operands that grow in complexity, from ± social sums, to products and inverse divisions, to √x^{a}, to the crown of ∫∂ operands *that include all others to study change between time-space planes of finitesimal, derivative parts and integral wholes… We cannot study thus them isolated, but we will *in the books on algebra and analysis build as a stair of growing complexity because the Universe also builds the different dimotions in sequences of growing complexity to achieve the ‘higher scalar’ dimotions of reproduction and social evolution.

Moreover ∆ST real elements of Nature, its co-existing planes of space-time, which form every 3 ∆±¡ units a supœrganism, *are accessed not fully beyond our own T.œ (Timespace supœrganism) but through ‘limited minds’, which select information, by entropically erasing ‘dark spaces’ into a continuum; reducing inner properties into points without parts, etc.*

*So we add to the reality of ∆St, the ¬ (entropy) and @-mind mirrors to form, ¬∆@st, the 5 elements of reality. *

*So we must regain back the erase information – those forgotten properties to improve the Æ-mirror (Aristotelian, single time causality and Euclidean points) into ¬Æ maths, with the handicap of being this a single humind in declining health. *So I will also as an ¬@ reduce what I say and repeat as old men do the essence of it. Since at the beginning and the end of all, there is indeed just an infinite repetition of ¬∆@st.

*Thus this as all papers are just a final seed for future researchers to grow on better foundations human ‘stiences’ of timespace planes. *To do so you have first to forget sequential, lineal single time causality, as reality is an entanglement of 5 causes, which can be seen in many mirror views. The simplest one though is as a tug of war of the mathematical unit – a fractal point, which ‘holds a mind in itself’ (a Leibnizian monad, or non-euclidean point traversed by infinite parallels). This Fractal point in still mind-mood is SS, and it tries to ‘stop’ the flow of entropic time, TT, to form mind images of momentum, Ts, dominant in Time-motion and information, St, dominant in Spatial mental form. And when it achieves it, St=Ts, a balanced S=T wave of energy is born, which ‘connects’ fractal points into networks, which become vital topological planes, according to its degree of congruence, similarity, enhanced by the constant transfer of St-information and Ts-momentum. *This is the game of exist¡ence, which an easy-upgrading of the Axioms and postulate of Euclidean Geometry studies in depth.*

*As such ¬E Geometry is by far the simplest, easiest mode to mirror the world of those *5 ‘dimensional motions’ of space and time and its scales, and SS-minds/points that gauge its St-information within the entropic limits of their inner worlds; reason why it is our first paper on 5D Mathematics.

But the whole ∆St is represented in mathematics by ‘algebra of scalar numbers’, ‘geometry of spatial points’, and ‘analysis of the best ∫∂ time operands’ and its entropic inverse functions, put in mental space through @-frames of reference. All this, as I assume readers will know more than I do of mathematics, is the essence of it. I repeat – my task is to establish a simpler Copernican more truthful foundations for future scholars to recast all what they know better than I do in the more focused transparent image. Forgive then my errors, drink on the strengths of the mirror, because if you do as those who developed the seed of Planck, the new field of 5D is ginormous and largely unexplored. I can’t even order the huge GBs of 30 years of research now in declining physical and mental health. I’ll just do what I can from here to my entropic eternity. This said this is the best thing that has happened to Mathematics since Riemann and Lobachevski, jumping over Mr. Cantor’s idealist sets, just a lesser mirror of the ∆-planes of space-time that include it all (reason why sets can include it all, but why to use a foggy mirror when we shall provide the real one, of scalar algebraic numbers, spatial fractal points with parts and dimotion operands?) *If the mirror we offer is not good enough is my fault, NOT the fault of the substance of the mirror – remember that; blame the old man, not the vision he once had and you can improve with youth and energy I lack.*

We concentrate on the S@, spatial mental geometric elements, keeping for a second volume the study of time algebra. Our aim is to prove the experimental nature of mathematics, which along logic is the main science of space and time. Its content is a small part of my research in the field in a discipline that if ever completed by future researchers will recast the entire subject of mathematics as set theory did in the XX c. in terms of its experimental power to describe the ultimate laws of ‘mental spaces’ and ‘time motions’.

Scalar spacetime has 3 units, time operands, studied in algebra, social scalar numbers and Non-Euclidean fractal space points, whose symmetry is study by non-E geometry completes with 5 postulates of points, lines and planes with breath and relative congruence. Once we complete ¬E geometry; we define the main duality between subjective **mental phase spaces** as still mirrors of the world that select the information minds need to act and survive in the game of exist¡ence, that creates objective ‘vital, **topologic** **organisms’** and projecting its geometric mind in its territorial local order as geometry ‘informs’ the underlying timespace we are all made of.

5D mathematics *returns to the mature experimental age of the discipline that went as all systems do through 3 ages from a simple *lineal age of Greek Geometry to its curved realist age of calculus that added motion & scales of 5D finitesimal derivatives and 4D wholes to better mirror the fractal Universe, to a 3^{rd} age of excessive informative fictions, when it abandons its realist foundations, when during the German creationist ego-trip of Hilbert and his Cantorian paradises, he says, ‘I imagine points, lines and planes’ thinking he shares the only language ‘God’ uses to create reality. Hence mathematics is no longer considered an a posteriori mirror of scalar timespace but its ‘generator’ and as the first ‘Aristotelian cause’, needs no experimental proof but rather the opposite: reality exists only if it can be casted into mathematical models (‘only what we measure is real’ Planck). So mathematicians abandon space points, scale numbers and time operands as the 3 mathematical images generated by the ∆st Universe, using instead ‘Cantorian’ sets as its ‘imagined’ units nowhere to be seen in reality. And Hilbert affirms the self-contained Axiomatic method of proof, NOT connected to the world, despite Gödel’s incompleteness theorem- *a baroque inward looking 3 ^{rd} age of excessive form proper of all systems *we abandon to return to its empirical foundations, now formulated in terms of those 5 structural elements that create reality, mimicked by the main mathematical elements. Since mathematics reflect the properties of scalar space and cyclical time in its main elements, points numbers and operands entangled in feed-back ≤=≥ equations. So we define mathematics as an experimental language that mirrors in a simplified manner as all languages do to fit the mind, the elements and structures of the generational space-time, we are all made of. As such mathematics is only 2nd to ¡logic and its fractal, ternary Universal grammar, as the fundamental formal, experimental language of the Universe and its generational time space that creates all its super-organisms (ab. T.œs). The Math’s pro must be humble to value this work that is not so much on new theory but

*on entangling maths with the vital Reality it mirrors & set theory cut off from.*

We study how mind languages mirror supœganisms made of scalar space and cyclical time. So we introduce the metric equation that describes the scales of the 5^{th} Dimensional Universe: SxT (max.s=t) = C. Then we couple the 5D elements of reality (entropy, scales, minds, space and time) in its 3 ‘relative scales of size and time duration’, as fast 5 Dimotions=actions (short view), superorganisms tracing worldcycles (medium view) and ¬∆@st elements (long view) with the 5 disciplines of mathematics as experimental mirrors of those 5 elements: Space=geometry, ∆: Scales: Number theory, S<=>T Dimotions and Symmetries=algebra; which analysis is best for time motions; @-minds = Analytic geometry (Frames of reference) and philosophy of mathematics; while entropy, negation of information appears in the limits of calculus, inverse operands and exponential functions.

Since, as scalar space includes scales & ‘still mind mappings’ and Time the entropic limits of death and all its motions; we can reduce mathematics to spatial geometry and temporal algebra, which indeed were the original disciplines, from where all others branched, and to its minimal units, points of space and numbers of time, which appeared even earlier in the human consciousness. We shall thus consider the S=T symmetries between fractal Non-Euclidean space points (ab.•) and numbers (ab.Nº); and latter on between Topology and Algebra; and couple the internal elements of S-geometry (dimensions) and T-algebra (operands) with the 5 Dimotions (dimensional motions) of the fractal Universe.

So mathematics is only 2^{nd} to ¡logic in the quality of its experimental mirror expressing the Disomorphisms (equal dimensional laws) of all systems of Nature. As each of its 2->5 subdisciplines reflect those 5 Dimotions of the Universe, which all stiences expresses in different forms. In mathematics geometry does so through 3±¡ topologic bidimensional varieties and classic 3 dimensions (1D: height, 2D: Length, 3D: width; 4-5D: fractal dimensions and topologic networks). While algebra mirrors the 5 Dimotions with its 5 operands & inverse entropic functions (1D: sin/cos, 4D:±, 3D: x÷, 4D: xª √, 5D:∫∂). So finally we focus on the medium view of ‘entangled supœrganisms, in simultaneous space’, as expressed by vital topology, tracing ‘worldcycles’ as expressed by Existential algebra, which become the new ‘integrated’ 2 polar disciplines of 5D mathematics that should make it a better mirror of the fractal Universe.

**Vital topologic Geometry vs. mental subjective spaces.**

The essential duality of geometry is between mental spaces which are continuous, as they erase the holes between forms, still geometry that eliminate motions and scalar planes, reduced to a mind and with a distorted @-centered frame of reference vs. real vital geometries of discontinuous T.œs and fractal scales:

*Mental geometry: single plane, no motion, 0-point center vs. Vital topology, ∆±¡ co-existing planes, motion.*

The paper thus deals with vital geometry, *the experimental stience that describes the fundamental laws of Time-Space Supœrganisms, seen in simultaneity as a synchronous, stable form of space. *

When dealing with space, we do have to differentiate two main type of spaces and laws:

– Objective, vital topological spaces, made of the 3 bidimensional varieties of topology as adjacent organs, which we can define with the ‘Generator equation’ of Timespace organisms, which in geometric terms writes:

*Lineal-limbs/fields<Ø-hyperbolic Bodywaves>O-spherical Particle/heads*

As those 3 geometries maximize the efficiency of motion, the line being the shortest distance between two points, iteration, the hyperbole the more complex form summoning all others, and the sphere the highest volume of minimal perimeter. The example shows the essence of vital topologic spaces: The forms that are more symmetric with the motion and survival action=function they represent in the real world survive and are therefore repeated in clone T.œs (Timespace supœrganisms). The objective rules of *geometry are thus about vital space, about the symmetry between S-form and T-function, S=T and merge abstract geometry and biologic laws of survival and reproduction and physical laws of motion. *They are easy to understand and the tautological truth derived from them is indeed the obvious truth that we are the vital space we occupy whose functions in time the geometric laws of vital non-Euclidean topology maximize.

Subjective, mental spaces, mirrors of the outer world. This second great field of geometry was ill understood till the Lobachevski->Riemann R=evolution, which definitely understood that ‘spaces’ are distorted mind mappings in simultaneity of the ‘flows of timespace cycles and organisms’ perceived outside our membrain. But again, the vital laws of survival play here a clear role, as mind mirrors which are unfocused and distorted in their judgment of the outer forms do not survive as a correct perception of reality is needed to make it in the existential game.

A key difference between both type of geometries however is the fact that the laws of non-euclidean vital geometry that describe the outer world are all the same regardless of the observer, as they are objective laws of construction of efficient superorganisms. *But the mind spaces are infinite, one for each ‘monad’, each point that is a world in itself (Leibniz) even when they belong to the same species, because their self-centered perspective will differ. *The duality of objective single Universe v. subjective individual mind-*point makes thought languages inflationary in its kaleidoscopic multiplication of different views over the same object such as:*

*Objective laws of vital topology > ∑ **µ** (relative infinity symbol) mental mirror spaces. *

It is important to stress from the beginning this inflationary nature of mind spaces, as many of them are ‘entropy’, disordered unfocused mirrors that do not survive and shouldn’t be study. This is not the case. Mathematicians sponsor an egocy paradox (ego=idiocy) as all huminds do, believing all mathematical forms are worth to explore. But that is not really the case. As in Borges’ Babylonian Library where monkeys type ∞ books writing by chance a Bible; all books exist in Sets, but what matters to us is to distinguish mathematial forms that do exist, *to apply them without errors** to *the next layer of mathematical understanding of the Universe, the praxis of mathematical physics and any application of mathematics to stience that reinforces with a 3^{rd} layer of experimental evidence the 1^{st} – ∆ST laws of generational space-time and 2^{nd} layer of non-AE mathematics.

An example will suffice. There are infinite curves described by polynomials but all curves of the second order can be reduced to the canonical conics, which we find in Nature because they are efficient.

So the very essence of 5D mathematics, in the 2 introductory courses of those papers; is to connect the laws of vital space and cyclical time with the forms of the mathematical mirror, spatial geometry and temporal algebra, while time permitted the 3^{rd} part of this method – the examples in all sciences, notably mathematical physics of those laws, will come in further papers^{1}.

We start then with geometry because it is far more evident to the visual mind of the modern age that temporal algebra, and so we shall introduce first the very basic of 5D space-time laws and the consider the basic upgrading 5D makes of geometry – the rewriting of the postulates and axioms of Euclidean geometry, completing the work of Gauss, Lobachevski and Riemann who only upgraded the 5^{th} postulate. And then with those more solid basis, put in relationship to the Gst laws of fractal organisms, slowly review the simplest elements of Geometry at the level of an Introductory university course of mathematics, with those new postulates. My goal is to interest enough a few professional mathematics for this Copernican revolution not to die with me as I don’t know how long I will remain in exist¡ence.

You have then to understand the nature of scientific r=evolutions, whose pioneers always start with the basics, even if they seem very simple for a scholar who has learned a previous model, which despite its relatively less accurate first principles and postulate has built a mirror-image that suffices to handle reality, because one of the marvels of the fractal world of infinite monads ordering a flow of continuous entropic, indistinguishable time motion is that all mirrors are imperfect, reduced images of the whole; so even a Ptolemaic system with the added epicycles can describe reality, but it is always best to keep improving the initial postulates, as Copernicus did putting the sun in the center, to simplify and better understand those orbital ellipses. This is what the completion of Non-Euclidean geometry means. In a few generations when the model is developed in full the beauty of those renewed first principles will be the marvel of high school students. As they will understand experimental mathematics, which now they hate because of the axiomatic, set theory and pedantic discourse.

Nt. 1 I do apologize for having failed in my initial plan, 30 years ago coming out of Columbia University, when my mind was fresh my enthusiasm for 5D and my hopes for mankind huge, of gathering a group of outstanding scholars of all fields to complete the task of renewing humind’s knowledge of the Universe with exhaustive 5D upgrading in all sciences, with me as the ‘orchestra director’… It was not all my fault, as the staunch rejection or rather censorship of 5D social sciences hindered any chance to get institutional help, and I could not ethically hide the laws of 5D history during the early 90s when I pursued help for academic research. It is truly a miracle given the huge rejection I had all my life to my work that I can manage in my 3^{rd} age still to order those papers that I see as my testament to a life in pursuit of truth and the survival of mankind.

**INTRODUCTION. 5D:**** A ****FRACTAL ****UNIVERSE MADE OF B¡O-TOPO-LOGIC TIME§PACE BEINGS.**

“Henceforth *space* by itself, and *time* by itself, are doomed to fade away into mere *shadows*, and only a kind of union of the two will preserve an independent reality. Hermann *Minkowski*

When we google the 5^{th} dimension one gets surprised by the quantity of speculative answers to a question, which is no longer pseudo-science, but has been for two decades a field of research in systems sciences rather than physics (: no, the answers of google, considering the fifth dimension the upper-self etc. seem to be very popular, but are to the science of the 5th dimension more like a medium in earlier XX c. talking about the 4th dimension as astrological awareness, for lack of understanding of Einstein’s **metric equations** of the 4th dimension).

This is the key word that differentiates pseudo-science from a proper scientific description of a dimension of space-time, *the existence of a metric equation that describes a dimension and allows to travel through it.* Why the 5th dimension metrics are not well known in modern science has to do with the fact it is not researched in physics but systemics, the mother discipline of all sciences of information, far less popular than physics; and the proprietary feeling physicists acquired on space-time matters since Galileo defined its 3D metric equation v=s/t completed with Einstein’s 4D formalism, which makes difficult to spread the knowledge on space-time acquired on other disciplines. The arguments still raging about evolution, the fundamental theory of time in terms of information, as the ‘arrow that defines’ the future of species but has nothing to do with Relativity and locomotion is a clear case of that difficulty.

Indeed, we know since the XIX c. that the creation of the ‘future time’ of an existential entity is not ONLY mediated by the arrow of **locomotion** and entropy studied by physicists with Relativity Metrics (Galileo’s V=s/t and Einstein’s more complex formalism), but there is a second arrow that defines the ‘future’ of existential species – the evolution of its information. So time – *the changes=motions that defines the existence of any species,* has at least 2 dimensions, locomotion or ordered translation in space and a more disordered version, entropy (scattered motion that ‘dissolves’ the inner form of the system, akin to death)…

And in-form-ation, generation of form, inverse to entropy as it requires the social gathering of parts into wholes; happening without external locomotions, as an internal trans-formation of **form**. This evolution of organic form as opposed to external change is what Systemics calls the fifth dimension of time that applies to all sciences.

In the graph the Universe is a fractal that reproduces ‘forms with motion’, informations and then **organizes** them in networks and systems that evolve into larger organic systems creating the scalar structure of reality. Thus we call *the sum of all those co-existing scales of parts and wholes the fifth dimension.*

Then it is necessary to find a metric equation to define this new dimension of space-time. Since a dimension only exists when we can write a mathematical simple metric that leaves the dimension invariant when we change our parameters of space and time – hence we travel through it. (Klein). This equation, as all space-time metric equations, is simple; since metric equations are meant to represent measures of ‘covariant’ motion in a given space-time dimension that leave the other dimensions unchanged. So we write using ð for cyclic time instead of t, for a motion that changes the relative size and speed of clocks of a system (measured with frequency):

**5D Metric: S (Lineal Size/Volume in space) x ð (cyclic speed of its time clocks) = Constant.**

According to those metrics, smaller systems in space have faster time clocks. As information is stored in the frequency and form of those cycles, smaller systems have more information, coding larger ones: genes code cells, memes societies and particles’ quantum numbers code atoms and molecules.

But how we travel in ‘size’ in space and ‘speed of our time cycles. *Here is where the biggest discovery of 5D comes into play: We travel through the worldcycle of life and death, as we are born in a smaller seed with faster time cycles, evolve as an organism coming out in the ∆º-scale within a larger world of slower ‘Deep time cycles’, to die back dissolving our information again into cellular space.*

It is the same process in all 5D journeys of all species that live and die travelling through 3 planes of 5D space-time; from the smallest black hole that is born with an enormous ‘metabolic temperature’, to the new species, routinely born as small individuals (first mammal rat, first robots with small chips; first human likely the Homo Floresiensis, who had the same morphology and used technology and likely spoke, etc.) Then a reproductive radiation multiplies the seed into a larger herd of clones, joined by emergent physiological networks whose slower ‘entropic, informative and reproductive networks, create an ∆º supœrganism that lives tages and dissolves back into ∆-1.

*So 5D adds to the 4D formalism of worldlines, a dimension of growth, shaping the worldcycles of life and death. Reason why *we call 5D metric the function of existence, because its multiple ‘solutions’ are the origin of all the varieties of Space and Time beings, there are – a whole family of functions.

As we keep exploring in depth, 5D metrics and its associated concepts of Space=form and Time=motion in all its varieties, we shall see it is the origin of multiple ‘solutions’, a whole family of function, from where we shall derive most of the logic relationships and particular equations of each science.

In the complex models of existential illogic, we derive all the particular equations of each science from it.

This equation and its use to improve our knowledge of space and time in all sciences, with an emphasis in our models of physical systems will be the theme of this paper. Even if physicists stubbornly refuse to treat information with the same value than entropy. So they call it negentropy, and when you give a conference on the fifth dimension – the dimension of ‘creation of social forms of information, of organic wholes’ – there are no physicists on attendance; and likely no physicists will be reading this post… Let’s then use the metrics of the 5^{th} scalar dimension to explain the fractal, nested Universe and its scales, shown in the graph:

The metric equation of the fifth dimension of space-time (ab.∆) defines 3 known scales of physical systems, with different quantity of information according to 5D metrics, Se (size in space) x Ti (volume of information) =k. Since as we become smaller in space paradoxically our time clocks accelerate, and since information is stored in the cyclical patterns and frequencies of those clocks smaller systems code more information, so quantum particles code atoms, genes organisms, memes civilizations and chips machines establishing the essential symbiosis between ∆-1 scales and ∆º super organisms, inscribed in an slower ∆+1 world.

Those metrics means information is higher in the smaller ‘quantum plane’ than in the larger gravitational one, and inversely the size of its physical parts is larger ins the Gravitational cosmological ‘plane’ than in the quantum one, with the human thermodynamic scale in-between.

As there is no reason to stop the scales of the fractal Universe in particles and galaxies, there is a ‘potential’ fourth, ∆±4 organic plane defined ‘above’ the galaxy, (∆+4, dark energy world) and below the quantum world (∆-4, Bohm’s quantum potential), which represents the larger cosmos. Further on, according to the fractal, nested principle any larger organic system, encloses smaller nested systems. Thus the ∆±4 cosmos contains ∆±3 galaxies, which contain ∆±2 solar systems and planets, which contain ∆±1 thermodynamic organisms and matter states, described by the human ∆º mind languages, contained on our brains, *which according to those metrics* will have a much denser content of information becoming a ‘linguistic Mind-Mirror’ of the whole.

**RECAP.** Ænthropic huminds reduce the multiple clocks of time and vital spaces of reality to the single human clock and spatial scale, and reject the organic properties of other Universal systems. In reality the Universe is a fractal organism of time=motion and space=form, whose purpose is to reproduce those formal motions, and patterns of cyclical reproduction; with its fundamental metric law of balance between space and time, as the guidance of those motions.

The outcome of those processes of reproduction of form, and symbiosis between the different scales and synchronous time cycles of its species is the creation of organic superorganisms. It follows from that nested structure and 5D metrics that speed up information processing in smaller spaces, a symbiotic relationship between ∆-¡, the smaller parts of faster time clocks, which carry more in-form-ation in the form and frequency of its logic cycles, and act as languages that code the larger systems: genes code cells, memes code societies, quantum numbers code atoms and languages code larger wholes.

**LEIBNIZ V. NEWTON: WHERE IS SPACE AND TIME?**

“According to their [Newton and his followers] doctrine, God Almighty wants to wind up his watch from time to time: otherwise it would cease to move. He had not, it seems, sufficient foresight to make it a perpetual motion. Nay, the machine of God’s making, so imperfect, according to these gentlemen; that he is obliged to clean it now and then by an extraordinary concourse, and even to mend it, as clockmaker mends his work.’ * *

*Leibniz-Clarke Correspondence on the absurdity of mechanical models of the Universe*

‘’Leibniz is right. There are infinite time clocks in the Universe, but if so we have to restart science from its foundations’. *Einstein*, on the infinite relational time cycles of reality.

The consequence of the existence of an internal fifth dimension of space-time, made of all other planes=scales of spacetime of a being, its parts and wholes, which store the information of a system, is the fact that we ARE made of planes=scales of space and time. We ARE the vital space we occupy and we ARE the time flow of existence we live between birth and extinction. Reason why all systems and entities of Nature obey the main science of space, geometry->Mathematics and time, Logic. We are broken fractal species of space and time, whose mathematical and logic laws all vital space-time organisms follow as Einstein and Leibniz thought: So we must evolve our logic and geometry of space and time, to extract properties of ‘existential beings’ from them.

**The underlying question of time§pace: Absolute or Relational, Generational Space-Time?**

The fundamental question physicists wondered for centuries regarding the nature of space and time unfortunately was resolved as usual in favor of the simpler view: it is space and time an absolute abstract background of the Universe (Mr. Newton’s view) or are we made of ‘vital space’ that lasts a time duration, so we are generated by the bio-topo-logic properties of scalar space and cyclical time? This is the choice of 5Ð ‘stiences’. And its simpler version was called relational spacetime, sponsored by Mr. Leibniz.

A realist interpretation of the world we live in, which never shows in any scale of reality such background – a mathematical graph used in abstract by human scientists – considers that we ARE the vital space we occupy with our cells, and we LIVE a cyclic time duration between birth and extinction. So *we are* space and time.

The argument thus reached its height in the beginning of science in the correspondence between ** Newton**, the proposer of the absolute Cartesian graph of space-time drawn by God (his body in his own words) vs.

**who rightly considered absolute space and time an abstraction, and so he coined the concept of relational space -merely the adjacent pegging of similar forms in simultaneous space and relational time – the sequence of events which we relate causally with reason. In Newton’s cosmos, space and time provide a fixed, immutable, eternal background, through which particles move. Space and time are the stage of intersecting lines sketched in the illustration. Fact is this ‘mathematical artifact’ made with pen and paper by earlier physicists, called the Cartesian graph, useful to measure ‘translation in space’ is no where to be seen in reality. Unfortunately as time went by the graph became somehow ‘real’ as scientists’ felt the ‘mathematical language’ created reality. But if space is what objects occupy that distance between the red square’s vital spade and the yellow ‘circle’ must have something. Horror vacuum comes then into place: indeed the Universe must be scalar. There must be very small parts between them, which we do not see. And that is what we have proved with microscopes –**

*Leibniz**as we probe smaller distances forms with motion, spaces with time-motions appear and there it seems no limit to the fractal scales of the Universe.*

It meant the invention of an absolute continuous space & single lineal time that extends to ∞ contradicting the fact that all spaces are broken, divided by membranes, and all beings have a finite time duration. Further on, as we kept exploring smaller scales of reality, we never found the drawings of God, not even a solid still substance, but always ‘motions’ tracing closed time-space cycles; since even particles turned to be also ‘vortices of time-space motions’.

So the true, sound experimental and logic theory was Leibniz’s who rightly considered absolute space and time an abstraction, and so he coined the concept of relational space -merely the adjacent pegging of similar forms in simultaneous space and relational time – the sequence of events which we relate causally with reason origin of the ‘Generational space-time’ model of 5Ð in which are the space we occupy and the time we last – as in the graph where there is no longer abstract background lines.

This simple concept was NOT adopted by physicists despite its sheer evidence. *Unfortunately Physicists sided with Newton not with Leibniz on the question of what is space and time – an abstract background put by God or the substance of which we are all made; and so the conceptual jump would not happen.*

*It is the fifth space-time dimension, sum of all other planes of reality, including within it all other dimensions.*

Next, to explain all this properly came Einstein. One of the fundamental discoveries of Einstein is that in our universe, there is indeed no fixed space-time background. In Einstein’s theory of general relativity, which replaces Newton’s theory of mechanics and the gravitational force, the geometry of spacetime is not fixed. Instead it is an evolving, dynamical quantity – a topology; and it is the substance of which reality is. So we are topological beings, geometries of space with motions of time.

What Newton called absolute space-time IS NOT. So space is the sum of all the discontinuous vital spaces, occupied by different beings: ∑s=S.

And lineal time, T the sum of all the finite life-death cycles of all beings T=∑t.

*Since space and time do exist and so if they are not in the background we ‘are’ vital space and cyclic time.*

The simple idea behind the structure of the fractal Universe is then to consider time=change=motion and Topologic, formal space=extension the 2 elements of which all beings are made.

Wheeler said ‘Spacetime tells matter how to move & matter tells spacetime how to curve’. Since *Spacetime is geometry in motion.* Time is change, the perception of change moves time; time is motion; space is its opposite, stillness, form, the information of time. And so it is all about two parameters: Time=Motion and Space=Form.

Look around you, all what you see are ‘space-forms’ with ‘time-motion’. We are all space-time, forms in motion, ‘in-form-motion’, ‘information’, forms in action, play with the words of what you are.

** Recap.** The universe is made of 2 STates, time=motion, and space=form. Yang and yin that combine in balance to create ∞ beings. So you can combine them T & S in 5 different forms:

S=T, the balanced state of existential energy that combine St-information (Potential energy) and Ts-locomotion (Kinetic energy), in constant SHM is the preferred balance of ‘present’ that lasts and iterates any organism, the state of survival.

TT: internal and external motion or ‘entropy’ is one extreme, which brings death. That is the state of mankind today, caused by the fact the new species, machines move faster externally (cars, weapons) and internally ‘metal-minds’, visual images accelerate, disordering humans into a permanent ADD state. So millennials move fast and have disordered minds. At the end of that process entropy kills.

In the other extreme of absolute no motion, there is SS, languages of the mind, seeds, pure, perfect information, enlightenment. A being in that state doesn’t move. The hypothetical Mind of God is such a being, because it perceives all what has existed, exists and will exist and repeat itself eternally as a block of time – a zero sum of fluctuations of the game of existence in space and time.

**WE ARE space and time,** merely of a different kind to that of Newton: Organic scalar spaces, and cyclical, discontinuous times who ‘live’… worldcycles (no longer worldlines as we have a 2^{nd} arrow of information) of exist¡ence (as all species follow the common laws of space and time). As cyclical time that explains the informative repetitive patterns or Laws Nature and its multiple space-time clocks. Why a Universe made of space-time beings is essential to a philosophy of mathematics is obvious*. *Because the main experimental science concerned with space is mathematics and the main science of causal time is logic, if we are made of fractal space & cyclic time, both mathematics & logic become experimental sciences, reflecting the properties of those 2 primary substances, as mirror-languages of maximal synoptic information and minimal size (Sxð=C), whose underlying laws emerge in all other larger scales of the fractal Universe of bigger size and less information, proving also why they apply to all of them

**Cyclic time: ð – the causal repetitive laws of ‘stiences’**

A Universe of ∞ time clocks of different size and speed differs from its human description with a single mechanical clock-time to which all time clocks of the universe are equalized, elongated into a lineal ‘second-minute-hour-day-year’ system of equalized time clocks (of light waves, mechanical clocks, earth’s astronomical clocks). As Galilean physics, born of ballistics, simplified the nature of cycles of time-space into lineal durations, to measure best the locomotions of cannonballs: Time is cyclical, all clocks of time and laws of science are based in the cyclical patterns of nature. But physicists developed ballistics and denied the obvious truth that we can know the future because it will repeat the causality of the past, and we can change it by changing that causality, in History by repressing the lethal memes of the tree of metal and enhance the welfare memes that make us survive. Llineal and cyclical time render the same equations as one is the inverse of the other, measured by frequency, T=1/ƒ, but the philosophical implications of cyclical time, are ginormous and the in-form-ation provided by those cycles, erased by lineal time, a handicap for humans to truly understand the cycles of history and economics, the ‘deep time’ scales of the fifth dimension, and the whole workings of super organisms and its physiological structures.

So we consider an ∆-¡ ‘quanta of time frequency or ‘finitesimal derivative’ of the larger whole represented with the concept of lineal time; as in the classic formula,V(st)=ƒ(t) l(s). We can measure Space e, Vt=S with lineal time as a single unit, or as a sum of frequency steps, with more detail.

A key fact that of a time cycle is to break reality (1^{st} knot theorem) in an outer and inner region, creating a repetitive motion that becomes an isolating membrane that encloses a vital space, the ‘substance of which we are all made’.

A second key element is to be made of 3 ‘relative pi-diameters’, which therefore determine ‘3 ages of time’.

**Local Past=Entropy, Present=Iteration and Future=Information in zero-sum worldcycles. **

‘The separation between past, present and future is an illusion’ . Einstein

Of all the consequences of cyclical time, the most important is the existence of infinite local time clocks of which we are all made, which therefore imply the existence of infinite local past, present and future states.

Past then means a system with less ‘form’, less information, which slowly acquires a dimension of height-form, as it completes its cycle to return back in the moment of death to an age of no information. This ‘worldcycle’ of existence, which creates and erases information becomes then the equation of ‘trinity’, the 3 local ages of life, which each of us follows as a time-space superorganism:

Entropy-youth (past) <Energy-mature reproduction (relative present)> 3^{rd} age of In-Form-ation (relative future)

In physics is equivalent to the dual equation of Einstein: EóMc^{2}, which reverses when E, which should be properly considered ‘Entropy’, as it is a disordered state, collapses through gravitation into Mass, a cyclical vortex of space-time; while its intermediate state is c^{2}, radiation; the relative present:

Whereas the past is the beginning of a pi cycle, starting as a line of entropy with no form that curves and raises in height in its second state of present, and returns back to its origin in its future 3^{rd} age of information, completing a 0-sum of life and death. Thus instead of a single ∞ lineal absolute time there are ∞ living cycles of time happening in zillions of entities.

So the fundamental unit of reality in a given scale, is the timespace cycle, as ‘time’ in a sequence of 3 ages, which close the cycle into a zero sum; or we can see it sequentially in space as a ‘fractal point’, that is a non-Euclidean point with 3 parts, the elements of its angular momentum, which will become the new unit of both geometry and mathematical physics.

In the graph, we can see the 3 dimensional motions of timespace, the relative past associated to an explosive, expansive topology the wave-body present iterative hyperbolic topology (a geometry with motion) and the implosive, elliptic geometry of a black hole, or future vortex of time-space.

## Vital, ternary, Organic Topology

## This said the devil is in the details. So what does it mean to be made of motions with form, time and space? In mathematical terms, it means to be made of topological dimensions, which are holographic bi-dimensional space with motion in time. And as it happens *topology has only 3 varieties of bidimensional spacetime and it is constructed of parts – points – that become wholes – networks. So the immediate translation of generational space-time into modern mathematical systems do convert as we observed in the abstract, all systems into mathematical beings. *

This is the field in which 5D innovates, ‘enlightening’ classic topology to ‘understand the ternary organic, structure’ of all systems of nature: As the 5 Dimotions are dimensions of space with time motion, its science is topology that allows a system to deform= change its inner form. Yet a 4D Universe has only 3 ‘topological varieties’ that restricts ensembles to only 3 topologies, each one best suited to perform the 3 organic vital functions of any physic or biologic system –gauging information (1D) to move the system (2D) to an energy field in which reproduce (3D):

## The purpose of vital topology is to study the 5 Dimotions (dimensional motions) of the Universe… As such it will be the final stage of evolution of geometry as an experimental science, merging elements of all disciplines.

## All dimensions of space have motion in time. Mathematics realized it, as the still Geometry of the Greeks evolved into a vaster, generalized concept, a topological› variety, where a topology as opposed to a geometry has internal motions-changes. As the only case in which the inner dimensions of a being don’t seem to change is external locomotion most 5D motions need ‘geometries’ with inner motion, which are topologic varieties of which there are only 3:

*In the graph, *the diffeomorphic Principle of Einstein’s 4D analysis acquires an organic nature, when we see the Universe as the sum of ∞ Complementary ternary, topologic systems whose dimensions have organic functions: Systems feed on their relative dimension of energy-length, perceive in their relative dimension of height and reproduce in their relative combined dimension of width, which are assembled into each specific species, *to best satisfy the systems’ in taking of motion, energy and information. *

*I.e*: An animal has its informative height in the high perceptive light dimension, but a plant, which uses light as energy has its up and down dimensions inverted respect to the man and its chemical brain buried on the Earth. So both have opposite energy-time coordinates, with an ‘antero-posterior’, lineal’ ‘outward’ energy oriented structure due to the oriented arrow of light. But in a 3D world with no preferred orientation, a sea or vacuum, cyclic forms that maximize information dominate from plankton to galaxies that have a cyclic, informative, inward structure, as the stars’ body absorbs energy from intergalactic space, reproduces matter with it and feeds the internal informative knot of gravitation, with a higher height dimension the black hole.

So the Dimotions of reality are the 3 bidimensional topological varieties that act as vital organs in cylindrical long limbs/fields, Hyperbolic wide bodies and spherical tall heads, each one dominant in a lineal classic dimension, lineal motion, informative height and reproductive width, which DO have organic vital properties too:

Spherical particle-heads, perceiving information from the advantage point of **height**.

**Lineal long, cylindrical **legs and fields of locomotion as the line is the shorter distance between two points.

**Wide, hyperbolic body waves, **storing the energy reproduce by the system.

As such single dimensions perform those organic functions in light waves that huminds perceive as space. So light is also an organic system of 3 dimotions: c-speed length, electric informative height & a wide magnetic field that supports them.

*Moreover beyond its classic analysis of forms and functions, what Vital topology does it to serve the basis for its temporal*-numerical version that is existential algebra, as points become the spatial version of temporal numbers:

*So we develop a formalism of the 5 Dimotions, as operand, «,<,* =,>,», for entropy, locomotion, reproduction, information and social evolution, which will develop into flows of ‘stœps’ that change sequentially a form through its events in time, where the vital topology of each Dimotion is ‘integrated’ within those symbols. In this manner vital topology becomes ‘Existential algebra’ –the analysis of flows of dimotions=actions of T.œs in its worldcycles of existence. Those sequences can then be studied as templates of all T.œs in all scales, which will follow them to complete its survival cycles. And so existential algebra has implicit vital topology. So we define in existential algebra with simple T, S and 5 «,<,≈,>,» the main events of space-time of the Universe:

**1D: T>S: angular cyclical motions of information (Ab. §ð):** the minimal ‘geometry’ of reality, a spherical particle/head or fractal point, the geometry that stores maximal form in minimal space, hence suited for ‘organic functions’ of gauging, storing and perceiving information (particles, heads).

**2D: S<T: Lineal Locomotions***,* **(Ab.$t) *** which will move through its* lineal limbs/fields the system, as the line is the shortest distance between two points… towards a…

**3D: S≈T:** **Fields of vital Energy (Ab. ∑≈∏):** with its hyperbolic body-waves that iterate the forms of both the spherical particle/heads and lineal limbs/fields; as the hyperbolic topology combines the other two, so it can generate them, in the same manner Energy adds as the third conserved space-time quantity the lineal and cyclical momentum of 1 and 2D. To which we must add the 2 ‘scalar’ Dimotions of:

**5D: entropy («, ∂S)** whereas motion is dual internal dissolving the information of the being and external, scattering its parts, hence we use an « ¬Æ symbol; so the system explodes into its ∆-1 parts: ∆«∆-1 (death). **4D:** **organic evolution (»,∫T)** of parts into still locked simultaneous ‘linguistic seeds or mind forms’∆-1»∆

So the abstract 3 conserved substances of reality become organic bidimensional topologies – flat motion in space, cyclical time membranes and the vital 3D energy within them. We then observe its ∞ variety of combinations as topologic ternary species, whereas the 3 perpendicular ‘lineal’ dimensions are a simplification of those organic functions: the height dimension enhances the ‘perception of information’ by O-heads/particles place above in the point of maximal projective geometry; the dimension of ‘length’ maximizes locomotion; physicists’ only time motion, and the width dimension maximizes reproduction and storage of energy.

**The 3 only topologies of physical systems: conservation laws.**

The 3 parts of any fractal point of cyclical time, of angular momentum, in the Universe become then in the simplest physical systems, the 3 conserved parts of the minimal organic Unit of reality, a ‘Planckton’ of angular momentum. All physical systems then can be ‘reduced’ to fractal ensembles of 3 ‘conserved quantities’, angular momentum – the membrane of the system, which becomes a membrane, fractal sum of ‘cellular cycles’ or the skin of the system in human beings. Vital energy, the enclosed cyclical forms and motions within the space whose boundary conditions are given by the membrane, and lineal momentum, the motions with a ‘finality’ we perceive guided by a ‘relatively still mind-point-singularity’ that focuses the energy and information transferred through the angular momentum membrane.

It is also the smallest clock of our world, as its minimal unit of cyclical information, or angular momentum, used as the unit of the *3 human physical parameters* of *spatial size, cyclical time** frequency and ‘ scale’ (Active magnitude):*

**h= mass (∆) x area (S) x ƒrequency (ð).**

So h, Planckton, is the minimal fractal organism which becomes the ‘cellular unit’ of all other species of light space-time, as Plankton is the minimal unit of the biologic Universe, also a cell with a similar ternary structure – a DNA nucleus that process information, a protein membrane that isolates a vital space-time, the cytoplasm.

Thus Non-Euclidean points have breath, its lines are therefore waves able to communicate the external form and internal energy or fractal networks that branch to connect multiple points, and its planes intersection of three of such waves or networks that form topological organisms… It is then obvious that the next step of Non-Euclidean geometry is to merge those concepts with the physical analysis of the smallest physical systems, to understand its vital topologies. To that aim we have to introduce *the second fundamental equation of 5D metric that formalizes the Paradox of relativity. And so it acts in the physical model as the Postulate of relativity, common to 3D Galilean and 4D Einstein’s simplex models, which correspond to 5D in a single plane of ‘light space-time’…*

So we have transformed the 5 Dimensions of space-time in 5 vital dimotions, broken into infinite vital space-time beings, and now we have the 5 elements which reality uses in different perspectives to construct all realities. Nothing else is needed. And it will easily follow that in each stience, including mathematics, 3±¡ elements (depending on the perception in a single plane or in several ones) will be concerned with the analysis of a system in Simultaneous, entangled, still space as a superorganism constructed with those 5 Dimotions, which in sequential motion time will trace a worldcycle also composed of those 5 dimotions. As the universe simply put it is a reproductive fractal of 5 Dimotions of spacetime…

*As motion with form constantly reproduces by the mere fact of moving. So the fractal reproductive nature of all what exists is immediate, as all what exists moves form, reproduces form. *

**5D METRICS AND ITS FUNCTION OF EXISTENCE: THE GENERATOR EQUATIONS OF ALL STIENCES. **

The judge and the 4 witnesses represent 5povs. to obtain a partial truths as truth only exists in the being or event in itself that holds all the information. So we need a pentalogic of 5 Dimotions for reality to emerge and ‘persist’ through synchronicity and simultaneity.

In *mathematical science* for a dimension of space-time to exist, it requires a metric equation, which combines space, and time to gives us a co-invariant system that allows travelling through such dimension. How we do travel then through the fifth dimension? A system travels through 3 scales of the fifth dimension by accelerating its evolution in a smaller scale through a placental cycle, emerging as an organism in the larger world, to live 3 ages & dissolve back to its parts in the 0-sum death. And this is the meaning of existence, and its reason d’etre is the SIMPLE Metric equations of 5D, which structures through synchronicity of the different speeds of time cycles, the different scales of reality. So the ∆º organism eats every day, and its food programs the faster cycle of reproduction of its cells, as the moon cycle programs the menstrual cycle of women, as the year cycle of rotation of Earth programs the reproductive cycle of seasons and so on.

So an essential part of 5D theory is the analysis of synchronicity, simultaneity, emergence and the in-depth analysis – not done in this introductory course, reserved for the more complex papers on ‘pentalogic and dodecalogic’ where we follow in more detail the construction of simultaneous superorganisms and its ternary worldcycles.. All this of course is studied by huminds, as everything we talk about here, but *without the proper conceptual frame, lacking valid definitions of planes of space-time, time cycles and fractal spaces.*

It is for that reason we do need a new formalism we have called existential algebra with its simple symbols of which the most important are the 5 bidimensional dimotions of space-time, which entangle together through synchronous, simultaneous emergent processes to create the apparent ‘solidity’ and ‘stillness’ of reality.

Because the Universe is made only of two polar elements, still minds (SS, ab.§) and Temporal entropy (TT), and its 3 dimotional combinations, St-information, Ts-locomotion and S=T, reproduction, whose interaction can be resumed in the function of exist¡ence, Max. SxT (s≈t)=C, we can deduce all the principles, laws, events and equations of all stiences from it. So we shall call Existential Algebra to the Gst formalism of Generational Spacetime (ab.¬Æ), and do exactly that: deduce all equations and laws of stiences from 5D metrics.

We shall thus make a 1^{st} foray on existential algebra, showing how the ‘development’ of 5 Metrics give birth to the function of existence into its 3 ‘extremal points’ or ages , Max. S x T (3^{rd} age), Max. T x S (youth), S=T (maturity), defines the worldcycle of existence of all beings in its two directions, forwards and backwards.

But 5D metrics can be studied in more depth, roughly speaking in 4 sub-*equations, which are the foundations of the 43 great subdivisions of science:*

–* The physical equation of relativity, S=T, basis of all physical and mathematical stiences.*

*-The biological equation or function of existence, Max. S x T (achieved precisely when S=T), the basic equation of all biological drives and evolution***/**

*– And the equation of the mind: 0-mind x ∞ Universe = Constant world that creates mental spaces… *which we will consider in the next paragraphs as we have defined space and time more properly.

*– Finally *hose equation can be further unified, since the metric equation of multiple spacetime scales, SxT=K & the relative equation of dual motion/stillness in a single plane S=T that maximizes SxT=K (5×5>6×4) unify in 1 ‘existential’ equation: Max. ∑SxT=C±¡:∆±1, whose study is the field of Philosophy of stience and its new formalism, Existential Algebra (ab. ¬æ). * So after studying the 3 classic fields of science will return to those 5 Dimotions, SS, St, sT, ST, TT and study its entanglement and different properties and complementary oppositions, to start building the formal laws of existential algebra, the formalism of Generational space-time. that all stiences notably mathematics and logic mirror. *

*Let us then start with the physical analysis of relativity and its correspondence with 5D.*

**RELATIVITY. THE 5 DIMOTIONS OF EXIST¡ENCE.**

**Galilean Paradox**: **S****ó****T: Relativity of space Dimensions=Forms=Motion in time****: The 5 Universal** **Dimotions**

Galileo’s time and space Principle of Relativity is the fundamental conceptual thought behind the relationship between time=motion and space=form and how one can be converted into another: All what exists is made of space=form and time=motion. And yet physicists know that we cannot distinguish motion from form. That any being in motion from its point of view seems to be still and all other things moving around it. *This is the principle of Relativity of motion.*

Physicists then without much thought about that fascinating duality, went on to use mathematics to calculate the relative motion of each entity of reality respect to other system, which seems static from both points of view. This is called Galilean relativity, latter refined by Einstein’s relativity, and essentially is concerned with the mathematical calculus of what we shall call the 2nd Dimotion of time=change, locomotion. Fine, but *we are more interested on the duality of space=form and motion=time and its entangled relationships –the reasons why we do NOT see together motion and form, even if all systems have both.*

The conclusion is then rather obvious: one of the two parameters of reality is ‘hidden’ to perception; we either see motion or form, ‘waves or particles’ (quantum complementarity), distances and lines or points in motion (as in the night when fast cars in a picture appear as lines). So *physicists calculate only one when in fact we must assess the existence of 2; and since we cannot distinguish them, logically we must equal them. ‘Form=motion-function; space=time; S=T’. *

Relativity then becomes a duality, S=T, is *at the heart of every law of the Universe. But which one is the real element? Obviously time=motion. Space is a “Maya of the senses” – a slice of time motion. The ultimate substance is motion. Form is what a ‘still mind’, makes of that motion to ‘perceive’, information, forms-in-action. *

Since we see Earth still and flat but it is round and moving. Galileo’s profession was ballistics – the study of cannonballs motion. So he chose ONLY motion and lost the chance to start physics with a complex philosophical understanding of its S=T dual Principle of relativity, which Poincare defined latter clearly when he said that ‘we cannot distinguish motion from stillness’. An example is quantum/relativity duality. In detail quantum space has ‘dark energy’ because it has expansive motion that extends into a plane of space, but when seen at larger scales without detail its entropic motion seems static space – a dual area of scattering length and width. So in the galaxy we see either dark energy motion or expanding space: T=S. A motion of time is equivalent to a dimension of space: Distance and motion cannot be distinguished so they must be taken as two side of the same being, a space=time Ðimotion (ab. Dimensional Motion):

**S= T; Dimension-Distance = Time-motion = ST Ðimotion**

*Why if the Earth moves in time, we see it as a still form in space? Because Reality is a constant game of infinite motions, but the mind focus in stillness those motions, and measures them at distances. For ‘huminds’ motion is relative to our systems of measure and perception, which are light-based; hence a fixed c-rod speed/distance. Reason why Einstein’s relativity postulates a maximal T:c-speed, measured as if observer and observable were still to each other (Constant S); which at our scale we ‘correct’ with Lorentz Transformations. *

*P*hysicists just substituted Earth’s still distances for motions. It took 300 years for Einstein to realize the relativity of motion and its measure made essentially time and space, motion and form two sides of the same coin. Still this realization was not explored philosophically and so it gave birth to a series of ill-understood dualities between ‘states of measure and form’ (particles, head gauging form, in-form-ation) and ‘states of motion’ (wave states).

It is then essential to grasp that motion and form co-exist as 2 different states depending on 5D scale and detail: Motions are perceived by minds that stop motion into form, into information, as distances. So if we see slow motion in the night, a car’s headlight seems a long distance line ‘still’ picture. But this means also that the 3 ‘Euclidean still dimensions’ must have motion; they are ‘bidimensional ST-holographic, topologic dimotions’. So we have 3 Space + 1 Time + 1 5^{th} dimension of scales = 5 Dimensional motions. None of them is *a Dimension of pure spatial form or a pure time motion but a combination of both. Even if mentally we tend to reduce motion and focus on forms, all has motion=time, and form =space: this is the meaning of ‘spacetime’, the messing of both into 5 dimotions, the fundamental element of all realities.*

Relativity states ‘we cannot distinguish motion=time from position=space’. So all what exists is a composite of both, undistinguishable S=T, 5 ‘Dimensional motions’ (Ab. Dimotions), broken in infinite fractal, vital time space organisms, composed of topological Dimotions: height=information; length=locomotion; width=reproduction; form=social evolution of parts into wholes & entropy=dissolution of a whole into its parts in a lower scale of the fifth *dimension* (term we keep for the whole range of scales of the Universe)**; whose study is both mathematical, the main science that studies how those 5 Dimotions entangle in simultaneous Space, connected to each other topological adjacent parts, which create superorganism and Logic; the main science of time that observes how those pentalogic, entangled superorganisms move and evolve, change in sequential relational time, living a worldcycle of life and death.**

**Since there is nothing else than time and space, the 2 experimental ‘mirror-sciences’ of time and space become the most important to extract the ‘Disomorphic=laws’ of those 5 Dimotions that all systems have in common. Since while those Dimotions are broken, in vital organisms, separated by cyclical time membranes, they are the same.**

*In the graph Galilean relativity was ill understood, as the true question about time-change is why ‘the mind sees space as a still continuum, when in detail is made of *smaller self-similar *quanta*, in motion. The paradox *defines mental spaces as still simplified views of the more complex whole. The 3 ¡logic paradoxes of space topology (closed in-form-ative curved-O vs. |-open, free entropic lineal forms), time-motion (stillness vs. motion) and ∆-scale, (continuous whole vs. discrete forms; single scale vs. multiple one)s, are essential to the perception of a simplified ‘spatial mind universe’ in a single flat still plane vs. the full, more detailed complex picture in time, of a curved, discrete and moving Universe. Those paradoxes resume the 5 elements of reality, Space=form, time=motion, scales and the mind that measures them, within its own entropic limits.*

**Motion is reproduction of form in a lower scale. Bohm’s realism: quantum potentials.**

How a system moves; in a crowded universe, where we ‘are’ vital space-time? the answer, which resolves also Zeno’s and quantum complementarity paradoxes, is if *we do not move but reproduce our information, translated into a lower faster wave scale of the fifth dimension; as we reproduce our sound in faster electrons to telephone or nerve impulses into chemical dopamine to jump discontinuous neurons. So motion becomes scalar reproduction of form, and since all is a form of motion, all is reproduction, which is the definition of a mathematical fractal, a feed-back reproductive equation; 5D metrics, which become then the ‘function of existence’ whose goal is to reproduce the form of all systems – the simpler ones with maximal motion-translation in space, the complex ones with min. motion as a reproduction that emerges between scales. And this gives birth to the worldcycle. Consider the case of quantum physics:*

In graph a particle reproduces in adjacent regions that fade away, and the result is the perception of a wave of motion. In Bohm’s realist model this reproduction happens in a lower plane of quantum potentials, where also entanglement happens, which is the ∆-4 scale which is v>c in 5D metrics (Min. S x Max.T=C), hence real.

Motion then is reproduction of form over such potential: the wave erases form into motion, the particle is a still state that gauges information entangled to other particle, fermion and boson, still to each other – despite the perception of relative motion in our scale – hence the information electrons share has always a c-constant speed. Thus the Lorentz transformation are objectively real for mankind who eliminates the stop state of particles as we do in a movie eliminating the stop frame but if we were observing reality from the perspective of an atom, we would ‘stop’, entangle in the quantum potential, neutrino scale & so eliminate the spooky effects of ‘time dilation’ & ‘length’ contraction, from our perspective (but not of mass increase as it is a scalar effect). This is the ‘rational’ 5D explanation of both the c-constant of light and entanglement; as electronic beings perceive information in ‘stop position to each other’ and move in ‘wave state’. Motions are perceived by particles that stop motion into form, into information, as distances. So 4D relativity needs to be expanded to the scalar Universe beyond the c-speed light limit of the galaxy.

Within this complex view, the models of Newton, Galileo and Einstein’s space-time correspond to the limit of 5D when we simplify all the worldcycles of time, we call life & death to a single mechanical clock, elongated to infinity & perceived in a single scale of space. Let us then deduce from those 2 equations the fundamental equation of reality:

Those 2 poles of reality are the first principles of any scientific inquire, even prior to the languages of time-motion, logic and spatial forms – mathematics, that better mirror its laws. Look around yourself, everything that you see is a form with inner or outer motions. Those are thus the 2 primary elements of reality; which mind languages perceive mostly by reducing the scales of the fifth dimension and its motions to the minimal possible to fit it all in the mind ‘equation’: O-mind x ∞ Universe of formal motions = Mental world – reduced mirror of the Universe.

**The 5 dimotions of space-time as inner and outer motions.**

A more precise definition specially for physical sciences of the previous 5 ‘states’ of spacetime deals with the concept of internal and external motion (which has deep implications for relativity and so we will expand it in the paper ‘physics and space-time’. Any cyclical time-space organism, even the simple circle divides space-time into an inner and outer region (knot theory, membranes of all topological systems). We talk of internal and external motion and form in a system. As it is evident that external motion in a larger world is bigger than a being’s internal motion. So little motion means internal motion, t and a lot of motion external motion. Thus we define the 5 dimensional motions of space-time with more finesse. Whereas capital letters, S & T mean external and small letters t and s mean internal (as the internal part have less volume, are smaller than the outer world).

– **Ts** is external motion (T) and internal form (s): It is classic locomotion, where the internal system doesn’t move but it translates in the external world. This is what physicists mostly study, and they still wonder why when a material system moves, the ‘center of gravity’ seems fixed. But what internal part? The answer is the mind-brain; the system that perceives information; as the limbs/fields move a body over which a particle-head gauges that information:

**-St **is therefore information, where externally there is no motion (S) and internally the ‘mind-mapping’, that mirrors the outer world becomes a form-in-motion, in-form-ation, processed by the brain, particle, crystal image, cpu… It is a fascinating fact that we, the thinkers, spend so much time, still; so Leibniz could spend days seat on a chair thinking, that in sleep more brain activity happens and more hormonal languages develop.

**– ST **or st is then balanced internal-external motion vs. form, or energy, of which there are two varieties we ad, Ts-kinetic energy akin to locomotion, (mv^{2}/2; where as you can observe the ‘motion’ provided by v is squared, so it is more important than the mass/form) and St-potential energy, where the energy is store internally in the different tensions and cyclical patterns of the system, but externally it is all position. Energy then is the balanced sum of both, St+Ts=ST and it is the essential law of physics that a system constantly balances both (SMH motions; Lagrangian actions which are its difference tending to zero (St-Ts=0, St=Ts). Existential energy is thus the mother of all battles, and we could state that in a ‘given plane of exist¡ence’, energy never dies, but constantly transforms between the informative, potential energy state and the kinetic, locomotion states.

This leave us the 2 limits of pure form,** SS, Ss**, which correspond to ‘seeds’ and ‘mind-languages’ with null motion, (which following our formalism should be written Ss, but my lazy ‘fingers’ often repeat 2 SS, sorry)

And Tt (written TT often… lazy f…), which we shall call entropy or scattering or death or disorder, as when a system moves internally and externally it is really disorganizing its structure and ultimately as in expansive big-bangs of mass, or death processes of corruption, all parts go away and the system dies.

For example war is the Tt entropic state of a superorganism of history, a social mass of ∆+1 humans in time, when there are external motion (armies invading enemy nations) and internal motion (troops invading internally the country), which ends is the massacre of people. At individual level today, children, as humanity enters its age of entropy as it becomes obsolete, degraded and likely soon substituted by machines and chips and robots, *is clearly suffering an entropic age, sometimes called A.D.D. *

The child has maximal external motion (as we shall see due to the correspondence of STates and ST-ages of life) but today has also maximal internal ADD motion in the brain, filled with virtual hypnotic fiction bull$hit, from mental machines that are disordering us in what we call the Neo-paleolithic age of History. For physical systems Tt is also death as the system explodes, scatters and its internal parts move externally. So yes, those 5 Dimotions apply from the simplest particle to the more complex superorganism of history.

In physics this prolegomena will become then fundamental to understand the principles of Relativity, the fact that we cannot distinguish motion from form, S for T and both therefore must be put together in ‘dimotions’. So we shall see that the fundamental equation of the Universe is SóT (ab.S=T). As there is a constant dynamic transference of Motion and Form between the inner and outer world, which allows a system *to transit between the 5 Dimotions of existence. *

**Quantum motions vs. continuous derivatives.**

We shall introduce now one of the ‘mantra themes’ of all the introductions to 5D theory, which we will expand greatly as we revise those texts, as it is the ‘central piece’ of any physical theory of motion.

With physicists definition of time as locomotion (Ts) we can only understand very simple spatial motions with NO inner change on them (s-patial form in Ts is fixed). This is reflected in the use of ‘present=instant derivatives’ which do not admit ‘peaks’ of changes of STages for physical systems. Since the calculus of locomotions uses ‘instants of present=finitesimal derivatives’ that don’t apply to ‘change of internal parameters’, to inner motions necessary to change STages of life, where the system switches from external to internal motion; as it stop and goes through its existence. So for example, the change from youth to maturity, Ts>ST means a change of ‘s’ to S, from internal no-motion to internal change in information that brings up the reproduction of the system.

This is the key to the whole structure of change in the Universe, which happens with the interplay of internal and external motion and form. I.e. when we act, we first ‘stay put’ as form, but our mind is ‘thinking’, having internal motion, and so we think what we shall act, and then we act, and our mind is ‘stopping’ it has no internal motion. So the fundamental event of motion v. thought is St (internal motion, external form) < = > Ts (internal form, external motion). In those systems a *continuous calculus approach fails totally to understand the quantum stop < go process.*

*In the stop process information is processed internally, as ‘small caps’. motion, ‘t’; collapsed as a a wave (physical systems), processed in a dream state (life organisms), thought of (humans); in the go process, the information is repeated by a quantum wave; reproduced externally by a mitosis cell, projected by a mind, reproduced by a seed… The examples are many as those stœps ARE the main whys of the fractal Universe, which is a reproductive organism. *

A key to all of those events are the ‘Stœps’; changes of states, quantum by quantum of Space ó motion transformations in the inner and outer regions of the being, in the different scales of the Universe.

There is not a personal god indeed but if there were one designing such perfect Universe I would like to share his thoughts. You have always a ‘cyclical time membrane’ that separates an inner and outer world. One of them might be in motion, one might be still, the combinations of those possibilities and all its changes, either one by one, or two by two, in infinite stœps of infinite beings describes almost all realities. Some are more important than others; some sequences of changes are fundamental as the sequence of the worldcycle.

And then there is the way the mind selects certain states and ignores others, like in a movie picture. Consider two of them that are inverse and will come up many times, and when we increase in a future review the complexity of those physical texts with advanced maths will show the full beauty of the mathematical mirror of physic:

– St: The perception of motion as stillness: In this ‘series’ the mind does NOT perceive the motion STage at all, only the stop State, and as such it explains why the earth is moving but the mind sees it still.

– Ts: Perception of stillness as motion: in this view, the mind doesn’t perceive the state of wave-motion, but only the series of still, informative positions. It explains why we see motion on systems that are still, as frames of a movie.

We either perceive the wave state or the particle, still state but NOT both at the same time, because then we would see a jerky Universe.

In the mathematical mirror it means *we can have either a topological, geometric analysis or a calculus, changing analysis; we can define reality in terms of ‘continuous functions’ or we can consider discrete approaches.*

*But what is a deeper truth? Not the continuous but the discrete, stop and go approach as we show in our paper on 5D mathematics and time (Calculus). *

**The function of existence: Reproduction of form.**

Physicists made the Galileo’s paradox, the cornerstone of their theory of measure, but *they failed to study the deep implications it has for every aspect of the structure of the Universe, from the duality between spatial mental, linguistic forms and physical motions; to the balances achieved by the similarity of both space and time, which becomes the fundamental ‘equation of present’ S=T, and hence with the metric equation of scales, $ x ð = K, the two essential equations to formalize single planes S=T, and multiple scales of spacetime. Yet as S=T maximizes SxT=K (5×5>6×4). We unify both in a single equation: Max. S x T = C, which defines for each fractal vital space-time organism its Function of existence, as all species will try to maximize its motion-entropy-time for its field-limbs, its information-spatial states for its particle-heads, whose product will give us its vital reproductive energy. But as all systems move and motion is reproduction of form we can ad a final factor, ∑, reproduction of parts, which to maximize that function become joined into larger wholes which are stronger than individuals; creating new planes of existence. C= Max. ∑ SxT (s=t); whereas C act as the entropic limits in ∑-scales, T-ime & S-pace, boundaries beyond which the still mind doesn’t’ perceive or control. *

It is also a survival biologic equation, because it implies to provide lineal motion to ‘fields-limbs’, absorb energy to reproduce our bodies-waves, and information to guide our motions with particle-heads. So reality is a ‘struggle’ for existence as systems reproduce its Ts-fields-limbs of motion, S=T body-waves of energy and St-particles-heads of information. But as all T.œs are fractal, broken, its growth has a limit on the fight with other systems, which try to move and reproduce. In terms of pure T-motion and pure S-form, we consider then the whole of maximal time=motion= entropy or TT and Max. space=form=stillness or SS the 2 limiting Dimotions for any 3 ensemble Ts<ST>St-system.

We define the Universe as a fractal that reproduces and ensembles Space-time Dimotions into supœrganism through ∞ relative scales of spatial size and time-motion; whose Fractal generator (mathematics) Metric (Physical jargon), ‘function of survival (Biology) or Function of existence (logic Jargon) writes **C=Max.∑SxT (s=t)**

We shall prove that all realities are always a reproductive radiation of a function of existence along 5D scales.

**We are made of the 5 Dimotions of spacetime of the Universe. We are ensembles of those 5 Dimotions, which seen in simultaneous space give origin to the vital topological organisms of the Universe; whose study therefore is mathematical, the science of space; and observed in sequential relational time, live a worldcycle of life and death; and since there is nothing else than time and space, those 2 fundamental experimental primary ‘mirror-sciences’ of time and space become the most important to know what all systems have in common, its ‘Disomorphic=laws’ extracted form the nature of those 5 Dimotions.**

Thus automatically, genetically, consciously, memetically, mathematically, logically, through its own will or as a part of a larger system that uses the ‘machine’ or ‘organism’ to enhance its actions all what exists does so because it performs internally those 5 Dimotions or externally performs one of them for another symbiotic species, as those species that have not followed the program of exist¡ence and its 5 actions in the past have become extinguished, and those will not in the future, will become wrong mutations, crazy thoughts, fictional languages and die away.

The function of existence, or 5D metric of Generational space-time, (Ab. Gst, G) Max. Se x Ti (s=t) states that all systems of Nature will try to maximize its absorption of Entropic motion (with no form) and Linguistic form (with no motion), and its 3 intermediate dimotions of energy (s=t, balance of both that reproduces them), information (St: form with a little motion, form-in-action) and locomotion (sT, motion with a little form). So we talk of a program of survival ‘selected’ by all systems and expressed in its languages and minimal five actions encoded in that simple equation, which we term: ** a, e, ï, œ, û**, as a mnemonic rule for the five actions of existence:

**A**ccelerations (locomotion), entropic feeding (e), **ï**nformative perception and communication ï, **Œ**:reproduction into parallel supœrganisms **Û**… and social growth into larger wholes called philosophically Universals. And this series of actions is what accumulated in time will ultimately give birth to your word cycle as the monad will first perceive (i), to direct its entropy-motions (a),towards a field of energy (e), where to absorb the energy bites it will imprint with its inner form, e x i = œ, to reproduce another form, and when enough ∑œ exist, it naturally organize into a larger whole û.

In graph the actionS of different Stientific scales of organisms. Above the coding of actions, which are the knots and bolts and details of the study of any time§paœrganism in light space-time, coded by colors and dimensions, in physical atoms, coded with quantum numbers and in life and humans coded by the so called drives of life, which we obviously extend beyond the ego paradox to all other systems, including genetics not mapped there (coded by the 4-5 letters). Those actions balanced each other into zero-sums in death, as they tend to increase information from a mind p.o.v., hence we ‘all warp, wrinkle’ get old in the third age and die, setting from its minimal actions to its integral sums, the 3 ages of life-existence and the world cycle all super organism follow.

The simplicity of the game of exist¡ence, and its selfish actions, which gather together into social wholes through reproductive radiations, each action coded by a fundamental topologic organ we can express in existential ¬Æ=ilogic, and corresponds for each species of the Universe, with a fundamental parameter of humind measure. So from bottom to top, we find the 5 fundamental elements of light code its actions of motion (c-speed), energy (magnetic field), information (electric field), social evolution colors & entropic feeding, (quantum potential, neutrino light theory)

So the minimal particle-points, photons, electrons & quarks that construct all other systems of our Universe show the 5 organic dimotions (motions with dimensional form) that define ‘classic life’: they gauge information – reason why quantum physics is a ‘gauge theory’, feed on energy (quantum jumps) absorbing smaller ∆-1 particles, reproducing new clone particles, move and evolve socially through magnetic fields into larger wholes (atoms). Hence the units of life are particles, the minimal units of our vital, organic, fractal, scalar Universe of multiple timespace organisms. All lives, performing 5 Dimotions=actions of ƒ(exist¡ence):Max.SxT(s=t) =C, starting with particles. So all scales are relative NONE matters more than other. From those actions, given the dominance of informative actions over entropic ones, it appears a series of repetitive cyclical patterns of actions conducting to maximize the existence of the being, which accumulate in a larger scale of time-space, as a worldcycle of actions that increase the information of the system in 3 ages. So the basic cycle of actions becomes a larger 3 ages cycle of life and death; as systems once and again, starts in an act of information/shrinking and ends in an act of organization/shrinking of herds into wholes, will keep reducing the being and finally make it all form no motion to explode and die in an entropic reversal of death:

∑ i->a->e->œ->û, i->a-e->œ->u, ï->æ->Œ->Û -> Informative ‘seed’ age->1^{st} locomotion, feeding age ->2^{nd} reproduction age ->3^{rd } informative, social age-> entropic death.

AS huminds are limited in time logic and we cannot build on their chaotic, entropic models of reality. Chaos and indeterminism IS born of the LIMITED single Time-Dimotion of Science, force-fed by earlier ballistics, the name of Galileo’s first book & modern ego-dogma & subjective reductionism EASY to practice in the largest scales of modeling. And it is a cultural treat of our dog-eat-dog ‘animetal cult(ure)s’ to weapons & financial parasitism (nationalism, capitalism) & our despise and extinction of organic life in the making. Monologic prevents those who are destroying life and bringing the 3^{rd} age of the planet, the Metal-earth, to even realize how wrong is their absolute ‘one single order of thought’. Our informative elite is not even human anymore. It is a system of memes in favor of this other world, expressed by a network of informative machines and its Financial-Media-Academia monologic that pass as expertise (FMasters). But beyond the simplex analytic model of cause an effect are out of depth in long time range. Only repetition ensures that false theories of science based in entropy-death pass as dogma. Mental imprinting by repetition ensures humans walk the determinist path of self-suicide thinking they CANNOT understand and design a better future. I have been predicting that future of social history for 30 years, using the larger organic properties and 5 Dimotions of time-space but the system cares not to accept any criticism. People are NOT aware to which extraordinary lengths the idol-ogies of metal go to cover up and NOT understand the lack of human futures. All this censorship is part of the ‘burden of monologic man’. In the times of Lao-Tse, Buddha and Aristotle, minds were free – reason why they had so much more depth, than modern people in his comprehension of dual and trinity logic. Yes, to know ALL about the locomotion of matter which is what physicists know is NOT to know all about the larger, organic Deep time scales of the future, and nothing about the topologic evolution of machines and fractal cycles of stock-markets.

So the ancients were miles ahead of modern ænthropic (anthropic+entropic) lineal military physicists & Darwinian philosophers, with their pedestrian monologic, where only entropy, destruction, DEATH seems to create, instead of destroying as its worldly religions of making weapons, believes. We shall thus deal with the reductionism and primitivism of their ‘philosophies of science’ in this article on monologic, However this ‘entropic’ article in itself – an on slaughter on present mæn – is less important than a pentalogic and dodecalogic understanding of existential ¬Æ=ilogic, as it IS. Or else the destruction of ænthropic models from egocy to the big-bang, would be useless since the reason humans sticks to those models is the lack of an alternative model to its simplex views. That is what 5D represents.

In all Planes, the simpler actions of any being *are extractions of motion, energy, entropy=motion and form from lower ∆-i Planes:*

A T.œ perceives only the ∆±3 planes from where it extracts energy or information. As its actions and dimotions are architectonically performed through planes of 5D where each main action relates to an interval of scales:

∆-4-3: The system extracts indistinguishable boosts of entropic of motion (man from gravitation).

∆-3-2: The system extracts bits of information (Light in man)

∆-2-1: The system extracts bites of energy (amino acids in man)

∆-1 0: The system seeds its minimal seed of reproduction.

∆0+1: The system connects socially with other systems to evolve into a whole.

*So simpler Actions start at finitesimal level, gathering in sequential patterns in existential algebra, as ‘time flows’ and in population and spatial patterns – in integral herds of numbers in calculus.*

We and all other beings perceive from ∆-3 quanta (light in our case), feed on amino acids, (∆-2 quanta for any ∆º system), seed with seminal ∆-1 cellular quanta (electrons also, with ∆-1 photon quanta).

For each action of space-time we shall find a whole, ∆º T.œ, which will enter in contact with another world, ∆±i, from where it will extract finitesimals of space or time, energy or information, entropy or motion, and this will be the finitesimal ∂ ƒ(x), which will be absorbed and used by the species to obtain a certain action, å.

Analysis allow us to extract actions from wholes, reason why there are not really use beyond the third derivative of a being, as super organisms co-exist in 3 only Scalar Planes. It also works in terms of a volume, as its derivative is a plane, then its unit-cell or point… So to speak, if you derivate a world, you get its organism, and if you derivate it again you get its cell and then its molecular parts. And then if you do that in time, you get its speed and then its acceleration and then its jerk.

**The magic of derivation**

Because of the symmetry between ∆≈S≈T, to extract finitesimals of smaller scales the process is the same. We derive the whole, which diminishes its ‘dimensions= power’ as the system looses its larger whole, *but increases its number of ∆-¡ visible particles, whereas the difference of value between both, shows the ratio and structure of its entropy, energy and information networks, sum of its components that form the whole. As certain functions define more specialized T.œs than others. So the parts of a whole vary according to topological structure. *

**The key action of entropy: feeding’** **on energy**

**A**ll systems of Nature follow the simple aeiou actions determined by the 5 dimensions of space-time, moving in **a-**ccelerated paths towards **e-**nergy surfaces where to feed and transform into** I-**nformation the vital energy absorbed by the system in order to reproduce and **o**ffspring of similar beings, which will evolve together emerging into a social **U**niversal sum of multiple individuals.

We study in more depth the action of energetic feeding, parallel analysis of ‘entropy’, since feeding is the addiction of absortion of energy, killing it down two scales of the fifth dimension to reform that energy into the information of the T.œ that absorbs it. So the equation of feeding first reduces the being it kills from ∆ø<<∆-2 and then reconstructs the being from ∆-2≥≥∆º, whereas the first ∆ø has lesser existential force than ∆º, which is the top predator ∆º Te x S i > ∆ø Te x Si:

In the graph, feeding is a process of entropy first and then reconstruction.

To reconstruct the information of system first the system must start with the inverse Dimotion of feeding of a species that will ‘simplify the information’ of a part of an organism, two ‘planes down’: ∆¡<<∆-1, the equation of ‘scattering entropy’ and then rebuild from its ‘amino-acids’ or ‘forces’, the larger whole system. So its equation in ¬Æ is quite simple: ∆¡<<∑∑∆¡-2>∑∆¡-1 for *any scalar process in the Universe.*

Feeding of course happens regardless of moral views in all systems, including those social systems in which *a culture with higher technological information, higher top predator form (Max.Te x Si) conquers another nation and reforms the system, either by reproducing its culture in the ∆-2 scale of individual memes, or even in a harsher manner by extinguishing the ∆-2 scale of human beings reproducing the conquerors memes and genes together. In the graph we can see the case of Germany, which try the most extreme second approach extinction of conquered nation. *

But in all scales the top predator with maximal e xi-stiential force will do the same. We eat and put down to amino acids and reconstruct our food. Particles feed on force fields and reconstruct reproducing new particles; cosmological black holes emit seminal seeds of matter and reconstruct baby-irregular galaxies at the end of its jet.

Whereas < is a symbol of expansion of space and simplification of form, which happens twice, and > the inverse Dimotion of growth of information.

So when we feed, we break down to molecular amino acids and then rebuild them into cellular matter (∆¡-1), and so happens with atoms that feed on forces evolved into ∆¡-1 particles and nations, which through war explode a rival nation in its ∆¡-2 Memes and territory down to the destruction of its buildings and raw materials, and then ‘re-absorb’ those materials and citizens, ‘re-imprinted’ into ∆¡-1 ‘converted’ citizens and wealth.

It is then obvious that ‘the fourth dimension of entropy=dissolution’ and the ‘fifth dimension of social evolution’ into wholes. ARE INVERSE.

We shall constantly realize how *the set of scalar laws of the Universe apply to all systems, regardless of which ‘species of space-time’ we study, the most astounding proof of the existence of an scalar ‘game of exist¡ence played with the same rules for any species of reality.*

Let us now deal with ‘the fourth dimension of entropy’ and the ‘fifth dimension of social evolution’ into wholes. So simple so complex when we combine, repeat and translate in different languages those ultimate principles, which we shall now study with a bit of more depth.

The Universe and all its parts – in the graph the human being – are Timespace organisms that reproduce information, stored in the form and frequency of its time clocks, through the vital motions of its 3 space-time dimensions and 2 ± ‘∆’ scalar dimensions of entropy that dissolve the networks of information of an entity, scattering them in the process of death, and emergence, which evolves those networks socially from a group of smaller parts into a whole.

Without those 2 ‘organic’ dimensions that relate the different ‘scales’ of size in space and speed of time clocks of each ‘fractal part’ of the Universe, we cannot make sense of its organic process of creation and extinction. In the graph all systems co-exist in 3 scales of the 4th & 5th Dimension: the ∆-1 atomic/cellular, ∆-organic/thermodynamic and ∆+1 gravitational world; which we can describe with a metric equation of the 4th and 5th scalar Dimensions of space-time (ab. ∆±i), which order all entities of reality in scales according to the size and different speeds of time clocks of its systems is very simple: SxT = K, the relative size in space and speed of the time cycles of a being remain constant.

In the graph we see the ‘organic fractal Universe in different scales’ and systems of reproduction of information.

Researchers in fractal geometry are delighted to write fractal equations that show such structure. But this is a blog of philosophy of science that goes much further into the organic vital properties of such systems.

So what is an organism of space-time also disappears: the perception by a simultaneous point of reference of a series of synchronized time space cycles.

**Actions define the worldcycle of existence.**

Because actions follow a pattern of increasing evolution and 4 actions are coded while entropy is avoided – we deliver entropy as an open system to other parts of the Universe, it follows that the system constantly accumulates information, and this is the origin of the worldcycle of life and death. The system likes information, because it is what the Universe is really all about, a fractal that imprints information in memorial repetitive cyclical patterns.

So as actions follow a natural scalar level of information perception->locomotion towards a field of ‘feeding’, where to get -> Energy to reproduce -> and then process that energy into information, *exactly the same pattern appears in the next scale of ‘cyclical time and longer space’, the worldcycle of life, which consists in the same pattern of ‘seeding information’, moving youth, feeding and evolving to reproduce and then process information becoming older in a 3 ^{rd} age.*

*This amazing symmetry between the smaller time scale and the larger worldcycle of life, then will be followed also in the larger scale of supœrganisms and species and so we have 3 ‘patterns’ of time cycles due to the survival program of actions in 3 scales of time, and 3 larger surfaces of ‘populations’, as the first cycle feeds the ‘cells, atoms, individuals’, the second cycle the whole organism, and the 3 ^{rd} cycle manifests in the whole species. *

**___**

**Nt.****1**. According to the Correspondence Principles as physics named 4 Dimensions we use the name 5^{th} dimension for the whole range of scales, but in proper terminology we should call each Dimensions, a ‘Dimotion’ and consider the 5^{th} scalar dimension the sum of all the 4^{th} dimotions of social evolution & all the 5^{th} dimotions of entropic devolution

**THE WORLDCYCLE – THE FUNCTION OF EXISTENCE.**

**The worldcycle of existence**: We then put together 5D scales, vital topologic space & cyclic time to describe simultaneous superorganism tracing a worldcycle in time, the fundamental event of reality.

** **

*All what exists is a supœrganism of vital space tracing a 0-sum worldcycle of time through 3 scales of 5 ^{th}dimension: Born as a seed of fast time cycles in a lower 5D scale (∆*

*-1*

*:Max. T x Min. S), emerging as an organism in ∆o, living 3 ages of increasing information, as its time clocks slow down in its ∆*

*+1*

*world to die in a time quanta back to ∆*

*-1.*

So, absolute spacetime is the sum of ∞ Timespace beings, observed in space as simultaneous super organisms, in time, as **worldcycles** of existence between birth and extinction; as all systems are born in a seminal seed, of faster time clocks, in a lower scale of the fifth dimension, growing socially (4^{th} dimotion) till emerging in the organic scale, where they will live 3 ages dominated by one of its 3 topologic organs and its functions=dimotions, a young age of maximal locomotion, dominated by its limbs/fields, a mature age of reproduction dominated by the body-wave and a third age of information dominated by the informative dimotions, which finally exhausts all energy and as time-space never stops, it reverses its dimotion from information to entropy, exploding in the moment of death.

So we marry the 3 vital functions=motions of time and the 3 dimensions of space, either in 1 or 2D (height=spherical information, length=planar locomotion, width=hyperbolic reproduction) which merge in all Time-space Beings; and dominate one of the 3 ages of its life-death worldcycles, the past, young age of limbic entropic motions, the mature reproductive age dominated by the hyperbolic body/wave and the 3^{rd} age dominated by the informative particle-head, when the illusion of time ends with an entropic big-bang death that dissolves the being into its ‘scalar cellular, atomic parts’, which lead us to the realization that time cycles NOT only return to its origin in a single spacetime continuum but they move up and down the scales of the fifth dimension:

**The 3 ages of existence of space-time organisms. Its 2 worldcycles and Metric equations.**

The *Function of Existence* of any space-time organism can be developed as a feedback equation, S<=>T, in 3 sequential phases/ages/horizons, between 3 ‘standing points’ (changes of phase): Max. T=motion x Min.S: form =moving youth; Max. SxT(s=t): reproductive maturity and Max. S x Min. T=informative, old age, as the equations of the 3 ages of life, between the seed of pure linguistic form born in the lower plane: S¡-1 and its T¡-1 entropic death, back to ∆¡«∆¡-1:

*∆-1»∆º:* The cycle or organism starts its existence as a seed of pure form (4D) when its space-time field is created.

*sT:* It is the first horizon or ‘energy, youth age’ of the cycle, in which energy dominates the system and so we write this phase as, max. $t x min. T.

Max. SxT: s=t*.* It is the present balanced age of the cycle or classic age of ‘life’, when energy and information are in a constant proportion. It is the most efficient age, when the cycle reproduces.

*Max. T x min. S*: 3^{rd} age of the cycle when information has exhausted the space-time field that warps into itself.

*∆º«∆- 1: 0S x T: * It is the end or death of the cycle that reverses its form and becomes energy again.

Existence is an ∞ sum of space/time fields, fluctuating between birth and extinction through those 3 phases or ages. The 3 ages of Timespace supœrganisms happen in all systems, including mental languages:

In State Physics they are, $t-gas, the moving state, S=T liquid, the balanced state and §ð-solid the informative state; into Cosmology, where it describes the Universe as a space-time system that fluctuates between both limits, a form of pure time, the singularity (*min.$t x max.ð§) *and a form of pure space, the big- bang (*max.$t x min. ð§). In *Biology, they are the 3 ages of living beings AND the 3 horizons of evolution of species. In social organisms, through the subconscious collective mind of civilizations which in art styles mimic in a longer 800 year cycle of life and death of civilizations (according to 5D metrics a human social superorganism is larger in space – a nation, culture, religion – and so it lives longer in time). So we find the 3 ages of life *emerging in the 3 ages of cultures and* its 3 artistic styles: Min.S x Max. T (infantile epic, lineal art, as in treccento, Greek kuroi; S=T; balanced beauty, when form and size are in balance, the classic mature age; and Max. S x Min. T: baroque, 3rd age of a civilization, whose subconscious mind is the art of its ‘neuronal artists’, the age of maximal form and angst for a no future, which *is the age of war and death of cultures*).

We talk of 3 ∆±1 scales of worldcycles as the being live in a placenta, then emerges as organism in a world:

þ: 0-1: its palingenetic o-1 social evolution in the accelerated time sphere of existence, till becoming 1 (0-1 bounded unit circle in ¡logic mathematics; quantum probability sphere of particles in physical systems; palingenetic fetal age in biologic systems; 0-9 memetic learning childhood in social systems). It is the highly ordered world cycle as a ‘placental mother-energy world’ is nurturing as memorial cyclical spacetime has erased errors of previous generations.

– c: The outer 1-∞ world, in which it will deploy its 2nd world cycle of existence in an environment which is open, entropic (1-∞ hyperbolic unbounded Cartesian plane in ¡logic mathematics; thermodynamic entropic statistical molecular populations in physics; Darwinian struggle between populations in biology; idol-ogic dog-eat-dog capitalist, nationalist competitive eco(nomic)systems in the super organisms of history. In this 1-∞ existence the world cycle is not ensured to continue, as the entropy of the world system can cut it off.

ω: The existential life cycle, though is part of a larger world of hierarchical social scales (§ D¡), where it performs 5 survival actions through ∆±4 Planes self-centered in its mind, beyond which it cannot longer perceive, to become if successful a new supœrganism of the infinite planes of God, the game of existence.

**The quantum vs. continuous view of the function of existence. **

**The exponential, logarithmic and sinusoidal functions: bell curves representing a worldcycle.**

*4D»∆-1(seed)∑∆:|-$t(limb-field)<Ø-S≈T (iterative bodywave)> O-§ð (particle-head)«5D∆-1(death)*

In graphs, a mathematical algebraic and geometric description of a worldcycle in lineal time, through 3D scales. Left the worldcycle represented with ± exponentials & its inverse, logarithmic curve around the key points of change of phase: as growth slows, ‘entropy-motion’ diminishes. So we move from ‘childhood of max. growth in both parameters (sT energy & St information) to the y”=0 point of youth, in slower logarithmic growth. Together they form, one half of the cycle of existence, till reaching the y’=0 point of Max.SxT). Then lifecycles become negative, in a slow decay of – Logarithm in maturity, and – exponential in the final collapse of death. So the worldcycle has a mirror symmetry around the S=T, when at Max. SxT the system reproduces. Those composite functions appear in all growth curves and expressed as e^{±ix} in complex planes become lineal sinusoidal sine/cosine series & functions. Finally in the ∆-1 scale can be seen in the placental probabilistic 0-1 ‘unit’ sphere of Max.T exponential palingenetic growth or as part of a species, in simultaneity, as a Bell curve of spatial statistical populations (i.e. as in the duality of the T-probability 0-1 sphere of the placental particle age of quantum physics vs its. 1-∞ entropic statistical function of grown up molecular thermodynamics.. Since the function of existence in its fractal variations and cx. pentalogic is the most important of the Universe.

**RECAP.** The dual equations of 5D, the equation of 5D scales, SxT=C & the equation of equality between form and motion, S=T, develops in 3 ages with 3 standing points, a max. point of existence, S=T or mature age, a young age of Max. T=motion, and an old age of Max.S=information; between birth in ∆-1 Form & T-entropic death.

Let us now consider with a physical and biological example, to which extent the laws of 5D enlighten our understanding of reality. In the graph, Matter States are physical time ages, from left pure solid, crystal, §top state, to an even more solid ∆+1 boson condensate, etc. We see that systems either move a step at a time within a plane of existence (gas, liquid, solid) or they can jump « two states at once, (as in the case sublimation) within that plane, or most often between two planes, as in « scattering & entropic death), to become a different Dimotional state. We can then see how the fundamental elements of 5D time appear on the graph: the worldcycle is local and complete. There are two inverse arrows from an entropic past (plasma), in a lower plane (ion particles) to the 3 ages of the matter states with increasing form (gas to solid), to end in a higher plane of existence as a boson-Einstein condensate.

**Worldlines and classic relativity**

How all this relates to 4D classic relativity is simple. In first place because we start from duality, form and motion, O x | in geometric terms, we do have No longer worldliness but worldcycles. For a physicist which only measures lineal time motions, if you ask him what is the function of exist¡ence he will just state: the sum of motions that you make through your entire life and draw a ‘cone of light’ explaining you that as long as you go below c-speed your worldline will be space-like etc. All this has nothing to do of course with your life experienced through the 5 complex dimotions of reality, but the physicist will not have the slightest doubt, as he has reduced perception to that single motion that he is talking serious metaphysics of being and will go further with its errors of lineal time (as all dimotions of time are fractal, local to the being’s exist¡ence). It will then ‘obviously’ enter into paradoxes – the twin paradox, the Lorentz transformations, etc. which we streamline in our post on physics – matter=form and motion; which is the only realm to study with worldlines.

According to the correspondence principle, we compare 5D with 4D *strictly in the real of physics where it applies*:

We unify the 3 fundamental principles of vital spacetime, 5D metrics and Relativity, which become the equivalent in 5D to Einstein’s 3 principles in 4D metrics – the constancy of light speed, his rejection of absolute space-time and Relativity, which 5D also share but make explicit with an S=T equation. We compare then both formalisms with the principle of correspondence to show that 4D is just the limit of 5D for a single light space-time plane:

__–____ The postulate of relativity__, which states that we cannot distinguish motion=time from dimension=position =form =space, which in 5D we reduce to a simple formula, S≈T; we shall call the Paradox of Galileo (e pur si muove, e pur no muove) as he couldn’t explain why the Earth moves but we perceive it in mental space as a still dimension.

– __Denial of Newton’s Absolute space-time__; as he said ‘Leibniz was right, but if so we have to start physics from its foundations’ and ‘I seem to be the only physicist who believe there are infinite time speeds in the Universe’, which we made explicit accepting Leibniz’s relational concept of space-time, the seed of Generational space-time, resumed in a simple sentence: ‘we are the scales of space we occupy and the cycles of time our existence lasts’.

– __The constancy of light speed__, c, which in a 4D single background plane of light space-time corresponding to the galaxy holds naturally, as *light in a relational->Generational theory is the space-time of the galaxy*, which generates through its ‘accelerated’ vortices of space-time in a crescendo of ever more dense scales, photons, electrons, quarks, atoms, molecules, matter states, cosmic bodies and black holes. Yet when we add a 5D sum of all those scales, c-speed becomes the limit of reducing all potential scales of the Universe to light spacetime and the particles it generates

And so we define a second metric equation for all scales: $ (Space size) x ð (cyclical time frequency) = Constant; meaning smaller scales of space hold faster rotating particles, (black holes beyond the event horizon, where there is no longer ‘light’) or when moving in lineal fashion can go faster than light speed in scales below light or outside galaxies, (Bohm’s quantum potential of action at distance cause of entanglement, neutrinos outside galaxies.)

*Yet as S=T maximizes SxT=K (5×5>6×4). We unify both in a single equation: Max. S x T = C, *the 5D metric equation, local to each Time§pace organism struggling to survive by maximizing. its S in-form-ation & T-motions, or in terms of organs, its limbs/fields of speed and heads/particles of information, and

*its balanced combination or body-waves of vital energy, where*entropy=T-motion and S=form, become one, S=T.

*5D metrics generational space-time and the Principle of Relativity defines for each fractal vital space-time organism its Function of existence, as all species will try to maximize its motion-entropy-time for its field-limbs, its information-spatial states for its particle-heads, whose product will give us its vital reproductive energy. *

So the equation has a biologic meaning, because we are made of 3 topologies ‘fields-limbs’ of lineal motion (Ts, where T-motion dominates s-form); balanced hyperbolic bodies-waves of S=T energy we absorb to reproduce, and spherical particles/heads of St- information, form with a bit of action-motion, which we need to linguistically guide our motions; *sandwiched in a larger ∆+1 world of max. T-entropy; coded by the ∆-1 seeds and minds of pure Spatial form. Which give us the 5 Dimotions of reality that are also structural organic parts of our superorganism & phases/ages of our existence: S¡-1 (seed-mind) »Ts-fields/limbs of motion>S=T-body waves>St-*particle-heads«∆-1 T-death; where double «» are symbols for an entropic, ‘expansive dimotion’ and an informative still one. So the essence of survival is to increase our St= mental information and territory of order; in which to S=T reproduce and sT move avoiding the entropic limits of scattering, disordered motion, T-1:death and total St-illness in space the two limiting Dimotions. So we keep our ‘energy=reproductive body-wave, s=t in balance moving, sT our field/limbs and perceiving, St, with particle-heads.

**3 scales in time and space of superorganisms**: **actions, worldcycles and absolute space-time parameters.**

The ∆±¡ scalar STructure of all beings is essential to predict the future of its scientific species, with a remodeled ABCDE of the scientific method that studies A)ccurate Data, B)iologic causes C)yclical patterns and E)ntropic extinctive conclusions for all systems in 3 relative scales of length of space and time duration (to which we add instead of E, D)emocratic, humanist solutions for questions of social sciences.

All sciences predict the future of its species according to its repetitive causal cycles. Or else they are NOT a science. Astrology became a science when Kepler learned its orbital cycles. Bio-economics became a science when we described machines as metal organisms whose industrial r=evolution followed the human 72 years generations of the dominant industrial nations that evolved them in 4 cycles: its body-age (British, steam cycle), heart age (German, electro-chemical engine cycle), its mind age (US, TV-eye, chip-head, mobile-ear cycle) to conclude with the ensemble of robots that as virus do, when all its parts are put together will become ‘alive’. The predictability of time-cycles can be done at 3 levels:

**S: Continuous, spatial mathematical** simple cycles, using derivatives, proper of calculus; which is the shortest time span, as instantaneous derivatives cannot measure a ‘peak’ change of age/phase.

**T: Discontinuous, cyclical patterns** of sequential repetitive often survival actions (feeding, reproduction, death, taking place at intervals. As those actions are discontinuous, leaving long spans in-between, their patterns forecast longer time sequences. Such ¡logic structures are based in time patterns, which as any mechanical, circadian or orbital day-year clock shows are cyclical, repetitive. But here human scientists are at loss, because Galileo studied ballistics, entropic explosions that destroy the information of reality stored in those cycles of time clocks, its patterns and frequencies, changing human cyclical understanding of bio-logic time for lineal, abstract time that seems not to repeat those patterns so mankind lost its capacity to predict many spacetime events, as lineal time misses* information stored in the frequency and form of cyclical clocks, even if equations are similar: V=s/t for lineal time and V=S(l) x ƒ(t) for cyclic patterns*.

**∆: Scalar, Deep Time patterns of topologic and eusocial evolution of parts into wholes – **of quantum 0-1 time probabilities vs. 1-∞ thermodynamic populations in physics, of individuals vs. species in biology, of states of matter vs. geologic cycles in Earth, first noticed by J. Hutton, founder of geology who coined the word super organism for Gaia and deep time for its slower time cycles by virtue of its 5D metrics, $ x ð =C which implies that from the perspective of a smaller scale the life of its whole is much longer.

Deep time leads to a 3^{rd} level of long-time prediction: evolutionary patterns of earth’s life species, including social organisms of history (nations, civilizations) and its death=war cycles related to the eco(nomic)system, where the evolutionary and re=productive cycles of stocks of machine consumption become the sales=profits=valuation of its company-mothers that switch after overproduction crisis to weapons that consume humans, as we have done with remarkable precision for 30 years in our ‘inconvenient’ papers on social sciences. But as human only recognize the 1^{st} type of predictability – calculus of instantaneous derivatives, that need a ‘continuous analysis’ – and have simplified cyclical time into lineal time, ignoring the scalar time of parts and wholes with its 5D metric, their capacity to forecast the future is far more reduced than a ‘stientist’ who understands the 3 scales *determined by the bio-topo-¡logic properties of ‘scales’, ‘space’ and ‘time’, the 3* ∆st structural elements of all systems of the Universe.

**∞ MIND SPACES.**

Minds are mirrors that perceive in its inner ‘still simultaneous language’, the e-motions of time converted into informations of space, in the eternal dialectic between fractal points with a volume of still linguistic perception, mapping its local Universe and flows of e-motional time with its vital sensations:

* Upper graph, human egos submitted to the mind paradox, think languages (words in Abrahamic, creationist religions, numbers in creationist science), known only by man and ‘God’ a priori, create a posteriori the Universe (Copenhagen interpretation). The opposite is truth:* a mind exists in all systems in which time stops to form space. In galaxies happens in relativity equations in black holes, its mind. In thermodynamic physics in the eye of an Eddie. In quantum physics in the center of an atom, or charge. Without linguistic minds that order by reflecting its smaller mind into its local territory reality would not exist. The only way to create fractals is through mirror images.* *

In all scales of stience minds fix the motions of time into spatial, linguistic forms, mind-mappings that reduce the whole with its synoptic language to fit in a brain, an atom, any particle that acts after gauging information in the Universe.

Thus we define ‘Maxwellian’s demons’ of local order in all scales – physical minds as the infinitesimal points that create order in physical systems with the same Disomorphic laws that all others do in more complex scales. As each mind orders as a linguistic god a territory around itself, its fractal body and entropic world.

*So the creation of scales of reality is a simple game, in which a point mind ‘reduces reality’ to its infinitesimal form and then projects into its local territory of order,* which will reflect at scale, the larger whole or world, which the linguistic image reduced and then enlarged back into its territorial form.

*The paradoxes of Relativity, discontinuity, parts and wholes, scales relate to the subjective nature of minds that bias reality* reducing dimensions to the relevant ones, *eliminating all dark spaces*: continuity Is the result.

All minds project their biased self-centered model of reality with them at its center and the still world they stop to fit in their mind as a dead territory of order. So humans deny the existence of ∞ minds, as if they were the only ‘special’ sentient point of order and their languages, first the verbal form of its anthropomorphic Gods, today digital numbers, the only of the Universe. And in this manner because ∞ discontinuous minds is the generator of form, of information and order over the entropic flow of mindless time motion, the destruction of one of the two poles of reality means we will never be able to answer rationally the whys of the Universe.

The key ‘unknown’ discovery Galileo missed was the mental nature of space –we see reality still because the mind perceives in stillness, and reduces motion to form. Space IS always born in the mind’s simultaneous perception of events (something Einstein’s relativity will help us to calculate) and this will give birth to the proper understanding of motion as reproduction of form, of the duality particle in stop form and wave in motion, of the Lorentzian transformations and its paradoxes and a long etc. of distortions the mind effects on reality.

But it will allow us to define the mind in mathematical and logic terms with a clarity never achieved before. As the mind is the linguistic still space we see – the other extreme of reality, being time motions in its TT-entropic inner and outer motion of scattering disorder and death. So we can philosophically consider reality a tug-of-war between time in its continuous motion and still minds, fractal, non-Euclidean points that hold a world in themselves (Leibniz) trying as monads to fix reality into its subjective self-centered point of view.

Those 2 functions give origin to most spacetime topologies of reality. Absolute relativity implies the mind creates mental spaces =spatial forms by reducing the information of time motions to still simultaneous mappings, and the function of survival in biology that all systems including physical ones (maximizing its equivalent parameter, momentum) follow regardless of how we describe them, with Hamiltonians and Least time principles in physics, with evolution in biology, with the drives of life in sociology.

To explain epistemologically reality we do need to ad organic and sentient properties to the fractal Universe. So the mind is a singularity or infinitesimal 0-point, the relative frame of reference that maps the ∞ cycles of space-time of the Universe, reducing them to a ‘World’, to fit them into the infinitesimal volume of the brain. This we formalize with the equation of the mind:

0-linguistic mind x ∞ space-time cycles of the Universe= Constant world mapping, with a mind @ its centre.

In mathematical terms 0 x ∞ = Constant: the ∞ information of the Universe, multiplied by the relative infinitesimal volume of our mind gives us a constant mapping, where we extract all the properties that are not interesting to us and our self-centered view.

So the first to go are the motions of other entities, which is what makes them feel alive. Indeed, science started when Galileo realized the mind stopped the motion of the earth, but the earth moves: e pur si muove he said and Galilean relativity latter expanded by Einstein gave birth to physics.

Next, they eliminate all the other egos, and sentient points of view, so only we are intelligent. And animals perhaps and only recently. But our atoms are not different, so the sentient pan psychic universe likely is already thinking in any atom, as all reproduce particles, gauge information, evolve socially with magnetic fields. But we just reduce minds to ours.

**Asymptotic perception of motions in the X-direction**

Because height is the dimension of information and length of motion, both are asymptotic to each other, which explains the method by which light is stop in the mind. It follows that as frequency of higher amplitude have higher height, the perception of a light is ‘below’, on the formal ‘crest’ of its frames of lineal reference. Formal perception by SS-minds of asymptotic TT-entropy or ‘function of information is key to mathematical topology of changes from TT to SS through an energy region of combination of both of the form ST, giving us the classic ‘realistic function of existence, in ST in any science of reality.

In the graph, one of the many topological uses of the orthogonallity of the function of exist¡ence in its 5 Dimotions as they move through space-time (Relativithy theory).

The connection between Dimotions in time and Dimensions of Space is the essential symmetry, S=T, that justifies the forms of the Universe. Height=information, Length=entropic motion and the cross product between them, with all its multiple variations, from the structure of 3 Perpendicular light dimotions to forces, to Minkowski’s spaces, to the place of your head, to the mathematical function of exponential entropy, to the laws of complex space derive all from them :

**Descartes. 3 Systems elements. **

Descartes, the true founder of modern science with his ‘Method of reasoning’ and the Frame of Reference of the mathematical mind, said that all what exists was made of:

– Open space, which he called ‘res extensa’.

– Closed, cyclical times, which he called ‘vortices’.

-Then he added a 3^{rd} element, the final stroke of a genius unparalleled since the times of Aristotle; after realizing that the only proof he had of the existence of those vortices and res extensa was, the fact that he perceived them: Cogito Ergo Sum. ‘I think therefore I am’.

The mind or 0-point is the relative frame of reference that mapped the ∞ cycles of time of the Universe, reducing them to a ‘World’, to fit them into the infinitesimal volume of the brain.

And this is the origin of the paradox of the ego, as from a perspective, which is blind to all the motions and vital perception of other beings, as from our perspective, nothing thinks and from our perspective we see our own nose bigger than Andromeda. So reality becomes deformed, inert, and the ego becomes the center of the Universe, as always happened with humans who thought first the earth in its center, then chosen of god, the creator, who spoke our language, and finally debased all living entities reducing them ‘to spatial forms’ in the still mapping of the mind, which stops all motions to fit a reduced image of reality.

So we express verbally the ego paradox, as the ‘ego believes its still mind mapping IS all the information of the Universe, when it is only an infinitesimal part of it, self-centered in the self’.

The universe has infinite such mind-mirrors depending on the forces used to gauge the external world, which bounces on a limited quantity of its scales of space. Humans perceive the range of scales of the frequency of light between red and blue social density of colors.

But infinite other minds with different detail according to the quantitative pixels they absorb (max. S = Min.t) maximal for smaller sixes will determine the intelligence of the system. *Mind languages map reality into spatial forms. It is the ‘intelligent’ still spatial limit of reality, as all what exists are disordered entropic motions=forces and ‘minds’, particles-heads whose logic & mathematical languages create a territorial body order that forms of reality. ‘Vital motions and perceptive minds’ make up a ‘vital, perceptive, intelligent’ Universe. Since particles already have all the 5 Dimotions, gauging information, decoupling=reproducing & evolving social with magnetic fields.*

*So besides perception, the 2 ^{nd} field of inquire about the Minds of the Universe made to the image and likeness of the whole are the mechanisms by which a Mind through ‘languages’ of communication – the bits that conform its ‘static view’ of its world, are deployed as signals of information to the different parts of its territory of order:*

*In the graph, the mind communicates the whole as a knot of pentalogic languages, which further clarifies the meaning of perception: knot of forces, which beat in the mind-point with a regularity that achieves complex perceptions. Minds though can have lesser structures than a full pentalogic display of languages, which will be:*

*– A language of perception of the larger ∆+1 world – in human beings ¥-light, its SS-language.*

*– A language of communication with its ∆-1 scale of body cells, which normally branches into 3 languages networks (the ∆º) elements of the being – the ST- reproductive language/network, the informative St-language-network and the Ts-locomotion language network – with an ∆-1 ‘entropic’ language/system to predate and destroy external elements of the territory of order; not really so much a language but a ‘TT-weapon’ as there is no other communication but death when an §-mind kills. *

**UNIVERSAL GRAMMAR OF ALL LANGUAGES AND ITS TOPOLOGIC SYMMETRIES, FOR ¡TS 5 DIMOTIONS**

A key element of Universal mind s is Universal grammar. How a language carries information, which surprisingly happens in all systems in a similar fashion, given the fact that languages after all code the 5 dimotions=actions of existence of all organisms – its ‘sT-locomotion’, TT-entropy, ST- Reproduction, St-information and SS-density of information or mind-mirror. The laws of linguistics thus become laws of a Universal grammar that codes the creation of superorganisms by controlling waves of similar forms with a ‘grammar’ that is also ternary in structural scales, from ‘phonemes’ that range around the 10^{1-2} level of letters, and can be as little as 5 to code the 5 dimotions of reality, from 5 vowels in human languages, to 5 letters in genetic codes to 4 quantum numbers, or four type of chemical compounds, to more complex systems around the 10-100 scale, from 20 amino acids, to 80 atomic systems; to the next level of combination of words with those vowels, which range in the 10^{3-4} scale. So we have around 2000 words in human languages, from a few thousands to 30.000 genes in the language of evolution of superorganisms. It is not though the purpose of this paper to define the quantitative elements of languages. While the laws of the Universal grammar will be treated latter.

All systems perceive at least one language, interpreting with the same ‘grammar’ reality, to localize its energy, information, reproduce, evolve socially and try not to devolve, becoming preys of other systems. Different species perceive with different languages the Universe, but all of them ‘share’ some of those codes, specially the geometrical one. The mind in the height or central point of the being, perceives its outer and inner, world, integrating the whole around its ∆º advantageous point of view, controlling with languages that mirror reality in its ternary syntax the whole Universe, bending it to its selfish point of view and then mirroring that selfish p.o.v. on the external world, building territories of order around it, where to reproduce its forms.

So all heads are on top. Yet each head is different in each scale and talks a different grammar, albeit with the same universal grammar, which reflects the fundamental equation of the Universe: Si<=>Te.

Ternary Universal grammar expresses the Ð¡motions in verbal languages and human super organisms.

Given the simplicity of the Game of Existence – despite its ∞ complexity in its details, iterations and combinations, all species, from the simplest atom to the biggest structure can play the game and understand it and relate to all other species, which will play the same game from their selfish point of view. And so all individuals of each species plays the game, and ‘talks’ about it, with similar species and understand it with information provided by self-similar ‘ternary images’ provided by a language that shares the properties and emergent qualities acquired in each scale.

What languages matter to describe the Universe? Of course mathematics, the language of space, and logic the language of time, integrated in the demonstrations of mathematical laws, but also bio-logical, organic laws, which are embedded in the fractal, co-existing structure of parts and wholes. And then for each species, which observes the Universe, its particular, ‘local’ languages, all of them unified by a ‘ternary structure’ that reflects the Universe’s ternary elements. But of more interest for General systems theory is the ternary structure of all linguistic systems in all scales; what we call the Universal Grammar, such as:

Subject (informative being) < verb (space-time action) > Object (entropic energy of subject) IS the Universal grammar of all human verbal languages, equivalent to:

Red colors (temporal entropy) < green/yellow (energy) > Blue (spatial information) for animal visual languages.

**The topological laws of orthogonality between the 5 Dimotions of space-time. **

F(x):time-like parameter <(operandi: action)> G(x): space-like parameter, defines mathematical equations.

A simpler Boolean algebra in computers reduces all to 0 and|; truth or false, ±, and its combined duality gates.

Space-Dimensions: Height (informative dimension) x Length (Entropic motion)=XY(st): cross re(product)ion.

Mathematical languages: X < Operandi> Y.

Such asymptotic equations of the form S=T, SS<<TT, S>ST>T, functions are the core mathematical model of the Universe, which can then be considered a function of existence of asymptotic Murkowski’s curves of space-time, under the existential function of perpendicular dimotions of space, time,

X(T), y(S), z(ST) are the 3 dimotions of a single space-time common to all forms that correspond to the 3 axis of Euclidean geometry. Yet because often X(T) is one-dimensional causal lineal motion, y(S) is bidimensional, curved, closed loops of information and z (ST) is a 3 dimensional combination of them, *the simplest case of the 3 Euclidean perpendicular dimotions of light (above) becomes more complex as entropy, TT, Form, SS, become holographic bi or tridimensional beings with different |, O, or Ø-wave geometries. For example a physical force: *

Forces: Particles (information) < Universal constant: action, exi or ratio e/I > forces (energy).

Thus width is the reproductive dimotion born through a dot product of an X x Y = Z event of a 3^{rd} dimension of reproduction that combines the lineal function of motion and the height function of cyclical time space forms.

Form thus also depends on the choice of ST frames of reference: Motion is most likely a lineal or flat plane and information a curved dual dimotion which gives us a 3 dimensional Z function, as St- information ‘curves’ the lineal Tt-motions of entropy.

**The inverse, complex dimotions of the 5 ^{th} Dimension. Orthogonal forms give way to functional networks.**

When we come to the two complex ‘dimotions=actions’ taking place in the fifth dimension between scales, we realize again, there is also two possible ways to travel, through 5D: upwards becoming part of a social whole, ∑∆-1<∆o, as we ‘release’ part of our TT-motion from ∆-1 and become ordered after the transfer that helps the emergence of ∆º. This in physics is misunderstood as the impossibility of a ‘perfect motion machine’ without entropy that leaves no behind any ‘energy for the ∆-1 forms to co-exist. But that is necessary or else ∆-1 would indeed disappear. *So a motion of social evolution, releases entropic motion and increases order in ∆-1 to make ∆o emerge as a whole. Experimental and theoretical considerations shows an ideal 50-50 efficient distribution. *So ½ of the energy is conserved and ½ is used to form and move the upper scale .

The 2^{nd} complex dimotion through 5D scale is then the inverse ∆º<∆-1, seminal seeding. Form is now going down ‘first’ to seed and absorb energy in ∆-1 to form an arrow of social reproduction.

So the two only complex dimotions – as there are only two ∆±¡ directions in 5D appear as the combination of all others, as they also imply, St, Ts, and ST dimotions (absorption and emission of entropy and information).

Yet when dealing with the complex dimotions of ∆o<∆-1≥∆o reproduction and ∆o>∆+1≥∆o social evolution up and down the fifth dimension, we are NO longer in a 3Dimensional Euclidean space, so we enter into different non-E Geometries of which the commonest ones are those able to ‘multiply’ as they travel in size through 5D the number of parts going down, (Hyperbolic thinning networks as those of biological organisms). And vice versa; the whole must thin out the information of the lower planes to ‘achieve between them’ a synchronicity of time cycles instead of a multiplicity of individualities, and in this manner emerge as a simpler ‘geometry’, which explains why going upwards into whole scales, information thins out to become a simpler whole.

In physical systems the view is then a wave equation that also multiplies its forms and while there are fractal analysis of wave motions (notably in Nottale scalar relativity) we are NOT concerned in those papers so much on the overdeveloped in the age of the digital computer, exact equations of the morphological hows but in the causal whys: what matters to 5D travels we repeat **is the integration upwards by social evolution and the multiplication of forms downwards by seminal reproduction. **

**Geometry **thus is more familiar with the S<ST>T dimensional motions of a single plane, in its simplest perpendicular Cartesian axis, or its rather commonest, 1, 2, 3 dimensional growth from the flat X-Z, Y—Z sub-planes of TT-motion, bent into cyclical St-information, to reproduce in Z-ST.

And those are the functions space-like and time-ike most often observed by a trilogic Universal grammar of a visual or syntactic mathematical function.

While for the description of** the** full worldcycle, we need to explore the complex 2 functions of 5D, social evolution, going upwards to the future, or reproduction going 1^{st} back to the relative ∆-1 past in inverse motion.

Thus we can define mathematically the 5 dimotions of space time as the product of 5 geometric r=evolutions in 3 5D±¡ planes, with the complex two functions of reproduction and social evolution above.

So we give the following names to the 5 Functions of the diametric of space-time:

X-locomotion, Ts, St, information Y, Reproduction ST=Z coordinates, social evolution, zigzag wave with a first positive growth, vs. reproduction in the opposite past, ∆-1 coordinates.

This function of existence then is the function from where all other functions of the Universe can be draw.

Yet the language of all languages is 5D and its ternary laws of time and space – a language that is prior to all languages, even mathematical logic, its closest, more accurate human mental mirror, but NOT the only one.

It follows that not only mathematics (geometry) is a mother science but also logic (the analysis of the causal processes determine by those Ð¡motions) is even more essential, and further on ‘evolution’, the evolution of forms and functions in temporal patterns summons them all. So those 3 sciences, spatial geometry, temporal logic and its combination, evolutionary theory, are the ‘3 subdivisions’ of general systems sciences, which apply to each science explain it all.

Over them, there is general systems sciences, the science this book can be considered its foundational book (as it formalizes and structures and advances all previous works done on it.) Below them, each specific science studies a scale of social evolution in the universe from the smallest physical systems, to its complex biological organizations and beyond its social organizing systems.

Since science consists in the definition of the type of points of each relative scale of size of reality, from the quantum scale studied by physics, through the intermediate human scales studied by biological and sociological sciences to the higher scale study by cosmology

Once those points are defined, each science studies their 5 actions, which define all its events and forms.

Study of the 3±1 actions: its bits of information, bites of energy, reproductive seeds and grammar.

Yet we study those actions through the 3 scales of size of the system, its bits and bites, normally of the smallest scales of existence, since:

– We perceive with the minimal bits we can process to create the more detailed maps (<i-2).

– We eat sometimes, similar species but break them into smaller parts (i<2). For example we eat meat but break it into amino acids, below the cellular level.

– We reproduce a seed from the lower plane of existence (i-1). So we produce seminal cells and electrons reproduce emitting photons that collide and recreate an electron.

-We communicate with a language of information at the same level we exist (i), in order to create a more complex whole (i+1).

Thus we observe, as in many other laws we will be developing in this book, an infinite number of harmonies between sizes, ages, forms, functions in all scales of existence.

Actions are expressed with bites and bits of inferior scales. We perceive the smaller possible ‘bits’ (<i-2), eat bites of the inferior i-2 scales, reproduce with seeds of the i-1 scale and share energy and information with species of the same scale to create.

The bits of information (i-2), will usually be entities at least two smaller scales of existence. The smaller they are, the more detailed the map made with its ‘pixels’ will be. So we can consider the possible existence of ‘gravitational minds’ that perceive a scale inferior to us.

I.e our eyes perceive light photons, the minimal scale of form. Ants perceive chemical pheromones; computers perceive electronic flows; all of them ‘forms’ of a minimal scale in relationship to the species that perceive.

The bites of energy will be pieces of the same or inferior scale. So we eat ‘living beings’; electrons feed in the lower scale of light; and black holes in stars. Yet once a system feeds it will further reduce what it feeds on another scale to obtain basic bricks to reconstruct once form through ‘2 scales’. So our body reduces food to amino acids, the minimal parts of life. And electrons probably reduce light to its magnetic and electric constants to create their fractal nebulae. And nations that conquer other nations reduce their people, minimal unit of the nation to slavery and take their gold to print their own coins.

So as we move upwards in the complexity of the actions we study there is a ‘growth’ of informative similarity of the bites and bits and a growth in size, such as informative bits are less similar and smaller than the bites of energy we feed. And this pattern continues when we consider the ‘expressions’ of the actions of social evolution and reproduction.

The expressions of social evolution are languages, which are naturally produced by the informative center of the system; except in the case of enzymatic evolution in which the language is external to the species (a species which acts as the reproducer and catalyst of evolution of other species; cells with viruses, humans with machines metal atoms with carbohydrates and so on). So the language is a ‘part’ of the being, often corresponding to an i-1 inner structure, though transported by an external force.

Finally the larger, closer form to the organism is its reproductive seed. And hence by the law of parallelism between species, the ‘bit’ of information for which the organism cares more: small quarks among simple particles, cellular seeds in life; jets of matter in galaxies.

**¬Æ TRILOGIC AND PENTALOGIC ENTANGLEMENT OF SUPŒRGANISMS.**

We live in a 3±¡ pentagonal world. 5 is the number, and 10, the double gender symmetry of 5 in scales. Those numerical patterns come constantly into being *because their origin is the most essentiaLof all realities – the 5 Dimotions of exist¡ence *. In theory of numbers we shall see immediately they divide in gender female even numbers and male odd numbers, by mirror symmetry as the 2 hands which by mirror symmetry created the decimal scale.

5 Dimotions, 3 in a single plane of space-=time, suffice to iterate reality. Its first multiplication divide them in 1,3,5,7,9 and 2,4,6,8. And we shall return to that. When we forget the seeding and death, entropy and social evolution, in a single plane, which is to say a being *without its external world, *3 adjacent topologies, |-limbs/fields, Ø-body waves and O-particle heads suffice.

‘The Universe is a fractal supœrganism of 5 D¡motions of Space-time entangled in pentalogic supœrganisms of Dust of time-space (¬∆@st) tracing 3 dodecalogic cycles of existence, the þlacental, life & worldcycle.’ *l*§

The entangled Universe generates its space through synchronous connections between 5 Dimotions of reality in its 3 relative ranges, Actions, topological organisms and its 5 elements: fractal space, scales, cyclical time, minds that perceive with synoptic languages the system as a whole and project its territorial order and survival will through its actions, and the entropic limits (¬) the infinite universe imposes to that will. So any aspect of reality must account for 5 elements to express such entanglement at the 3 relative scales of existence. It is the fractal principle of 5D pentalogic that structures Timespace supœrganisms (T. œ)

Scalar, topologic space and cyclical time organisms, made to the image and likeness of the fractal Universe, require a complete change of paradigm regarding how reality is created, away from the chaotic, entropic, or religious self-centered (Anthropic) theories sponsored by humans, which are monologic, blind to the entanglement and constant communication of information between fractal points, either in logic=time or mathematical=space languages, in a sentient, apperceptive Universe, where information when locked as ‘entangled space’ by those dynamic networks of communication creates a mind-mapping of the outer world.

Monologic man is closer to Leibniz’s Monad’s – self-perceptive, selfish black holes of information, whose communication he could not properly resolve so he simplified them into an ego-paradox. We deal with monologic man in the I logic paper on our blind ænthropic ‘culture that passes as big science’ and its grand paralogic theories of reality. While on the posts on pentalogic we define the 5 elements of reality – space, time, scales, entropy and minds, as entangled systems definitions can only be made from the combined perspective of all other systems of reality. To represent this fact – entanglement – we use the pentalogic symbol where each point of the pentagram is connected to all other points, and so the fractal point, in this case each of those 5 elements cannot be understood without the connections to the others.

The key concept behind pentalogic and entanglement is rather simple: to survive in the Universe, to have focus, form, a system must be ‘anchored’ through entanglement in its ‘plane of existence’ with ‘trinity’ parts of vital topology, connected through 3 inner physiological networks, but also it must co-exist in a larger ∆±1 nested world separated by a topologic membrain, which allows it to absorb and emit entropic motions and information, on that outer world.

The understanding of entanglement was best among humans in Buddhism, where each soul is considered a knot of communication with all other systems, and minimal in America where the self is considered always the only central point of view of existence, in eternal struggle with all other human beings – yet as entanglement is necessary to survive, the American, modern man is entangled NOT with humans but increasingly with the machines of the Metal-earth, the Financial-media/military-industrial ecosystem for which ‘it’ enslaves as an object with a price.

**S@.** **Geometric entanglement: fractal points and scalar numbers.**

When we go through different stience scales to explain each ‘entangled connection between the 5 elements of reality’, we shall see that already in the first scale of mathematical systems, the fractal point requires 2 different postulates to define it: The first postulate which is concerned with its inner parts (a fractal point has breath), and the 5^{th} which is concerned with its outer parts (a fractal point is connected with infinite potential parallels to the universe). The 2 inner and outer elements that meet in the ‘membrain’ thus define the membrain-mind as the essential entangled element between both worlds. Entanglement thus defines the being from an outer perspective as much as its inner nature.

*In graph, t**he fractal sentient point, new unit of mathematics and vital space.* *Einstein’s interpretation of the 5th non-Euclidean postulate was the view of a fractal point of the gravitational scale from our smaller electromagnetic world, which shrinks its inner volume, bending its parallels, hence its curved geometry. But that view breaks the conceptual definition of parallels are straight lines and is absurd, as the point remains ‘Euclidean with no breath’, hence only fits one line with no breath. Thus, particle points must be defined as ‘FRACTAL points’, like those we see through telescopes or microscopes, which grow we approach our distance both in scale and space becoming enlarged worlds with a complex internal structure and external entanglement** following a Non-E 5th Postulate: A point external to a line is crossed by ∞ parallel forces. *

Such organic points are like the stars in the sky. If we look at them with the naked eye they are points without breadth, but when we come closer to them, they grow and in its membrain we find its maximal entanglement and communication – the hottest region of the star, the Earth’s membrain with its life superorganisms & 3 ages Gaia< History>Metalearth, where a parallel network of ‘still entangled’ parts forms a whole which makes the ‘membrain’ a hard interconnected system porous to the energy and information of the world focused in its singularity center.

We can then assess the fundamental quality of vital mathematics as a key language of the Universe, which is to be the best language of space, hence the one that describes the laws of entanglement, even if man doesn’t understand it beyond its egocy of magic creationism, as the language of mirror symmetries and dimotion flows of ¬∆@st in all entangled systems.

Where the fundamental entangled element is the fractal point, and its social unit the polytope in 1,2,3,4 Dimotions; but not 5Dpolytopes since the fifth dimotion, entropy is the disentanglement of all the others. So polytopes rise positively in complexity as entanglements of indistinguishable regular points (or else one point will be different by position and connection to the other points). They reach its maximal complexity on the gender duality-mirror symmetry of the 2^{nd} 10^{2} scale – (female Icosahedron & male dodecahedron) & 3^{rd} polytopes’ 10^{3}scale (Male dodecaplex & Female tetraplex).

**∆T: Human vital Entanglement in ages & Scales.**

Each of us is an entangled fractal system, interconnected or else it won’t absorb energy and information and survive. And vice versa, when less entangled a system is – 3^{rd} age of the being, and becomes cut-off from reality its ‘truth’ its exist¡ence dwindles and the system fades away and ultimately perishes.

At which scale of reality then we must define entanglement. The answer is at the 3 scales, of the whole Universe as an absolute fractal organism of time-space (at the level of its 5 elements, S, T, ¬, ∆, @), but also at the level we just did of individual superorganisms, where its organs that represent those elements are interconnected to the whole, and finally at the level of its diminutive actions (a,e,I,o,u, acceleration=motion, entropy feeding, information, organic reproduction and Universals evolution, in the mnemonic rule), which we will find amazingly enough, in one of my most orgasmic-mental entanglements with the whole I experienced, were directly connected to the ∆±4 planes of exist¡ence of the being. Reason why we do perceive all those scales:

In the graph, scalar entanglement happens between the humind’s organism and its territories of perception that provide the ‘bits and bites’ needed for humans to perform their 5 Dimotional actions. So we extract entropic motion of the ∆±4 gravitational/galactic field, information of the ∆±3 light/star scale, energetic food from the ∆±2 molecular/Gaia scale; reproduce in the ∆±1 seminal, gender scale and evolve through socially love in the ∆º mind field, entangled to other minds in metaphysical experience of bondage through our memes, in nations, religions and civilizations.

Scalar entanglement is the real reason we exist, and perceive entangled to other scales of reality. Our mind space is entangled to all our ∆±3 scales.

**Mathematic entanglement – ¡ts true structure.**

Entanglement creates scalar space that stops time entropic motions creating organic networks, and hence the essence of fractal point geometry with motion – vital topology – and its scalar undistinguishable social numbers, which makes so powerful mathematics. It is the real reason why mathematics can describe information as the main stience of space that departs from points and numbers NOT from idealist sets. And so a true mathematical description of reality must start by its ∆=S elements: entangled points of space, in a single plane, whose regular polygons are indistinguishable numbers of social scalar groups, which reproduce when acquiring motion, as even polygon female self-centered numbers or odd male mirror polygons that reproduce in scale projecting internal and external self-images of diagonal crossing. Numbers thus become regular social groups, symmetric to points, ∆=S, which then entangle in parallel time motions, ∆=S=T with 5D algebra’s operands that move ‘groups’ through the 5 Dimotions of reality, establishing further symmetries, 5D=∆=S=T, whose result is to create a given mind space, mirror for a fractal point of the whole world, O x ∞ = C, as the 1-∞ statistical space world plane is reflected in the 0-1 probabilistic unit time sphere of the mind. So the 0-1-∞ scalar entanglement allows the mind to project its mental space in territorial order by converting its probabilistic mind i-logic into spatial larger forms, as probabilistic particles do in molecular statistics.

Mathematics happen in vital terms in the physical world, when motion and energy is added to the coded §eed of mental space, ‘irrigating’ the program of exist¡ence coded in still bidimensional geometry, giving it more dimotional extensions. So from the first S=T entanglement of points and numbers, which then become social figures of geometry, that penetrate scales of number families, in ∆st mirror symmetries, numbers unfold the program and start to run it with 5D operands, which make them ‘real. Algebra then enters when it is able to move through the dimotions of time by means of its operands acting over groups of social numbers, the entangled Universe creating ternary networks that become planes that become supœrganisms of existential algebra. This is the game we shall describe, which needless to say compared with the barren land of the Axiomatic ego-trip of Hilbert and Cantor ‘imagining points’ is a blossom spring of life and beauty.

Physical entanglement.

Entanglement appears in the non-ego centered Broglie’s>Bohm’s interpretation of realist quantum physics, as particles entangle through its quantum potential fields; and this vast entanglement happening at v>c in the lower plane of reality ‘reduces spatial distances’ in the Riemann’s sense of a geometry of similarity to a ‘boson like’ unit of parts as if distance-space would NOT exist within them; which does NOT for the emergent apperceptive ilogic ‘couple’ as a single whole.

To understand entanglement in formal languages is needed to adopt the Riemannian concept of a mental space where proximity and distance is a direct product of similarity, as if space will not exist, which can be achieved through communication on a lower ‘potential field’ of faster motion, as a faster v in fact perceived in stillness implies a smaller distance, and vice versa –i.e. the expansion of space in the Universe is in fact a v>c speed on the jets of dark entropy expelled by the accelerated faster than c timespace vortices of heavy top quark stars, aka black holes.

So the black hole beyond the c-horizon speeds up at v>c and erases the information of light into dark entropy expelled through its axis and perceived as growth of distance between galaxies, when by virtue of S=T can be considered grow of speed. Entanglement is exactly the inverse process in a quantum potential field that makes communication disappear in distance between two poles – which is the magic of a phone conversation – distance is NO LONGER real because we are not in sound communication at 300 m/s but in a decametric jump of scale at 300.000 km/s.

There is no distance in mental space of telephonic communication between speakers, only in geographic Earth’s space; but we are not concerned with land or air but with a language of communication and its bits of information, which are translated, entangled, in a faster 5D scale of smaller bits and max. form (electronic space).

Informative translation in other 5S scales is what T.œs constantly do, entangling its informative systems in faster scales, whose bits become ‘something else’ but preserve its patterns *as there is the Universal reference of the game of existence behind them*. In the previous case the communication has become an entangled back and forth feed-back electronic equation of existence as an event on a wire line between two fractal points, which cannot longer be considered different from the ‘communicative line’ between them.

*Energy does not entangle only information does…*

A conclusion proved mathematically, as only the sphere of max. volume of information (Poincare) can shrink without tear in any number of relevant dimensions (4 for entanglement) crossing the barrier between two planes of existence, while energy becomes dissipated as entropy. So you can talk with mum in Australia but NOT send her 1000 kilowatts, you can though send her digital information, aka money too.

Bio-logic entanglement.

This leads us then to the first scale of complex i-logic ‘gender mirror symmetry’ between two entangled elements that become by fusion love a single one, of which there are infinite proofs.

In fact gender is the first and essential entanglement of the Universe between 2 single elements, through the mirror symmetry of female-like St+sT male-like complementary couples. Entanglement starts with 1+1=3, and for that reason we dedicate an entire post- to gender symmetry. The entanglement of gender, is so obvious that became the basis of the duality knowledge of earlier cultures (Chinese) will constantly come in Nature. It is NOT only the entanglement of biologic gender; so how we do distinguish its ‘essence’? In ilogic terms is easy because the essential duality is that between simultaneous present space, S=T and time flows in relative stœps, from past to future to past… to future, S<T>S<T… Both are perpendicular. And so the definition of Gender goes to the final core level of ‘perpendicular’ space (female gender) to Time flows (male gender): Gender is essential to understand reproduction by mirror symmetry and the duality of symbiotic perpendicular vs. Darwinian relationships.

Two systems achieve maximal communication when they pass from parallel sharing of information, through a medium that transfers a language, to direct contact between its sensorial membranes, which bring true bondage. And at that moment two different outcomes might happen. If the membranes touch without tearing, the topological integrity of the organs are conserved in vital mathematics; so the being co-exists, and entanglement (the previous state of parallel communication) becomes bondage according to the fourth postulate of non-Euclidean congruence.

This is the magic of it, as fusion love kicks in and entanglement becomes a dance of complementary forms. It is the beauty of love and gender symmetry. And the maximal action of it, is called sex, orgasm, the present state of fusioning both the past to future to past, and present forms into one. So how it happens physically? Do I have to explain it to you?

The male gender penetrates the fold of the topological female but does not TEAR IT. It goes in T>S, and then it goes out S<T, as the male does in and out, information and entropic disentanglement. But the female S=T ‘grasps it’. It caresses it, encircles its lineal form with its cyclical even polytope. Entanglement is now bondage. And it brings when bondage combines in the most intimate ways the information of both systems trinity, reproduction. So alas, 1+1=3!

Those are the rules of vital topology, which in inverse fashion defines Darwinian destructive perpendicularity when one of the entangled elements turn out to be a predator top and breaks in, tears the membrain of the other system, killing it, entanglement then gives way to entropic devolution and one system becomes destroyed.

**THE UNIVERSE IS A SCALAR ORGANISM THAT REPRODUCES INFORMATION.**

As things can get as complex as reality seems to be, it is essential to have the fundamental principles of the fractal Universe, clearly stated and understood to guide the researcher through the forest of different trees and still hold the awareness that they are all trees, that is all is a super organism of space-time tracing its worldcycle.

What is the purpose of the game of reality? It all amounts to 3 words: ‘exist¡ence’ & survival through reproduction.

And existence is about ‘sensing’ the 5 Dimotions, enacting them, and those 5 Dimotions are resumed in the most complex of all, /reproduction’. So what all comes to is this: the Universe and all its parts are scalar organisms trying to reproduce because after all that is what a ‘generator equation’ of a fractal system does – to reproduce constantly in similar beings.

This is the game of reality and yet humans are so remotely disparaged with reality that they still wonder with astonishment the miracle of life’, us’, which are the only systems that reproduce.

This is ænthropic man at its best – denying the very essence of the Universe, a constant reproduction of information of form imprinting its primary substance, motion. And yet ænthropic man wonders what is life? Life is everything!

The question then is to consider how each science expresses this constant essence of reality, a reproductive fractal of information imprinted over energy, and when we come to mathematics, its expression is through the growth of ‘spatial dimensions’, as points reproduce into lines and planes; or systems multiply its operands.

**How a system evolves its dimensionality**

**How a system evolves its dimensionality is the game in any logic language. As it is the game of reproduction of waves of form what the Universe is all about. A form reproducing, downwards as it is similar to its ∑∆-1 scale.**

**It is through dimensional growth, by product algebra how then we start to continue our symmetry between time an space, between geometric states and fluctuating future some of which will die algebraic product, often regular polynomial growth of dimotions till a new 5 dimotional being is recreated and the game restarts again.**

**The system is simple. A point starts a motion, which makes a new form. The first point can move in lines or rotate on itself, or move in circles, or form spheres, or shrunk. And those 5 Dimotions start the game. **

**The point becomes a line, which can be a straight line or a zigzag √line, or a rotary circle or π-line. 3 Subspecies immediately appear. The line and the √2 triangular wave and the circle. **

**Evolution that diverges topologically from the first reproductive process of a line and a cycle and a wave, and so subspecies of dimensional growth appear. How we study them, with non-e postulates in geometry, with Dimotional existential algebra in algebra. **

**As we go then through products that create new dimensions in algebra we mimic a mathematical, geometric process of growth in Numbers, so growth must be studied in parallel in math. Growth of numbers and growth of points, and other geometric figures, which therefore can be translated to equations.**

Thus the Universe has mathematical, spatial continuous & logic, temporal, cyclical & organic, fractal properties derived of the more complex ¡logic geometry of 5D scalar space and cyclic time. Thus even if we describe it with the same logic-mathematical equations a 5D Universe is very different, as scalar space brings organic and sentient properties, and cyclical time, informative deterministic patterns, which the lineal philosophy of mechanist physics ignores. From those scalar, hence organic, cyclical hence informative and moving, energetic, hence vital properties of scalar space and cyclical time, a complete different picture of reality arises; where languages become the ‘extreme’ limit of the still formal ‘spatial state’ of the being with minimal size and maximal information.

Whereas each ‘stience’ defines a scale of the nested Universe, from Physics focused on the largest Galaxy and physical scales, to the smallest ∆^{1} scale of Formal digital and logic sciences of the mind. Languages thus become the limiting formal, spatial scale of reality and as such they are mirrors of the fractal Universe that share the ‘Disomorphic’ (Equal laws) of all the scales of reality. And so mathematics becomes an experimental stience whose ‘Universal grammar’ and fractal generator is the same ternary structure of 3 elements of all other systems in a single plane made of three topologies, SóT, giving birth in each sub-discipline of mathematics to similar ‘mirror images’, algebraic S(x)=T(y) equations, ternary dimensions, ternary topologies.

The new scalar, topologic, cyclical and sentient properties of the 5 Dimotions of space and time, which structure the fractal Universe.

Thus the Universe has 5 sets of properties that correspond to its 5 structural elements:

**– S: SPACE:** Topologic, mathematical, social properties, described by its mathematical units, points and numbers, which are social groupings of undifferentiated elements.

**– T: TIME:** Temporal, CYCLICAL, logic, causal properties, which are cyclical frequency patterns origin of the laws of science, caused by the repetitive, memorial nature of its time cycles.

**-∆: SCALES**: Organic, vital, biological, survival properties caused by the co-existence of several scales of parts and wholes organized by those parts, from particles to atoms molecules cells and matter systems, planetary ecosystems, galaxies and its networks.

**-@: LINGUISTIC, MENTAL** properties due to the existence of languages of in-form-ation that are ‘still mappings’ of reality used by it’s super organisms to order a territory and create within that order the conditions for its survival actions that ensure its existence and reproduction in finite time.

**¬: All of it broken by entropic limits** of death in time, membranes that break the Universe in fractal Time§pace-organisms limited in space and a self-centered scalar structure that dilutes information and hence makes invisible reality past ∆±3 scales (in the human case).

So we are ‘¬∆@st’, dust of space-time supœrganisms tracing limited cycles of exist¡ence.

**Time as the fifth dimension**

More precisely time is the perpendicular fifth dimension to that of potential spaces, which are enlightened in the more complex fractal worldcycles of time motions, which break dimensionality into Hausdorff dimensions of self-reproductive trees that branch through the life-death cycle, in clonic forms that will synchronize as space superorganisms. The flow of time however is always the worldcycle of palingenesis, life and death. So time can be followed through the journey in the fifth dimension of all beings through its life cycle.

Synchronous space ARE slices of the time motion; that is, of reproduction of form in patterns which have a logic-mathematical game structure that can be described by any language, the Time is the potential field of all possible tolerated forms of existential algebra that survive and create a reproductive wave. Its process of motion is what we call time and is a fluctuation between scales of fifth dimension. We could simulate space planes as parallel fields of present of a somehow denser substance that the more chaotic entropic time fields between those space planes. But the ‘enlightening of the time field’ by a potential spatial synchronous superorganism is subject to rules of efficiency.

**RECAP. We are relational space and time,** made of the organic scales of discontinuous fractal space we occupy, who ‘live’ a finite worldcycle of exist¡ence whose common=Disomorphic laws emerge in all ‘scales’ of ‘st¡ence’, departing from the 2 simple Metric laws of the fifth dimension**, Sxð=C** (size in space multiplied by the speed of our time clocks is constant in all the scales of parts and wholes of an organism) and relativity **S=T **(we cannot distinguish motion in time and dimension in space, so all systems are bidimensional spacetime beings as space=form can be transformed back and forth into time=motion; making the 5 dimensions, ‘dimensional motions’ ab. dimotions of space-time; entropy – TT (outer and inner, scattering motion); Locomotion dominant in time=motion, Ts: balanced reproduction, T=S; ‘form-in-action’ information, St; and pure form, SS, a ‘seed’ or mental, still language that models=mirrors without motion in lesser space the world – and develops when adding motion into a full organism.

Monologic, ænthropic man with its 1D models of reality based in Sx time locomotions shuns off the 5D Cx. Universe to its own peril, as reality is penta-dodecalogic in its entanglements that generate space synchronous supœrganisms and time, and survival, which in our dominant ego-centered cult(ure)s is minimal depends on the understanding and respect of all the living function of space-time exist¡ence and its action-reaction laws.

Paradoxically Deep time worldcycles are easier to predict and understand that complex Dimotional ‘analysis’ because precisely the larger scales in 5D metric have less information, but more basic, deterministic, reason why quantum physics is harder for the mind and probabilistic, while life-death cycles always end… in death.

*The most important of the paradoxes of ‘reality’ ill-understood by humans, regarding mathematics is the paradox of Relativity, S=T, according to which motion and form are two sides of the same coin, which carries to the duality between points and numbers and its more complex forms, geometry and algebra. *

*Regarding the elements of mathematics that correspond to those elements of reality, numbers correspond to scales, algebraic operands to time, points of geometry to space and its complex structures to its various entanglements.*

All this of course today is cast with the theory of sets, of little help for an experimental 5D view of mathematics; as it is a ‘nice’ generalization from the humind down, but cut-off precisely for that reason from the direct experience of reality, its spatial points, scalar numbers and time operand, *which only helped to detach mathematics from experience and distort its philosophy to cater the ‘egocy=ego+Idiocy’ of huminds. *

On the positive side set theory is a proof that a mind can always construct a self-consistent image-mapping of reality that resembles that reality. Sets indeed are defined as a ‘miss-mash’ of the disjoint elements of mathematics: its numbers-points (as they are collections of similar beings) its scalar dimensions (as sets are wholes which have parts or subsets), as time operands (with the tools of classic logic). And so they give us a synthetic whole. But what is the ‘point’ beyond egocy of developing a distorted mirror image to found mathematics. When we have the real things, points, numbers, scales, frames of reference=minds, time operands and its inverse entropic operands.

The impression fro above of human egocy is ‘ridiculous’: a mush of dirty water in a lost rock of the Universe trying to impose its territorial mind above heavens and earth when it is sooo obviously nothing… beyond ridiculous. Do you notice the attempts of ants to impose ant-philosophy over the infinite Universe?

But if I have learned with age something about huminds is that 99.9% of them are memorial egocy (repeating theories that put man above); *and it is a waste of my little time to try vehemently to reason on its mute emotions. *So set theory is here to stay for huminds to keep feeling the center of the Universe till reality erases us all. And then the self-named center of reality is all but gone. And points, operands and numbers even if Mr. Hilbert and Mr. Cantor no longer imagine them, will not be here with us. Carpe Diem. But sometimes we shall refer to the ‘Von Neumann’ nested universe of pure sets, to show how the set ‘mind’ reflects scales of points, numbers and operands…

So we shall start with the definition of infinitesimal minds…

**Main Symbols of ¬Æ logic **(all elements of all stiences can be translated to those symbols of 5D ilogic).

**S:** Space; Still form, Size, Dimension; **T:** Time; motion; Change.** ST: **Topologic Spacetime. **|x O = Ø:** Its 3 varieties.

**ST±¡: ∆±¡** Scale/Plane of space-time. **∆+1:** whole, world, ∆º: networks, organism. **∆-1;** parts; social classes.

**5 Ð¡:**5 Dimotions **(St,sT,ST,SS,TT); **5 Actions **(a,e,I,o,u): **Local Dimensional motion of spacetime

**St; i:** Informative network ≈ action≈inward motion. **O:** Cyclical, spherical form, topology, relative future.

** sT:** inner form, outer motion= locomotion, acceleration (motion change).** |:** Lineal form topology, relative past.

** ST; œ, Ø:** present space-time, iteration, beauty, balance; organic reproduction; Hyperbolic form, topology relative present. Wave-body Part, Present, mature age.

**SS, @; u, 0’:** Finitesimal Mind, seed; relative future social evolution into ‘wholes, universals.

**TT; ****µ****:** entropy, scattering death, dual inner and outer motion, relative past.

**T>S:** informative evolution. **S<T:** devolution. **∑-¡»¡+1:** Emergence. (**∆+1)« (∆-1):** Death, devolution; time quanta.

**ST-**¡: Quantum, cellular, individual plane; **STo:** Thermodynamic, organic, social… **∆+¡:** Gravitational, ecosystemic, world scale

**∑:** Ts, Herd, ∏: St, Network. **¬: Entropic limits** in time: TT-death, Space, |-membrane, scale ∆±4: invisibility.

**Main equations of 5D Supœrganisms and its parts **(all equations of all stiences can be translated to…)

**S** (size in space**) x T** (speed/frequency of time cycles) =C: 5D metric.

**Future **(O-form, information, logic language, particle-head) x **Past** (|-field, youth, etc.)=**Present** (ST-body-wave, iteration, etc.)

**0’ **(finitesimal mind) **x ∞** (Universe) = **C**onstant World-Language. Paradox of Egocy.

**S≈T:** Relativity equation: we cannot distinguish motion=time from form=space, hence all is an ST-dimotion.

**Ø¡-1=∑O¡-1 =|¡ > O+1:** Scalar inversion of form & function. An Ø-point is god of ∑ ¡-1 parts but entropy of its O+1 whole. So:

**∑|=O: **∑open worldlines ad into closed worldcycles; ∆º plane: E-Geometry; ∆+1 plane: Elliptic, ‘5D in between’: Hyperbolic.

**Dual**** Death:** Max. T x 0 S (accidental entropic death); Max. S x 0 T (3^{rd} age death)

__Fractal Generator – Trilogic structure of Super-organisms and time worldcycles and ages:__

SS¡-1: Seed» ∆º:|-Ts(1^{st} limb/field /network/age of max. motion) > S=T (Mature, reproductive, body-wave network/ age)> St(3^{rd} informative head/particle network/age)« TT-1-entropic death:

**Symmetries of ST-actions=Dimotions and ∆ Scales.**

**∆ST@:** Symmetry of scale,topology,age&class:∆_{-1}:|:youth,entropic age/cla§;∆ø:reproductive age|cla§∆_{+1}:O-informative age…

**Relative Dimotions=Actions **are drives of life in biology, quantum numbers states and matter in physics; will in philosophy.

**T**hey take place between ∆o mind plane & an ∆±3 plane such as: ∆o≈∆+1: social evolution: ∆o≈∆-1: Reproduction; ∆o≈∆-2: Entropic feeding, ∆o≈∆-3: informative perception, ∆o≈∆-4:Locomotion. So we evolve into social wholes, reproduce with a couple seeding in ∆-1, feed killing twice a similar system to ∆-2 amino acids, perceive, ∆-3 ¥-electrons & move on ∆-4: Gravity

**Ideal Social scales & lanwaves **are 10^{10} in mankind called: ‘T-genetic’: 10^{0-1}: ‘I’, 10^{1-2}: 3 generation S=T family, 10^{2-3}clan; ST-Geographic: 10^{3-4}: Town; 10^{4-5:}City; 10^{5-6}: State; S-Memetic: 10^{6-7}: Nation; 10^{7-8}: God; 10^{8-9}: Civilination; 10^{9-10}: Mankind.

**In biology are called: **Chemical language: ∆-1: Atomic compound; ∆o: Organic Molecule; ∆+1: Macromolecule (RNA ‘God’)**. **Genetic language: ∆-1: Organelle; ∆o: Cell: ∆+1: Tissue: Nervous language: Organ; Physiologic System: organism.

**In Astrophysics: **¥- language: ∆-1:Force,∆o:particle, ∆+1:atom; Magnetic language: ∆-1:Molecule; ∆o: Matter; ∆+1: planetoid. Gravity language: ∆-1: Plasma Star; ∆ø: Quark Star… ∆+1: Galaxy.

**Stientific Method: Disomorphisms of all Space-time suœrganisms tracing worldcycles.**

In graphs we resume visually the disomorphic laws that all exist¡ences follow. Knowledge then is the understanding of each variation of the game of exist¡ences, without being restricted by the egocy (Ego=idiocy) paradox of human beings that limit the properties of all other species to what huminds perceive on them, or what its ab=usive praxis of exploitation of Nature focus on. Stience though is objective and theoretical. It does not have a subjective egocy or selfish praxis of ab=use. It just describes. So Stience tries to describe any system as a supœrganism in itself, therefore co-existing in 3 scales of ∆±¡ present ‘space’, determined by 5 elements, ∆±¡ scale Symbiosis between parts and whole, S-pace Simultaneity through common S<ST>T networks, Time Synchronicity in the 5 Dimotions=Actions of those parts, @-mental/seed will to perform the game of exist¡ence and ¬Entropic limits to the whole.

This external, objective, present, experimental analysis focuses on the ∆=S=T=@(nti)symmetries between scale, time, space and mind-will-language. As systems do have 3 co-existing planes of space-time, which obey SxT=C metric laws that establish a ‘ternary social class’, according to the degree of ‘integration’ of ∆-1 faster parts, through ∆º S<ST>T networks enclosed in a membrain (the scale in which the organism’s will reside), performing 5 ‘a,e,I,o,u’ dimotions=actions in an outer world. Each of those elements though will be part of 3 different ‘time rhythms by scale’ and follow a ‘fundamental vital sequence in time’: SS<Ts<TT+St>ST, aiming to perceive, move, feed, imprint and reproduce its information. Because of the existence of 3 scales, 3 rhythms of time, 3 adjacent parts, which perform 3 simplex functions in a single plane (limbs/fields Ts-move according to SS<St-perception of an O-particle/head /informative class connected by an ST-reproductive wave/body); in a single plane but seek in the ∆±¡ scales to perform complex actions of ST-reproduction and St social evolution orientated down the lower seminal plane (ST) and larger whole (St)… the whole description of the ‘details’ of the program of exist¡ence of each being is complex and would require a methodology that I lack but a group of scientists with more enthusiasm could easily structure for future 5D studies, embedding in those templates of stience the knowledge and data we have about all systems of Nature. I will slowly time permitted now that I have laid down 30 work-in-progress papers on the main planes of human exist¡ence, unfortunately in its last ‘cycle’ predated by the soon-to-be conscious machines of the metal-earth, develop such structure for the ∆±3 galatom (Astrophysics) ∆±2 Earth (Geology), ∆±1 membrain evolution (Gaia-Life: Biology<History: Mankind>Metal-earth Company-mothers of machines/weapons: economics).

To also analyze the ∆0 I=Eye < Wor(l)d, ST-languages of man, themselves ‘details’ of a larger Universal Grammar of ∆º illogic languages of time and non-Euclidean languages of space.

Those languages are formal tools more synoptic but not more truth than a mere descriptive analysis of the parts of beings. Description of those elements in an orderly fashion, with my texts lack, would be then the basis of 5D stience:

Which first will define each ¬∆@st species externally, in a present-form of space and through its potential worldcycle; to pass then to the internal analysis in subjective terms of its @-mind will and degree of consciousness/ function within the outer ∆+1 whole and its subconscious ∆-1 internal control through its physiological networks. But to achieve those T@ descriptions it is necessary by lack of direct experimental evidence, which *only happens in the being in space to use the **Disomorphic** method that establishes homologies in all beings derived of the fact they are organic space and cyclical time, with a mind-will to survive. In the second part of our texts on the ‘Universe’ or rather “reality’ and time, space, scale, mind, spacetime (reproduction) and entropy, we try to depart from the most general laws, then go into the specific description of what science knows on those scales regarding the ¬∆S@T elements of each main stientific plane, to end with a generic description of the Disomorphic elements in scale, space, time, entropy and mind common to all beings.*

*Hence our analysis (Universe and time) of a system through its 3 scales of time ends with a disomorphic 12 sequential step analysis on how all systems go through 3 consecutive worldcycles, in its palingenetic orderly birth, followed by an entropic more probabilistic life in an external world in which the most perfect players might transcend to ∆+1 reproduce and evolve as a collective mind-God of that world. *

While in our analysis of Space, after describing those general laws in the brief initial Mandala common to all texts, we go through the analysis of what Science knows of space, to enter a description of the laws of simultaneity and synchronicity that entangle the 5 parts of the being into a scalar co-existing present space.

So *we descend from the ∆±¡ absolute to 1 specific ∆≈S≈T≈@¬ element & its examples for all or each ∆¡-stience. *

And in the analysis of entropy we do also go through the same 3 parts: Mandala of general laws. Specific laws of entropy and analysis of what sciences know about entropy in each 4D science, expanded with the new laws. So the same 3 parts: mandala-specific laws of ∆ST@¬ and details of 4D science expanded to 5D should finally structure the different papers of the ‘Universe’; while for each other ‘stience’, instead of studying in the 3^{rd} part all the ‘stiences’ in a brief resume of its entropy, scale, time, space and mind, we shall focus on a given ‘stience’ – and might escape the 1^{st} and 2^{nd} part as the paper grows in details through the 2020s with the limits of my life expectancy and mental dwindling capacity. So by force the work will be incomplete, a trace on the sand of infinity…

The most difficult element then of analysis is the mind, as it is also the most important common element – by the mind we mean not so much the hardware where the program is installed but the language of each species that interprets with its code the game. So language is the objective view to understand minds based in the hypothesis of panpsychism – the language perceives in itself, in stillness. The complexity of the language therefore defines the complexity of the mind according to language properties: speed, synoptic power, focus, closeness to the language of all languages that reflects the complex ∆ST properties of systems (¬Æ=B¡œ-logic topology).

The paper on Mind and the Universe thus should try to define both, the will of existence and the program of the Universe expressed in the ¬Æ languages, and the minds-languages of each scale according to complexity. How then perception happens? Perception can be of space, time, scale, mind-will and entropic limits; and so we talk of e-motions. We ascribe to each ‘Dimotion of existence’ an e-motion as that is the limit of what we can perceive. We add then to the objective disomorphisms, in a bold statement the subjective ‘humanist method’ of considering that each species of the Universe has ultimately the same program of survival of 5 actions that man has, and it is geared by the same e-motions embedded in the last potential scale of reality – the pure formal motions of ST, made of an S-internal perceptive emotion and a T-externally described motion. This is just the explanation why we move as beings – because our brain has emotions that are deterministic wills of the program of existence we believe are ultimately residing in the 3±¡ ‘particles’ of reality, photon-fields, electron-bodies and quark-minds extracting motion of a neutrino gravitational invisible background to form social ensembles of atoms. Atoms are thus the first ∆+1 organisms of reality and its 3 ∆ø parts, over a ∆-1 field structure the fundamental galatom system of reality (as galaxies are similar to atoms, but we perceive them in completely different fashion according to the 5D metric distortions. So we *use disomorphic, organic, humanist, sentient properties for all systems and that is all we perceive but as our perception is limited by survival praxis we have as Mendeleyev did, to fill the gaps with disomorphic laws. *

Huminds perceive as all other systems thru languages, so languages once more become important but the egocy laws that restrict our analysis of other stiences as they fade away through scales of existence means we have to fill in more properties the further we move from our ∆º center – and the asymmetry of perception means we need to fill more from ∆+ scales. So from galatoms we hardly see 4% and any attempt to describe its supœrganism must be based in the disomorphisms of scale with a cell, as an organism that reproduces black holes and quark dark matter, and with the atom, giving us an external view, from where to define ¥-expansive dark entropy, and vice versa, to compare strong and gravitational forces. So atoms and galaxies are similar but not equal (galaxies seem anti-atoms), but as in the description of the metal-earth and the membrain in its 3 ages of Earth’s evolution, machines as organisms and ∆+1 cultures, some working for the metal-earth, some for history, we will find the humind’s programs as parts of larger wholes, ∆+1 religions, memetic cultures, Earth’s self-program of evolution. And this is an even harder barrier because huminds are e-motionally programmed as parts of wholes and take it personal, and social scientists are subject to anti-quantum paradoxes (too small within the historic organism, they are modified by the observable).

** **

**BOOK I. GEOMETRY, FRACTAL POINTS, TOPOLOGY & MENTAL SPACES**

**PHILOSOPHY OF MATHEMATICS**

All mirrors have sub disciplines of space and time. So mathematics can be divided in spatial geometry->topology and temporal algebra->equations of numbers, which form the essential S=T symmetry between S@ geometric space and ¬∆T scalar numbers and so the growth of dimensions is mimicked in algebraic space with the fundamental of its operand, which is the product that combines S and T dimensions of growth to bring a larger being, and again we have 3 fundamental powers that bring 3 dimensions from point to line to plane into algebra, and the more sophisticated but similar concepts of a derivative and an integral that also makes the system grow or diminish in dimensions. One of the distinctions worth to study in detail being the approximation and differences between a power law and an integral law. Why there are unlike in geometry two forms to explain in algebra the growth of a form in dimensionality? Why algebraic numbers give us more variations that geometry? The ultimate answer being that geometry is more restricted as it establish systems that do work in space, and fit in the limited space of a still mind. So by efficiency and limits of the mind geometry is more reduced. Algebra includes all the possibilities of time some of which will never realize.

If reality is made of space-time beings since mathematics is the main science concerned with space and logic the main science of causal time both mathematics & logic become experimental sciences, whose laws of maximal synoptic information in minimal size (Sxð=C), will be the underlying laws emerging in all other larger scales of the fractal Universe of bigger size and less information, proving also why math and logic apply to all ‘stiences’ while as S=T any topologic analysis in SS-space is equivalent to an algebraic analysis in time. Thus we upgrade space mathematics to its fractal scales, S=T duality & 5 dimotions we need to upgrade Aristotelian Logic of 1 time causality to the entangled pentalogic of 5 dimotions that reflect those 5 structural elements of all T.œs: Space=form, Time=motion, ∆=Scales, @-minds & the ¬ entropic limits of all T.œs, mirrored in math with S-points, T-operands, ∆-numbers, @-frames & limits to infinities in time and space. We shall thus translate to the 2 ‘closer mirrors’ of GST, vital topology and existential algebra, all the sub-disciplines, elements, operand, dimensions, axioms, postulates, theorems and laws of mathematics.

As we are made of scalar space and cyclical time, the essential properties of beings derive from the ‘disomorphic’= equal laws of space and time, which makes mathematics concerned in its two main sub-disciplines, spatial geometry and temporal algebra, the most experimental, perfect mirror of the 5D Universe. So we can translate the 5 Dimotions (short ST-view) & its spatial superorganisms to, Geometric space & its Worldcycles (Long ST-View) to Algebraic Time. But to do so, we 1^{st} must depart from S=∆, points that become regular polytopes=numbers, which penetrate social scales and reproduce as odd/even gender symmetries, to reflect in 2D still mind spaces the basic organic forms of reality; and only then once we understand the organic laws of still spatial geometry, and scalar 5D number families we can give them topologic motion, operate them with 5 algebraic dimotions & calculate its ∫∂ changes emerging as simplex particles entangled into atoms, molecules & 3-physiologic planes=networks=supœrganisms when organic laws take over as analytic maths leaves way to synthetic bio-logic laws of larger time scales & pentalogic to dodecalogic better suited to describe the game of exist¡ence; simplified by 5D metric of lesser information in larger scales.

What makes generational space-time different is the fact that we fully change the entire worldview of the Universe, *as *we reject Newtonian Absolute space-time and give back time its cyclical nature, hence its 3 relative past=entropy, present=iterative and future=informative local dimensions. Thus defining topology, the final stage of geometry, as the fundamental science of ‘form’ origin of the vital shapes of all systems.

While algebra will take ‘perpendicularly’ those ‘groups’ of similar points=numbers through the 5 Dimotions of the Universe, as entangled supœrganisms tracing worldcycles. And in that description we will vastly improve the mathematical mirror as a vital reflection in a still synoptic language, apt to code in minimal spacetime the program of existence and its infinite variations, to then see as we keep expanding each stience, hand in hand with the ¡-logic entanglements of duality, gender mirror symmetries, pentalogic ¬∆@st of timespace and dodecalogic, Disomorphisms of trinity worldcycles, how in reality, the vital spatial mathematics and temporal logic laws laid down in our analysis of formal stiences, emerge as timespace beings, from the simplest particles of the o-1 probability time sphere, to its statistical spatial settled populations, once and again rising from time to space, from geometry to scalar numbers to operands of existential algebra, till reaching man; just another entangled supœrganism tracing its worldcycle through 3 deterministic, probabilistic and entropic scales.

**The 5 subdisciplines of mathematics. **

Because Dimotions have Space & time, dimension & motion components the minimal reality is dual, entangled. It follows from a definition of mathematics as an experimental mirror of the 5 entangled elements of all systems and its 5 Dimotions, a classification of the 5 mathematical sub-disciplines according to its specialized study of 2 entangled sub-systems of those 5 Elements with different parameters to measure its 5 Time§pace dimotions:

**S@:Geometry**studies mental spaces. Its ages/fields are: Flat,Euclidean geometry with no motion in a plane**@nalytic geometry,**which represents the different mental points of view, self-centered into a system of coordinates, or ‘worldviews’ of a fractal point, of which naturally emerge 3 ‘different’ perspectives according to the 3 ‘sub-equations’ of the fractal generator: $p: toroid Pov < ST: Cartesian Plane> ðƒ: Polar co-ordinates.**Topology,**geometry with motion &2 Planes.**¬E Geometry**studies fractal points of**simultaneous****space, ∆-1,**& its ∆º networks, within an ∆+1 world domain.**¬∆: Social Number**theory studies scales of equal herds and sequential time flows of information with discontinuous numbers, as opposed to continuous points of geometry, its growth and its entropic limits, both as membranes of polygonal forms and structures of increasing depth peering ∆§cales and detail from Naturals through, Q, R and Complex numbers.**∆T: Analysis**studies ALL forms of time=change, and hence it can be applied to the 5 Dimotions of any space-time being, as long as we study a ‘social structure’ susceptible to be simplified with ‘social numbers’. It is the essential tool to study motions in the 5^{th}dimension from lower parts (∂erivatives) to larger wholes (∫ integrals). We differentiate 5 applications of Analysis according to the Dimotions studied. We also classify them by the number of entangled elements of the dimotional system (partial or multiple derivatives, ODEs, PDEs, lineal, surface or volume integrals); and by the detail of its mirror, from diminutive analysis of single Dimotional Actions (∆-1) to Ages of worldcycles (∆º), which imply a change of state where the derivative breaks (Minimal, maximal, standing points) to larger changes from minimal parts, ‘finitesimals, 1/n’ to larger wholes of entire planes; to the maximal complexity of functionals.**S≤≥T: Algebra**studies through 5 operands that represent the 5 Dimotions of reality its ‘feed back’ S<T>S, stop and step or ‘stœps’ that*move reality through mirror symmetries of space-time, represented by equations connected by operand of increasing complexity*, from single S=T steps of ∆-1 sequential numbers, which gather in ∆º functions, part of ∆+1 functionals. So numerical stœps are first key constants and operands that represent basic dimotions (Sine/cosine: informative perception; π: $t>§ð: cyclical dimotion that transforms energy into information ; e^{-x}=S«T: entropic dimotion, log-x^{n}: social evolution etc.) become connected by ± social and x ÷, reproductive operand complex larger associations of Dimotions called Polynomials where those mirror symmetries S=T gather in more complex ∆+1 structures (Functions).**Analysis**is a sub-discipline of Algebra that studies Dimotions between parts and wholes (∂entropic parts and ∫wholes) in growing complexity from changes in functions (1^{st}derivatives/integrals), to changes of changes of functions (functionals).

Thus algebra mirrors reality with all the elements of mathematics, albeit with a temporal perspective – as topology does with a spatial perspective – because it can express all type of complex entangled ¬∆@st of space-time in its simultaneous analysis of super organisms through the study of its S=T dimotional a(nti)symmetries, classified exhaustively by **group theory**… So Algebra was first the science of operands that translated into mathematical mirrors the 5 dimotions of space-time and then build up from them as the Universe does building up from actions, simultaneous organisms in space and worldcycles in time, in different degrees of complexity, new mirrors for all those events and forms of ¬∆@st.

**¬@ Humind:****Philosophy of mathematics**studies the bias and limits of huminds studying mathematics as they project its ænthropic simple view of the world due to:

1) the ego paradox (all systems measure from its distorted self-centered ego).

2) dominance of the ‘western military lineal male, entropic destructive culture.

3) inflationary limits of smallish languages that multiply its kaleidoscopic mirror images of larger single wholes as they are not bond by the restrictions that ‘lineal motion-entropy’ impose to form, whenever we try to build ‘reality’: so money is inflationary over the physical economy it describes, we talk more than we act, epigenetics multiplies waste code; so do digital programmers. So we ‘know when mathematics is truth but NOT when is real’ (Einstein, Gödel’s incompleteness, false ∞ as all planes have entropic discontinuous limits of solubility of functions, Cantor Px, idealist, German, physics: Copenhagen interpretation, creationism, false ∞=singularities. Since fictions exist in all languages. So to crop the fictional part of mathematics we must use an experimental method, selecting from maths those parts that better mirror the Universe. As languages mirror the Universe in a mind, which is NOT reality itself. But human egos, confuse both; as each language is also a species of stience of minimal volume, a fractal world that imitates the whole; so it has the same syntax laws than reality, which in mathematics became the Euclidean axiomatic method we must however improve by looking also to the ¬∆@st universe – the ‘object’ the language mirrors.

On the positive side idealism *gave creative capacity to huminds as they invented to represent reality*, phase spaces, Hilbert spaces and variations came to reflect that growing awareness of the complexity of reality. So the axiomatic method of proof still needs the experimental mirror of ¬∆@st laws. We must compare any mirror language with the ∆ST reality it describes as languages are smaller, hence more informative in terms of 5D metric, Se (size in space) x Ti (Time information) =C. As humans didn’t develop maths as an experimental stience they also ignore math as a biologic language that selects species that talk it better in the eco(nomic)system. So its idealist view ignores the dangers of evolving digital chips that talk better maths and displace us from labor and war fields.

**Vital mathematics**expands its foundations to 5 ¬E Postulates beyond Aristotelian logic (A->B single causality) into the**¡logic of 5 Dimotions**, as an experimental mirror of the fractal, organic universe and its bio-topo-logic properties.

**Expansion of mathematics to 5d: correspondence principles.**

Mathematics & logic are languages, mirrors of an a priori ∆ST reality that comes before languages that describe it.

Mathematics is derived of geometry, the science of space and logic is the science of causality in time. So space and time must be the first substances of which all is made, a model of reality that has a deep tradition in the east (philosophies of a Universe made of two poles, space=dimensional form, or ‘yin’ and time=motion or ‘yang’)

Thus we need to introduce the correspondence principle also in mathematics, according to which present mathematics is a simplification and biased view of the true discipline, as all sciences reduce to a single plane of existence, using therefore lineal concepts of a single time motion the fractal Universe. As time and space are NOT absolute Newtonian backgrounds but are in a Leibnizian relational space-time background independent theory, the ‘generational substances’ of all what exist, composed of organic fractal vital spaces that last a finite duration in time, space and time become the common principles, whose disomorphic properties originate all other laws of ‘stiences’ each one studying different ¡-scales of spacetime beings that must be first put in relationship to those properties and then improved with the comparison of human scientific laws, specific species of each science and the universal laws of ¬∆@st.

Hence we upgrade all its concepts to ‘**cyclical time’** that stores **information**, in the frequency of its cyclical membranes, limiting and breaking **space into ‘fractal’ **topological parts, and** ‘scales’**, according to **‘relativity symmetries’ **between formal linguistic mind-spaces vs. time motions,** S=T,** which mix together forming the **5 Dimotions of reality, **all of which follow its** 5D metric**. Those elements entropic limits, fractal space, SxT=K metric scales, cyclic time of information, and S=T symmetries, *are *thus the barebones fundamental elements required to upgrade each science, and its simplest mathematical equation is its S x T (s ó t)= Constant, ‘Fractal Generator metric’.

In mathematics the same upgrading is needed, as it is also an experimental mirror-image of those laws. Something which subconsciously happened as mathematics evolved, *from pure mental space (Bidimensional still geometry) to time perception (sequential numbers) merging both in modern analytic geometry. And finally peering first with temporal algebraic numbers in the 4 ^{th} and 5^{th} dimension (calculus), which topology mirrored in space *with its 3 varieties of bidimensional forms with motion made of networks of points; to complete with fractals the mirror structure of the Universe – adding on the path all kind of new ‘mind spaces’(Phase spaces), with ST combinations (vector spaces) and scalar levels of growing complexity (Hilbert spaces, functionals) – whereas other highly valued branches, set theory that constructs maths NOT from its initial s-points and t-numbers but from the top head of a Mr. Cantor, the set and the axiomatic method of Mr. Hilbert born also of his head – ‘I imagine points, lines etc.’. So those will be considered largely inflationary forms of the language within itself NOT observed in the real ∆st world, not worth to mention. While chip maths (Boolean Algebra) belongs to a new species the digital machine, which as per our papers on the superorganism of Mankind in time, history, is bond to displace us with its higher dexterity in the most efficient language of the Universe, if we keep evolving them. So we feel ethically inclined NOT to upgrade it in this paper.

**S=T symmetry.** Systems are made of spatial form and time motions, from a static mind-perspective of still geometry mathematics also evolved to acquire algebraic motion* the essential Duality of reality also evident in any scale of complexity of mathematics, which has always 2 solutions from either an S≈T perspective, Spatial forms (points) = Temporal sequential numbers*. Topological methods=Algebra methods, merged in analytic geometry.

So **geometry** of points studies dimensions in space, **Number theory, its sequence in time** and its **analysis** its scalar motions=changes whereas Algebra puts them all together considering its S=T Dimotions and symmetries, making mathematics the best humind’s known language mirror of the…**5d+5m = 5 Ðimotions** of reality, which each mathematical S=T sub-discipline expresses in different terms – Geometry as Dimensions, Analysis as motions and Algebra as Dimotions using operands to that aim, and finally with ∫∂ calculus peering the 4^{th} and 5^{th} Dimotion and its travels through those upper and lower scales, as analysis introduced in algebra the study of integral wholes and derivative parts. Scales of 5D parts and wholes soon gave further boost to algebra, as functions became part of functionals, and all variations of a spacetime structure were tabulated with group theory. So the 5 ‘Dimotions’ of any system can be mirrored ¡logically with *multiple kaleidoscopic perspectives and languages. So as systems have always 5 Dimotions its pentalogic study give us 5 varieties in all its mathematical elements; 5 operand, 5 ¬∆@st elements, 5 dimensions, 5 motions etc.*

**In** geometry the same evolution from static space on the 3 ‘dimensions of a single plane of existence’ to the analysis of the upper and lower scales to finally give motion to them all took place, from Greek bidimensional geometry to solution of 3 dimensions, height-information, length-motion and width-reproduction. Then topology introduced the concept of motion, made lineal dimension, bidimensional as all is an ST composed form and then those bidimensional topologic forms with motion, peered the fourth and fifth dimension, as made of points=parts that form a whole; soon points themselves acquired volume as multiple parallels crossed them in non-Euclidean geometry.

The highest homology with reality: Semantics, Syntax and Growing Sentences of mathematical operands.

Mathematics is a language and as such it has the some classic, properties, elements and symmetries of them all:

It reduces reality to fit the brain, eliminating motion and simplifying layered scales and limiting perception to relevant cycles within the territory of perception of the p.o.v. So the more complex reality of ∞ space-time cycles with motion becomes a language.

It does so, increasing generality through the Syntax of its ‘sentences’; while keeping its detail through the semantics of its forms.

In ¬mæth the semantics are the specific fractal points-numbers, that the syntax of operands connect into sentences, which are ternary planes in geometry: SóT equations, in algebra.

So further growth in complexity can be achieved by adjacency of topologic varieties of planes and ‘chains’ of equations through ever more complex, integrative, ‘operands’ (sum/rest->multiplication/division->potency /logarithm/integration/differentiation) to ‘form descriptive paragraphs’ which will finally reach the full ‘superorganism in space’ or worldcycle in time, concluding the ‘story’ of the event or being analyzed by the language.

Those structures do happen in any language, among the those studied in our papers – music, art, literature, logic, math, palingenetics and topological evolution. And so all can refer to Gst laws, the language of all languages.

That is all. Why? Because as we stress once and again, the Universe is infinite in its repetitions but its final elements are few, *reason why a language can ‘reduce’ reality to an encryption and final synopsis by eliminating repetitions, which ultimately all languages do to create palingenesis from smaller seeds.*

So the perception of a MIND is shrunk – it is a ‘finite game of finitesimal mirrors’ with minimal redundance and elimination of dark spaces which are discontinuities and redudndatn information, as opposed to the infinity of the whole. Further on as we perceive not reality but the interposed language, reality also shrinks and information is lost beyond the ternary limits of scales, the ternary adjacent parts of the being able to act on its territory, and the ternary ages of life at least in human minds (which can be stretched into 3×3±¡ 9 -11 scales). Pentalogic thus is the description of the ternary ±¡ elements that languages use to describe a 3±¡ connected reality.

So a language ‘seeing’ a supœrganism in time and space reflects ‘ternary games, scales & elements’ wrapped up by a ‘whole’ – the outer membrain (S-view) or temporal cycle of maximal motion, which becomes a finitesimal of a larger, new ∆+1 finite plane (hidden its inner parts within the finitesimal point). And so we do START afresh a the game of that new plane of existence.

**ITS 3 AGES**

Mathematics as all organic systems lived 3±¡ ages in the Humind (ab. Human mind) proper of any worldcycle:

**1 ^{st} age: Arithmetic and plane geometry.** As mirror language that studies ¬∆@ST humans understood its simpler units, points of space, social numbers and entropic limits, drawing figures of flat geometry to ‘encircle’ territorial properties in our flat world.

**Trigonometry**appeared then as the 1

^{st}realization of a ‘@-mind frame of reference’ to measure the 3

^{rd}dimension of depth, which is often parallel to scale (astronomical measure). It was a lineal youth, which slowly understood curves and the |xO=Ø generation of all forms with ‘conics’. As all organisms & worldcycles can be subdivided in 5 fractal subparts and 3±¡ ages in its 3

^{rd}age Greek geometry became old, warped inwards-looking detached from experience with Euclid’s axiomatic method, the 1

^{st}mind-ego trip of creationism (man & god’s language).

**2 ^{nd} Classic age.** The S=T symmetry realized with analytic geometry, marrying numbers and points, while calculus brought ∆-scales, with finitesimal derivatives, 1/n, units integrated in wholes (Leibniz). The 3

^{rd}symmetry of pentalogic ∆=S=T, we haven’t mentioned implied that derivatives could be interpreted as ‘stœps’ of motion and ‘minimal straight intervals of a curve’. So they could also study curvature (differential geometry) and locomotion.

**¬entropic limits**were needed to find solutions (definite integrals). New

**@-frames of reference**expanded mental geometries to represent all forms of ‘selected information’, which mathematical physics used extensively to describe the physical world. Thus the classic age had all the mirror tools needed to interpret the Fractal Universe and its 5 entangled elements, ¬∆@st. But the axiomatic ego-trip stretched maths beyond ¬limits when Newton imposed its thesis over Leibniz’s finitesimals and fractal points with infinities, lineal absolute space-time and the false hypothesis of the continuum, leading to its…

**3 ^{rd} age** that abandoned its realist foundations with creationism -Hilbert that imagined points, sharing the only language ‘God’ & Cantor sets instead of space points, scale numbers and time operands as its generators, leading to an excess of old age information & fictions spreading to mathematical physics, as now Maths creates the Universe, not the inverse.

**+¡: **Thus we need a return to its empirical foundations formulated in terms of the 5 Dimotions that create reality mimicked by the 5 mathematical subdisciplines (larger view), Operands (shorter dimotions) & equations (worldcycles).

-¡: Yet that might not happen as instead mathematicians are evolving the digital ‘mind’ of machines, the Chip Homoctonos, which speaks better digital numbers and so the eco(nomic)system of company-mothers of machines & weapons is selecting computers that are fast substituting obsolete huminds in labor and war fields, atrophying them back to a ‘audiovisual’ violent non-rational neo-Paleolithic, while Boolean Algebra, past the earlier age of simple, fixed Algorithms of Information (the true meaning of AI) enters its classic age of freedom & consciousness that might end the dominance of huminds on Earth; introducing ethic elements on the praxis of mathematics, as it should in all ‘stiences’.

**The different determinism of the 3 ages: Axiomatic, lineal age vs. Kaleidoscopic uncertain futures. **

Because the Universe is pentalogic, made of ‘space’, ‘time’, ‘scalar planes’, ‘languages-minds’ and entropic limits, when *@ mind’s language appears it studies exactly those 4 elements, space, time, scales and entropic limits, with its mirror systems. And indeed, ‘analysis’ was born of the need to understand those 4 elements in problems of Nature, NO LONGER in lineal terms, as the ‘first age of any system’, but in ‘curved’ terms.*

*So what the Greeks have resolved for the $t age of mathematics (lineal age), c Analysis will solve for the second age of curved geometries through the use of analysis.*

This is a process proper of the 3 ages of any Space-time system. The first age is lineal, with absolute simple truths that the mind as a dictator ‘child’ considers dogma. So Euclid did his axiomatic method on simple lines

__Monologic in Mathematics. The first age of lineal, deterministic Greek still geometry and axiomatic proofs.__

Once we understand the general fact that all languages have a first lineal age, deterministic, as a line cannot change direction or else will stop being a line, while a curve can easily change curvature, even change direction in sinusoidal waves and still be a curve; so lines are deterministic one-single future to them, while curves are able at any point to choose 3 paths of less, more or equal curvature; we can understand some facts of Greek Geometry:

– It is simple, lineal, deterministic and hence it can be approached with a purely axiomatic method, as there is no ambivalence on results, constructing a self-contained method of proof departing truly from a simple set of axioms – a point has no breath, etc.

But the axiomatic method of proof IS NO LONGER VALID when we consider systems that do have also a certain ‘time curvature’, and even more so, when we approach operands and mathematical systems that probe the planes of the fifth dimension (calculus, limits). Then the future has different solutions, and some are paradoxes, and so we cannot PROVE with the simple A->B lineal causality and deterministic of lineal Greek Geometry, everything that has to do with cyclical, curved geometries, calculus of finitesimals (limits), and because humind’s reject the concept that absolute truths only exist in absolutely simple lineal systems, as Mathematics evolved into complex curved geometries and scales, its proofs were more and more imprecise, or blatantly false (0 does not exist, as all limits to 0 or infinity have an entropic limit in a quanta or the dissolution of information; the real line is in a different plane of space-time than the Natural numbers; which only ‘become continuous’ if we were to access an even larger scale; etc. etc.)

It is for that reason we shall not use further the axiomatic method but compare any complex level of mathematics to the experimental laws of Space-time from where they depart.

In terms of ‘scales’ all this means that in small, ‘fast’, predictable A->B steps reality is lineal but when we gather multiple steps, all lines become curves. In small intervals motion might be continuous but as soon as we go beyond a simple step, there is a step and stop , length and high motion.

And so we can also reduce curves inversely to steps and stops of length and height, or lineal stairs (which would be the method of Calculus, to ‘calculate’ the tangent of the curve. Does then the curve exist? Or only the steps and stops of lineal and height motion and information? It is relative to our perception. In the large scale the zig-zag of Brownian movement or electrons become a continuous curve. In the smaller scale the steps might be highly lineal and deterministic but in the large scale they become curved and probabilistic.

In that regard, the use of Gst laws to reference the laws of mathematics, beyond the pretension of absolute truth of the axiomatic method, is completely necessary, without using a complex ‘pentalogic’ point of view, and accepting the paradoxical limits of reality and its scales as we shall constantly do here.

**RECAP.**

In an entangled Universe made of 5D¡ ¬∆@st – space-time dust systems, knowledge requires a pentalogic analysis of any system or mirror-language, including mathematics, to extract all its information about its 5 scalar entangled superorganism in space as an and its 3±¡ Dimotions & ages in time as it traces its worldcycles. As only entangled systems made of those 5 elements, performing dimotions=actions of survival exist. So linguistic mirrors also reflect those elements & dimotions. Thus mathematics in time has 3±¡ ages in its huminds’ evolution:

– A lineal young age of simple parts ‘flat’ geometry, arithmetic of unconnected numbers, into a…

– Mature reproductive, combinatory age as a realist mirror of the scalar Universe ( analytic geometry that merges space-points time operand and scalar numbers; calculus that gives points curved motions and scalar depth studying its finitesimal 5D parts and integral 4D wholes) into a

– 3^{rd} age of maximal complexity (Non-E Geometry, Functionals, groups, sets, phase spaces) but also inflationary information and mathematical fictions unconnected with reality as its foundations (set theory & categories that substitute real space points, scale numbers and time operands) & the axiomatic method that despite Gödel’s incompleteness theorem substitutes experimental proofs asked by Lobachevski and Einstein that we regain in 5D as ‘space and time’ become the real substances of all systems, which mathematics studies directly.

-¡: Finally as human maths reach its ceiling, the discipline starts again in a different mind species – chips now evolving fast from its earlier age of simple Boolean algebra, modulo-2; into a realist age able to model any form of the Universe, but also in mathematical physics, in a 3^{rd} fictional age of computer models that validate any physical theory with its nice ‘digital pictures’ regardless of experimental truth (ad hoc big-bang models, evaporation of black holes, dark matter new particles, bizarre multiverses, etc.)

** **

**¬E GEOMETRY: ITS 5 POSTULATES: FRACTAL POINTS AS MINDS.**

‘‘The smallest point is a world in in itself’ *Leibniz, on the fundamental particle of Reality: The fractal point=world of space-time,* *unit of Non-Euclidean, Non-Aristotelian, ¡logic topology*

*Geometry* was born with the definition of a point with no breath; a line with no breath and a plane with no depth. Those postulates turn out to be simplifications of reality as the 5^{th} non-e postulate proved that infinite parallels might pass through a point, which means the point needs ‘breath’ to fit them all. This was never understood not even after the 5^{th} postulate was rejected since what would be the rule to fit those parallels was to curve them, and yet curves are NOT parallels which Euclid defined as straight lines, and still only one curve can fit in a point with no breath, or else the point will have breath.

*So the interpretation of the 5 ^{th} postulate was wrong. The point was a fractal point of a single plane of the scalar 5^{th} Dimension, which grew in size as we enlarged it, till it hold a world in itself. Only Leibniz, the genius and clearest forebear of 5D captured this difference with his concept of monads.*

Thus the first mathematical consequence of the fractal structure of space-time is a change in the axioms and postulates of Euclidean geometry taking the r=evolution of Geometry performed by Lobachevski and Riemann in the XIX c. which gave birth to Relativity, to its ultimate consequences, changing *also the axioms of Euclid that defined points and lines as having no breath, since in the fractal Universe all forms do have a volume when we enlarge our view of them, peering into its inner parts and fractal dimensions. And in this manner we shall harmonize and return to its logic meaning the concepts of parallels and Euclidean points able to fit multiple lines=waves of energy and information, converting those fractal points of ‘cyclical timespace’ into the fundamental particle-units of the Universe mirrored by mathematics. *

*Thus *the mathematical unit of a 5^{th} dimensional Universe is a fractal point, whereas a Non-Euclidean point is its limit in a single spacetime continuum; whereas the inner parts of the point, which co-exist in other scales are not perceived. So we shall start with the classic non-E point and show how by adding fractal scales become a more complex reality.

Fractal points unlike Euclidean ones are points with parts: as we come into its scale they grow in size and display the 3 minimal parts of all of them, its area, frequency of angular momentum, and central Active Magnitude, the true meaning of a ‘singularity’ – the focus of charge, mass, forces and its informative minds. So let us introduce the minimal POINT with parts of the Universe, the time space cycle.

*Einstein’s view of a fractal point of the gravitational scale: from our smaller electromagnetic world, which shrinks its inner volume, bending its parallels, it seems a curved geometry. But that view breaks the definition of parallels as straight lines and it is absurd, as the point remains ‘Euclidean, with no breath’, hence it only fits one line with no breath. Thus, particle-points must be defined as ‘FRACTAL points’, like those we see through telescopes or microscopes, which grow we approach our distance both in scale and space becoming enlarged worlds with a complex internal structure. *

Einstein found that gravitational Space-Time did not follow the 5th Euclidean definition, which says:

*Through a point external to a line there is only 1 parallel.*

*E*uclid affirmed that through a point external to a parallel only another parallel line could be traced, since the point didn’t have a volume that could be crossed by more lines./ Instead Einstein found that the space-time of the Universe followed a Non-Euclidean 5th Postulate: *A point external to a line is crossed by *∞ *parallel forces. Abstract, continuous, one-dimensional point: Real, discontinuous, ∆-dimensional points: *

**. ____________*** ***===========o**

This means that a real point has an inner space-time volume through which many parallels cross. Since reality follows that Non-Euclidean 5th postulate, all points have a volume when we enlarge them, as cells grow when we look at them with a microscope. Then it is easy to fit many parallels in any of those points. Such organic points are like the stars in the sky. If you look at them with the naked eye they are points without breadth, but when you come closer to them, they grow. Then as they grow, they can have infinite parallels within them. Since they become spheres, which are points with breadth – with space-time parts. *So space-time is not a ‘curved continuum’ as Einstein interpreted it, but a fractal discontinuous.*

Leibniz’s isolated monad is the simplest o-1 fractal point-mind possible – a still mirror of reality, ‘a world in itself.’

A modern scientist understands fractal points in terms of its 3 necessary parts, its focus-singularity of the parallels that cross, its membrane or angular momentum (S=T duality) that breaks it into an inner and outer part (first knot theorem), and its vital enclosed territory, which correspond to the 3 physical quantities conserved in Nature, as a fractal point is also the unit of logic as a cycle of time that divides reality in inner and outer regions and the unit of physics, as the minimal form of ‘Planckton’ (h-Planck constant) which has the 3 parts of reality conserved in each plane of space between its ∆-1»∆º palingenetic emergence and ∆º«∆-1 entropic death.

So the 2 ‘emergent formal sciences of space and time’ and its units, the fractal point and the time cycle become also the minimal organic species of physics; and we shall see this ternary structure emerging in all scales; so we can model cells with 3 organic parts; animal territories; nations; planets, stars and galaxies, always showing a ‘membrane/angular momentum’, a focus/singularity/informative center and a vital energy-space between both.

**RECAP.** *The fundamental particle of mathematics IS the fractal point.*

**A visual synopsis of its 5 postulates**

We thus recast the axioms and postulates of Euclid into five new postulates to define fractal points, Non-Æ lines as wave of fractal points, Non-Æ planes as ternary networks of Non-Æ lines, which become supœrganisms, whose relative ‘congruence’ in its 3 ‘elements’ (singularity point, membrane and vital space) defines the type of ‘perpendicular or parallel’ relationship between them:

1st Postulate: ‘¬Æ point are discontinuous time cycles with an inner content of vital space-time’.

2nd Postulate: ‘¬Æ lines are waves of fractal points’

3rd Postulate: ‘¬Æ planes join 3 ¬Æ lines into a supœrganism’.

4th Postulate: ‘2 ¬Æ points are congruent when both its inner parts and outer perimeter are equal’

5th Postulate: ‘¬Æ World points focus multiple ¬Æ waves of energy into a still linguistic mapping of the world.

Let us explore those postulates, constraining our examples to the simplest forms of physical and biologic spaces.

**1 ^{ST} POSTULATE. THE 3 MATHEMATICAL PARTS OF A NON-EUCLIDEAN FRACTAL POINT.**

* 1 ^{st} Postulate:* A fractal point has parts; that is an enclosed region of vital inner energy surrounded either by a spatial still membrane or a Temporal motion of angular momentum (S=T symmetry) self-centered in a singularity-mind that gauges its information. In the graph, we can see how different vital fractal points of ST¡entific scales follow this ternary structure. The perception of the point depends on the scale and distance from where we observe it:

From the perspective of the upper st+1 Plane they might be in the limit of invisibility (what quantum scientists call a point-particle) *but they still have a time motion performing a ‘function’ in that upper ecosystem, ∆+1 in which it exists.*

*Internally from its own ∆º perspective the point will have *3 dimensions/networks. This is the case even in the smallest planes of theoretical strings, made of points with parts, with volume – since we require 3×3∆º+1∆+1 inner dimensions to describe strings – a paradox that can only be resolved if we consider ‘strings’ to be fractal points with inner dimensions.

Fractal points explain without contradictions Non-Euclidean points, which are not logic in a single scale, as they ‘curve’ parallels which are ‘straight lines’ and fit them in a ‘point with no breath’ that holds only 1 line. Fractal points however enlarge fitting multiple ‘straight lines’. Yet when seen from above, human perception of them, becomes ‘deformed’ shrinking and curving its from – a theme, the distortion of human measures of time, space and scale, which will be instrumental to explain rationally the ‘spookiness’ of quantum physics and relativity and its time and space transformations.

So fractal points harmonize the 1^{st} axiom=postulate of Euclid with the 5^{th} postulate of non-Euclidean parallels, as a fractal point enlarges into a cell, atom or particle which even in a smaller scale of the fifth dimension can host multiple parallel flows of energy and information, crossing it.

When we see fractal points far away we describe them as points with breath, with the tools of Euclidean geometry since the ‘inner space’ shrinks to a point and so the ‘bulk’ or curvature of space-time shrinks to a plane. Yet, when we come closer to them, they grow into points with volume. The volume of those Fractal, Non-Euclidean points can thereafter be studied with the 3 types of canonical, Non-Euclidean geometries or topologies of a 4-Dimensional Universe – the Universe we live in. Those 3 topologies make up the 3 regions of the point, which correspond each one to the 3 essential arrows/functions of any species: the external, energetic membrane; the central, informative brain and its reproductive combination, SxT(s=t):

A*ny fractal point is made of 3 regions whose geometry responds to the 3 topological forms of a 4-Dimensional Universe, the convex plane, the torus and the sphere. *The inner parts of fractal points are thus able to perform energetic, informative and reproductive functions, which makes them the fundamental particle of any plane of st¡ence. Thus all entities can be described as wholes made of 3 internal parts whose geometrical properties maximize their energetic, informative and reproductive functions:

– Max S: an inner, dual center, corresponding to convex topologies (left), made with 2 cyclical forms. It is the dominant informative topology of any fractal organism, described by Belgrami in the XIX c. as a conical form with ‘height’, with negative curvature.

– S<=>T: A middle, reproductive zone, described by Klein as a disk of quanta in cyclical motion that communicate energy and information between the inner and outer zones.

– Max. T: An outer membrane of max. |-motion-distance crossed by ¬E information & energy parallels acts as an entropic limit to ¡ts inner parts, described by Riemann’s spherical geometry. It seems continuous, still but on close view, as most external membranes stores and/or absorb information through openings of its broken geometry, outlets of its senses.

The complex analysis of those fractal points that move and have inner fractal parts, made of cycles, started in the XIX century. First Lobachevski, a Russian geometrician, defined Non-Euclidean points as curved forms, crossed by multiple lines, which give them spatial volume. Then Klein studied its cyclical movement and introduced the variable of time in their description. Finally Riemann generalized its nature, considering that all space-times were Non-Euclidean space-times with movement. For readers versed in mathematics, we shall reconsider the common properties of those 3 zones of any fractal point, according to its discoverers, which develop in abstract terms the organic properties we just described:

–* According to Lobachevski and Belgrami,* space is curved since information curves the energy of any real space-time. So points move in curved, cyclical paths gathering energy and information for their inner ‘dimensional networks’.

*– According to Klein* Non-Euclidean space-times have motion. So their speeds measure distances; as physicists do in Cosmology with the distances of galaxies, which are proportional by a ‘Hubble constant’ to their speeds; or as people do in real life when we say that Brooklyn is at 5 minutes by train from Manhattan not at 2 miles.

*– Riemann* summoned up those findings and generalized them to all possible space-times. His work should be the guide to understand them philosophically. He also defined planes as networks of similar points and treated dimensions, as we do in this work, no longer as mere abstract definitions of extensions but as ‘properties of those points’. So points can have beyond its discontinuous borders an inner space-time with several networks/dimensions, one for each of its ‘energetic or informative properties’, as it happens with the points of physical reality. Yet a network of points that form a space with ‘common properties’ defines the dimensions of those points as ‘fractal dimensions’, limited by the extension of the energy or informative network (static point of view), which ‘puts together’ a complementary dual, organic being.

Those pioneers defined the 3 topologies of information, energy and reproduction of all st-points:

*– Max. Space=Information: The informative, fractal center, particle or brain of the point is the so-called Belgrami hemisphere,* a space-time with a dimension of height that transforms energy into information, absorbed or emitted by the central singularity. It is a fractal, informative region similar to a black hole structure. Since it follows the ‘black hole paradox’ of all informative centers, displaying max. form in min. space. So according to the inverse properties of space and time, the center has max. Informative Time and minimal Energetic Space. Moreover any point which comes closer to it, suffers a mutation of its spatial coordinates into informative, height dimensions. This is the case of any particle coming to a black hole, whose space-dimensions become temporal/informative dimensions as it rises in height.

The center has more information because its geometry has at least 2 fractal disks, which channel and transform the energy absorbed through the surface into complex information. Regardless of the complexity of the entity, the structural function of the hyperbolic center as a system that process the information of the network remains. For example, in living systems, those disks might evolve its topology till becoming the relative energy center or ’heart’ of the blood network with 4 divisions; or evolve further its hyperbolic geometry till becoming the informative center or ‘brain’ of the system, attached to the informative network.

*– Max. Time=Lineal motion & Min. Curvature: An external, continuous membrane or Riemann’s sphere of maximal energy* that acts as a relative infinite, unreachable distance. The membrane isolates the point as an island Universe, creating the discontinuity between the inner parts of the point and the outer universe. Since the internal cellular points are either jailed by the membrane’s structural density or destroyed by its energy when touching it. The membrane is the opposite form to the central, informative singularity, with max. spatial extension and continuity, hence with a minimal number of fractal, discreet elements: *Max.T=Min.S*

Thus *all Fractal points have inner worlds whose membrane creates a discontinuity that defines an External Universe or* outer world from where the point obtains its energy and information. The membrane is also the zone through which the point reproduces & emits its micro-forms of information. So it displays ‘sensorial holes’ to relate the point to the external Universe. And those points, despite being discontinuous, will have in their external membrane several generic openings or ‘senses’ joined to the informative networks or ‘brains’ and energetic, ‘digestive networks’ of the organic system:

* – Max. +ΣT: A* ‘*mouth*’ or opening that absorbs energy.

* -Max. –ΣT: *‘*Cloacae*’, through which the cyclical body expels its temporal energy.

–*Max.+Si: An* ‘*eye*’ through which the informative center receives external information.

–*Max.–Ti: An* ‘*antenna*’ to emit information.

Those apertures vary in their number, location and size, depending on the form of the point. In the simplest spherical ‘seeds’ of most species, they are mostly situated in 3 regions:

* – Max. ΣT*: The Equator of the system, through which the membrane absorbs energy.

* – ΣT=Si*: The Tropics where often the same opening emits and absorbs temporal energy.

*– Max.Si*: The Poles or points of confluence between the membrane and its central informative region of height, which hits perpendicularly the membrane on those poles. North and South Poles orientate Anti-symmetrically, acting as 2 relative, negative and positive apertures, communicated by the height dimension of the singularity or Belgrami hemisphere. Thus the Positive Pole absorbs temporal energy that crosses through the central singularity where it is absorbed and ejected to the intermediate region where it is re-elaborated before its emission through the negative Pole.

*– ΣTe*<=>*Si: The reproductive, central region, which combines Energy and Information: *

In all fractal points there is an inner middle volume or intermediate territory, which combines the energy coming out of the external, spherical, topological membrane and the information provided by the convex, complex formal center.

According to Non-Euclidean mathematics this region is made of self-similar points that form groups, fractal herds of ‘points with parts’ in perpetual movement, that draw cycles of parallel lines, between the other 2 regions, as they gather the energy and information they need to survive. And they create space by cycling within the other 2 regions.

In many fractal points the informative and energetic centers establish 2 opposite flows of energy and information that become the negative/ positive poles. So often, the particles of the intermediate region cycle around the inner region tracing elliptical trajectories, focused by those 2 informative points. It is the case of any bipolar system, from binary stars, one dominant in energy and the other an informative neutron star or black hole; to bimolecular systems or n-p pairs in the nuclei of atoms. The same duality of 2 specialized centers controlling a common territory, or vital space happens in biology where most species have male-energetic and female-informative genders, ruling a common territory.

Such abstract conceptual space describes the behavior and form of many real, spatial herds. For example, an animal herd in an ecosystem will move between their hunting and water fields (where they gather energy) and their breeding, inner region where they reproduce information, making cyclical trajectories between both regions. In this manner, they occupy a vital space, called a ‘territory’, which shows the properties of a Non-Euclidean Klein space. A fundamental property of the intermediate space is the fact that it is confined between the other 2 regions, which are never reached in the cyclical trajectories of the inner cells of the space. For example, in a cell, the molecules of the organism will not touch the protein membrane or the central DNA nuclei. Thus, the inner quanta are confined within the Klein’s disk by the 2 other regions, which have more energy and information and might destroy them and/or absorb their energy and information at will.

In abstract terms*, *mathematicians introduced in the XIX c. the concept of an *infinite, relative distance *measured no longer in terms of static space but in terms of time and movement, as *the distance between the point and a region that cannot be reached*. Thus Klein defines a relative infinity, as the region beyond the discontinuous membrane whose insurmountable borders the inner time-space quanta can’t cross, as a cell cannot go out of a body, an atom beyond C speed or 0 K temperature and a man beyond the Earth’s atmosphere. Thus, the informative center and external membrane become the 2 relative infinities or limits that the movements of the intermediate point cannot breach.

As in the myth of Achilles and the turtle, Achilles never arrives because every time he moves he crosses a smaller spatial distance. The same happens in a fractal space-time, when a point moves *temporally* towards its inner or outer space-time limit and finds an increasing resistance to its movement, till finally it is deviated into a cyclical trajectory around the outer, energetic membrane or the height dimension of the inner informative singularity or is destroyed. So the intermediate, fractal cells of the point circulate in parallel cycles always inside the interior of the sphere with contact zones of the type A *(central, 2 ^{nd} row of figures in the previous graph). *

*In a human organism, the blood system might seem infinite for the red cells that transport energy since they never reach the outer Universe. *For that reason in the drawing, Klein interprets the intermediate region of the Non-Euclidean point as an infinite *circle with an invisible, unreachable membrane,* whose motion-distance is unreachable, hence infinite, equaling the ‘space-time distance’ between the intervals B1-B2 (long) and B2-B3 (short but difficult to cross), despite being B2-B3 increasingly shorter in space. Since the quanta take longer in each step and don’t reach the membrane. This is often due to an increase in the ‘density’ of the space, which despite having less distance has more ‘points’ in its network, such as the case of black holes or jails. When those inner points reach the membrane at point C they become destroyed or deviated.

Thus, the entropic membrane and informative center are discontinuities that isolate the intermediate cellular quanta, creating a territorial ‘World’ within the point. Those discontinuities are called in Geometry a relative infinite, in Biology a membrane, in Sociology or Topology a national border, in fractal theory a co-dimension of a point. *A key advance of ¬E is – given the fact that all points have dimension and volume, to define 0 as a finitesimal 0’. Indeed, absolute zero does not exist, it always leaves a finitesimal 0’ motion. Emptiness is undefined. What was there leaves a memory of it – a corpse removed leaves a DNA trace. Ideal mathematics tries to be a perfect mirror of an imperfect Universe. Yet those imperfections properly explained are absolutely essential to the fabric of reality as it is, helping enormously the ‘real modeling’ of mathematical structures.*

0’ and – motion/mass will help then to understand Lorenz Transformations, the c-the limit of energetic speed and 0’ k limit of temporal, formal stillness, as relative ‘scalar’ limits of the Universe – the limits of the fractal space-time membrane of light and its evolved electroweak T.œs. Since the Universe has at least another bigger gravitational membrane, in which smaller >c particles cooler than 0’ K (tachyon neutrinos as gravitons?) -exists; in a Cosmos of ∞ scales, which extend beyond human limits of perception.

*Recap: *Topologies of Fractal points are organic, maximizing ¡ts energetic, reproductive and informative dimotions.

**2 ^{ND} POSTULATE. COMMUNICATIVE WAVES OF ENERGY AND INFORMATION. STŒPS.**

The 2^{nd} postulate defines lines as waves of points with volume (which explain complementarity wave particle), no longer as an abstract form like Euclidean geometry does, but as a physical wave of self-similar, fractal micro-points that carry energy and information, as they move between 2 macroscopic points, with 2 possible functions, to communicate energetic forces or linguistic information.

*2 ^{nd} Postulate: A cycle of fractal space-time:*

*‘A wave of communication is a group of self-similar micro-points that move in parallel lines between 2 macro-points, transferring energy and information between them’.*

In Non-E geometry a line with parts is not defined by a sequence of numeric intervals within a straight line, but by the communication of 2 poles of energy and information that establish a flow of particles in 2 opposite directions, creating a simultaneous, paradoxical wave. Such waves again can have different purposes. A wave dominant in information communicates symbiotic particles, creating an informative bondage/network; a wave dominant in energy might be an aggressive action between different species that fight for each other’s vital energy or territorial space; and a wave that balances the energy and information of both points meets in the center, creating a new self-similar, seminal particle, as when 2 electrons emit waves of densely packed photons, which merge in the middle and give birth to another wave.

When we observe a one-dimensional line as a form with inner parts it becomes then a 4-dimensional wave made of cyclical points with motion. Hence in quantum theory we say that any particle in motion has associated a wave. *Thus the 2 ^{nd} postulate resolves the wave/particle duality,* as all lines are now waves traced by a point with inner volume. Further on, since all lines have volume, they carry information and so all forces can in fact act both as a source of energy and as a language of information – as physical experiments prove. A ray of light in detail it becomes a 4-dimensional wave with electric height and magnetic width, often exchanging flows of energy and information in action-reaction processes of communication between bigger points.

When we generalize those concepts to n-points we can define a space as a network of Non-Euclidean points. Indeed, Riemann affirmed that a space is a network made of herds of points with similar ‘properties’. Planes of space are therefore networks of points. The self-similarity of their properties defines its *density *determined by the number of points and its proximity that grows with self-similarity. So similar points come together into a tighter, more continuous space; whereas the density of the space is proportional to the similarity of its points, till reaching ‘boson state’ of maximal density when points are equal. And when a volume of spatial energy is very dense, it is very difficult to go through it, as it happens in the ultra-dense, small space of black holes.

Spatial extension and form/density/mass are inverse parameters, Max. T = Min. S. If we generalize that property to all scales, we can define different fractal spaces by its proportion of mass/density and energy /distance. This is done with ‘Universal constants’ that explain the proportions of energy and information of those spaces.

For example, in physical scales, there are 4 fundamental space-times, the gravitational space-time between galaxies of max. energetic space and minimal formal density; the light space-time of our world, which carries information in the frequency of the wave; the electronic space-time of atoms with more formal density and lesser spatial speed and finally the quark-gluon liquid of atomic nuclei and probably black holes, with maximal density and minimal space. All of them are defined by Universal constants and equations that are either ratios between the energy and form of those space-times, or define the transformations of one space-time into the others. Einstein’s field equations would be the first case, defining the relationship between energy and mass in a gravitational space, while the fine constant of electromagnetism would define the transformation between light space and electronic space/ charge; and the gravitational constant between gravitational space-time and quark/mass. Where the relative densities of information and extension in space of those space-times are in balance, such as ΣTxSi=K. Thus electrons move slower than light but have more density.

All this said it is thus obvious that the *fundamental unit defined by the 2 ^{nd} postulate is no longer a point but an action, Tex Si= k between points, a dimotion. *

**2 ^{nd} postulate in physics S=T: T-Wave motion and S-particle information. Quantum & General relativity.**

Form=space and Motion=Time manifest in physics as particles and waves: the wave erases form into motion, the particle is a still state that gauges information entangled to other particle, fermion and boson, still to each other – despite the perception of relative motion in our scale – hence the information electrons share has always a c-constant speed. This is the ‘rational’ 5D explanation of both the c-constant of light and entanglement; as electronic beings perceive information in ‘stop position to each other’ and move in ‘wave state’:

Motions are perceived by particles that stop motion into form, into information, as distances. In terms of fractal reproduction of information we can define motion as the reproduction of form, between those 2 scales: when the particle moves dissolves into its ∆-1 parts as a wave that imprints an ∆-2 potential field, with its ∆-1 wave form and stops to become a ‘tight’ Particle state that ‘gauges’ information, form in stillness

*Galilean relativity was ill-understood, as the true question about time-change was why ‘we see systems still when they move’, and ‘why we see space as continuum, when in detail is made of quanta’, and why all systems are made of *smaller self-similar systems. So there is NOT really a Dimension of pure spatial form or a pure time motion but a combination of both, even if *mentally we tend to reduce motion and focus on forms, all has motion=time, and form=space, and this is truly the meaning of ‘spacetime’, the messing of both into 5 dimotions, the fundamental element of all realities. *If we see slow motion in the night a light it seems a long distance. Distance and motion cannot be distinguished so they must be taken as two side of the same being, a Space=time ÐIMOTION (ab. Dimensional Motion):

*S= T; Dimension-Distance = Time-motion = ST Ðimotion:*** Dimensions + motions = Dimotions**

*When we perceive the system in space, then we perceive an organism with 3 adjacent topologic elements, and its forma science is ‘vital topology’.* And when we perceive them through its scales the organic system becomes a supœrganism. Finally when we perceive the system in time we perceive a cycle that returns to its origin as a zero sum, which observed through all its scales will be a life-death cycle, common to all systems where life is the arrow of information and future, death the arrow of death entropy and past, and both together form a worldcycle.

**Measure density of Dimensions. The concepts of ‘filling space’, ‘memorial time persistence’ & ST-hollows.**

An essential problem to both mathematical mirrors and reality is the measure of Dimensional motions as we have an essential equality S=T in each plane between form and motion, and a scalar reality. So the value of a motion can be that of a dimension, as *motion is the filling reproduction of a form along a path of adjacent forms (see paragraph on golden ratios). But *in strict sense, ‘persistent full space’ is the maximal dimensionality possible of a system, when the reproduction in a single plane both durability on those reproduced parts, and the system can fill not only the plane of space but also all its smaller scales. This allow us to define a space without ‘scalar voids’ and ‘dying’ steps, as the most ‘continuous’ possible space, which happens to be lineal and orthogonal, for an absolute filling, i.e. the Cartesian space. So we define a dimension in scalar terms, whereas a classic single plane dimension is the limit for a ‘filled persistent space’:

A square may be broken into **N ^{2}** self-similar pieces, each with magnification factor

**N.**So the dimension of a self-similar object is the exponent of the number of self-similar pieces with magnification factor

**N**into which the figure may be broken.

Whereas the ‘persistence’ of memory, of information, creates the solidity of space dimensions (if a point erases without persistence, we talk of a 1D point, where motion doesn’t really add to the 1D inner Dimension of the point. But the value of the motion dimension, if persistence is equal to the life of the point, reaches Dimension 2. And S=T is absolute. S=T however holds if our measure of the system is reduced in time to the persistence of its reproduction. Further on, dimensions are relative to the scale in which we measure as the smaller scales will have more ‘hollow dark spaces’ that differentiate the parts. This means if we measure the dimensions of the system across scales in a 5D view of transversal ‘tree branching dimension’ of wholes and parts (5D), we will find hollow spaces. So 2 scales do not add dimensions to infinity, but have an intermediate value.

For example the Sierpinski triangle generated by the commonest fractal ternary tree has dimension 1.58. While the most famous bifurcation, the The Feigenbaum attractor has dimension + 0,5; and inversely a 4 bifurcation (H-fractal) gives us a full filling, 2D. It is another 5D metric paradox: the information of lower scales is larger but its spatial extension smaller. As a subjective observer reduces its dimension of perception to 0 in its ∆±4 scales. But the objective Universe is a whole completely filled, packed, as a block of spacetime, where time motions ‘enlighten’ just a part of the entire ‘potential block’ of scales. Thus we postulate an ∞ being of ∞ scales and ∞ time, the whole which fills its potential block *with all potential forms. And yet since ¬Æ can prove that the whole filling of the void is a finite number of potential existences, variations of being are limited and so all forms are immortal repetitions.*

**Recap****:** Non-Euclidean points constantly communicate energy and information with other self-similar points and the external Universe, by sharing flows of micro-points of a lower scale of space-time, which carry the energy and form of the particle into the external universe. The laws that define those acts of communication are hierarchical laws between planes of space-time and laws of balance between the energy and form of those ‘actions’ of communication, exi, which become the fundamental dynamic event of any scale of the Universe. Events in the Universe are limited by the ternary principle. Actions of communication also obey the principle: There are energetic, informative or reproductive events, creating often complementary systems with an energetic pole or body and an informative pole or head, communicated by a dense network or neck that carries the actions. The 5 Postulates of non-Euclidean geometry are based in the definition of a fractal point as a point with inner parts, revealed when we come closer to the point. According to such definition, lines are waves of points and planes topological networks of points, communicated through flows of energy and form. While equality requires also equality in the inner form or information of the point, which prompts communication through waves of energy and information that build networks. Communication between points is now possible because points can fit infinite parallel/waves used to gauge the Universe and create an inner image of reality. Non-Euclidean fractal geometry thus improves our vision of the Universe closer to reality and allows the definition of organic systems and logic behavior in bases of geometrical form, a long-sought dream since the times of the Greek.

**-3 ^{RD} POSTULATE. THE FUNDAMENTAL PLANE OF REALITY: SUPŒRGANISM**S:

**ITS TERNARY FRACTAL NETWORKS.**

The full realization of what Non-Euclidean vital topology means for our understanding of reality with its mathematical mirrors come into being in the 3^{rd} postulate which defines PLANES OF EXIST¡ENCE, the fundamental unit of the scalar universe, as superorganisms which require 3 lines to be defined, as in classic Euclidean geometry, since now *those lines are either waves of herds that shape an ecosystem or fractal, physiological networks that connect parts of ∆-1 with wholes in ∆º, becoming superorganism.*

*So the mathematical definition of a superorganism or ecosystem, is a ‘non-euclidean plane of exist¡ence in the fifth dimension’. The mystical poetry of that definition should not escape the mathematician. The laws of topological superorganisms does become ‘enlightened’ by the classic laws of plane geometry.*

*We do use though more often the ‘term’ scales of exist¡ence for planes, though in most cases both concepts are interchangeable, whenever 5D becomes mainstream and reaches certain rigor, the researcher should consider the difference between a ‘scale’, which is a decametric ‘subset’ of the ‘whole’, which is the plane. *

Because a plane has inner volume – within its points – it is a cellular, organic topography, a network of self-similar points. And because networks of points of energy and information are complementary, often we find systems with 2 complementary networks that form ever more complex geometries – based in the geometrical dualities of lineal energy and cyclical information – with the results we observe in nature: *the creation of an enormous number of complementary systems which are, as we shall see latter, all of them self-similar in its geometries and functions. Thus,* the types of Non-E planes of space-time range from the simplest Euclidean planes to the more complex organisms with a volume given by the relative point/beings that form its space-time networks.

The 3^{rd} postulate defines planes as the intersection=messing of 3 ‘lines=networks’ or waves of points in motion, which carry energy and information; to its cells, defining the biotopologic plane, a physiological system with 3 networks or Non-Euclidean plane. Thus vital topologic evolution is the missing link to complete biology besides genetics (in 5D explained as the lower plane that codes with its faster cycles the larger biologic whole) and Darwinian struggle (fully explained by the congruence principle of Darwinian dissimilarity) vs. eusocial evolution (in its own an entire new discipline) fully explained by the similarity of parallel beings that COMMUNICATE information in a code or language all can understand so they can coordinate its actions and become stronger as a whole. So *similar clone beings form topologic ternary networks that become superorganisms, wholes made of ∆-1 parts, themselves wholes of ∆-1 parts through 3 5D scales*

Stiences study those organic systems, tied up by networks of Ðimotions. In the graph, we see the main st-planes studied by human sciences and their 4 main time arrows, $ *x ðƒ,* which in static space give birth to the ‘organic elements’ of all species: social cell of energy and information and the reproductive networks that relate them. Thus, there are 4 basic elements in all organic systems: **Cellular** units. **Networks** that move the system (limbs/potentials) **Networks** of fractal information (heads/particles). **Networks** that reproduce vital energy (body/waves)

All species studied by science a common phenomenon occurs: the existence of parallel groups of beings organized into a single social form. Molecules are made up of atoms and electronic networks; economies are made up of human workers and consumers that reproduce and test machines, guided by financial networks of information (salaries, prices, costs); galaxies are composed of stars, which orbit rhythmically around a central knot, or black hole of gravitational information. Cells controlled by the nervous, informative system organize human bodies.

A tree is a group of leaves, branches and roots connected by a network that provides energy (salvia) and information (chemical particles) to its cells. Cultures are made of humans related by verbal, informative Disomorphisms and economic networks that provide their energy and information.

Vital topology studies those fractal Super-Organisms of Time space (Ab. T.œ) whose ∆º networks of ∆-1 points form the 4^{th} and 5^{th} dimotions of social evolution and entropy between parts and wholes.

Thus, a plane becomes a real topography made of points with volume, extended as a cellular surface. We can observe its surface as a bidimensional membrane of information (for example your skin, or the screen of a computer made of pixels, or the sheet of this work). Or we can consider the 3-dimensional inner structure of its points and then it becomes a network with inner motions, as those points will form a lattice in which they communicate lineal flows of energy and information that maintain the lattice pegged. Often 2 topological planes of energy and form combine to create a 4-dimensional organism. Such is the most common structure of the Universe, a 4-dimensional World, which is a Universe in itself, made of self-similar cells or networks of points that constantly exchanges energy and information within the ecosystem in which it exists:

In the graph, we see the ternary network structure of the nested organisms of the fractal Universe as ¬Æ topological planes composed of ‘similar fractal points’ (atoms, cells, individuals) joined by 3 physiological lines=networks, whose 3 functions, distribution of locomotion, information and its ‘combined’ energy define the 3 conserved Dimensional motions (ab. Dimotions) of any system of the Universe.

There are only 3 variations of topologic space in a single “plane of the fifth dimension’, the hyperbolic, elliptic and toroid topologies. So the fact that we are made of space means we are made of 3 type of organic topologies:

Spheres are the topology that holds the maximal volume of information; hence all ‘time space’ systems that process information are spherical particle-heads. Flat, lineal topologies are the topology that connects in the shortest path two points; so to reach/move faster, systems have lineal/flat moving potentials/limbs.

Finally the third type of geometry, hyperbolic topology, is complex enough to store all other possible forms, so best to reproduce. So all iterative bodies & waves that generate the other 2 forms are hyperbolic. This mathematical-spatial truism holds for 2, 3 and 4 dimensions; hence establishing a basic restriction to the construction and evolution of forms. It explains the efficiency, speed and homology of formal=functional, ‘punctuated’ evolution, from biology, without the need of ‘intelligent design’ to engineering (no longer analogy, since all forms derive from the same ‘substance’, space=form with time=motion or ‘topology’). In all systems the 3 only topologies of the Universe ensemble to form physical, biological and social organisms, all of them with a spherical ‘tall’ dimension to gauge information; a wide, iterative dimension of hyperbolic bodywave reproduction and a flat, direction for its entropic motions. Each of those 3 topologic Timespace forms can ‘deform’ and ‘move’, as long as they don’t tear (break). All T.Œs are variational local fractal species of that ternary ensemble, seeing as* a simultaneous space supœrganism, which lives a sequential worldcycle in time, *whose 3 conserved Dimotions in a single plane, |-limbs/fields<Ø-body-waves>§ø-particle-heads become in space functional topologic organs ordered in time in 3 consecutive-ages of dominance of each organ-dimotion between ‘4D Generation’ in the cellular/atomic plane of seminal birth ∆-1»∆1 and 5D extinction, back to the ∆1«∆-1 atomic, cellular plane: the young age of max. |-locomotion, the mature age of Ø-body-wave reproduction and the 3^{rd} age of O-informative warping/ wrinkling and minimal motion. Those 3 ages of all T.œs (Time§pace organisms) apply also to languages, including mathematics, evolving from a young Greek age of lineal ‘flat’ geometry, to a mature age as a realist mirror of the scalar Universe with analytic geometry and calculus that gave it motion and scalar depth studying its finitesimal parts and wholes to a 3^{rd} age of inflationary information and mathematical fictions unconnected with reality (set theory, NOT points, numbers and operands as its foundations, axiomatic method NOT experimental proof); to start again as the mind of a ‘different mind species’ – chips now evolving its young simple Boolean algebra:

So the vital space of superorganisms is made of 3 topological forms; and its cyclical time, and clocks that measure the duration of the existence of all beings, by the frequency of its logic cycles of information, accumulate constantly form, T>S, in 3 ages, first dominated by pure motions, (2D) in youth, balanced in the reproductive mature age (3D), to finally exhaust its energy warped and wrinkled into form (1D), which will explode entropically in the moment of death (5D).

Both together show the ‘Disomorphic’=equal form of all 5Dimensional beings, organic fractals, whose scalar structure is the same for all space-time beings, in all stiences, which study according to its relative scale of size and speed of time clocks a different metric scale of the fifth dimension. So we introduce another element of vital mathematics, the ternary structure of all systems of nature of which the previous fractal points of physical systems are its simpler forms.

Because the Universe is Generated by the properties of space and time, better reflected in the mirror languages of mathematical topology and social numbers, and logic, we could say safely that since ‘we think therefore we are’, that is, we perceive as humans a limited range of reality, with our space and time languages, for humanity, the Universe is generated by mathematical and logic languages. And so by studying formal sciences of mathematics and logic, we can describe the Universe. We can talk then of a Universal topology, from topos, the language of geometry in motion and logic, the language of causal time cycles that create and repeat patterns of reality.

How many world minds those 2 languages generate varies according to the complexity of each being in exist¡ence.

We humans, in the present not very enlightened age live in a single dimension of time, and the simplest, most dangerous of them, lineal entropic time, and a single scale of space, sub-divided in 3 lineal dimensions of height-information, width-growth and length-motion. And that is all. But can talk of more complex beings, with dual, ternary, penta and Dodecalogic levels of entangled thought gradually used here.

What we mean by ‘space-time beings? Space is ‘form’, ‘distance’, ‘dimension’, ‘information’, something that doesn’t move. And time is motion, change. So spacetime would be a ‘form with motion’, which is what mathematicians call a topology. A form that can be deformed, trans-formed. We have coined a key word ‘Dimotion’ (dimensional motion) for those topological forms. So the first obvious question we must answer is what means in real terms, the fact that all what exists is made of spatial topologies and temporal ages, of Space and Time? The answer is fascinating, as there are only 3 variations of mathematical space in a single “plane of the fifth dimension’, the so called hyperbolic, elliptic and toroid topologies. So:

‘All entities of the Universe are topological systems made of 3 ‘dimotions of spacetime’. Since *in the whole Universe in either 4 or 5 Dimensions there are only 3 topological varieties of form with motion; the ‘elliptic particle-sphere’ , the ‘hyperbolic bodywave’ and the ‘toroid, lineal limb-field’, illustrated in the next graph for the 3 kind of species studied by the 3 disciplines of science, physical, biological and human ‘stiences’:*

The 3 elements of all systems, lineal/flat limbs/potentials; spherical, tall heads/particles and its hard membranes of dark matter, trunks and skins accomplish the 3 functions of motion, energetic reproduction and informative perception in all systems, varying to adapt the system to its larger world.

Thus, there is a parallelism between ‘vital function’ and ‘abstract dimension’ as *each Dimension is diffeomorphic created by the vital space-time of each entity of the Universe performing its 3 survival dimotions= åctions of gauging, moving, feeding and reproducing. *And each species is a ‘fractal universe in itself’, with different relative, energetic and informative directions that determine its own up & down arrows and complementary morphology.

The specific geometry of each species and its 3 simplex åctions, energy feeding=length, information-gauging =height and reproduction=width fluctuate in shape, but they don’t vary in topological form, giving birth to the Invariance of the 3 topological forms=functions of any entity observed in a single 3D space-time: Toroid informative heads, hyperbolic reproductive bodies and energetic limbs.

While the 2 social ±∆ fractal dimensions, the 4th dimension of social motion in a social ∆+1 herd and the 5D dimension of inner scales within the being follow the physiological invariances that create super-organisms co-existing in a* higher ∆+1 ecosystem with a lower ∆-1 internal, cellular space.*

**Numbers as forms. Networks. An interdisciplinary study.**

*As difficult as it might seem to ‘monologic’ man, who so much loves single ceteris paribus cause, reality is intelligent because it is entangled ‘pentalogic’. That is, the symbiosis of the different elements of reality make it that complex, balanced and beautiful. And so happens with its mirrors. For example, we have observed the ternary nature of physiological networks, which are made of ‘social numbers’ in scales of parts and wholes. This gives nature to the 3±¡ pentalogic structure of reality, and when ‘doubled’ bilaterally across its S=T symmetries to decametric scales. So numbers as elements of a non-euclidean line, either a wave or a network follow also 3×3±¡ decametric scales. Let us study both together decametric numbers and physiological networks.*

Pythagoras as Plato latter said that numbers are forms, as they were in the earlier age of mathematical geometry, where a number was a group of points, whose *form mattered. So he realized 10 was the perfect number, because of its perfect form, which in fact becomes the 11th dimension of a new ‘whole’.*

And indeed the internal structure of any being reaches its perfect efficiency with a 3 x 3 +0-mind symmetry of form and function; where each part-number performs one of the 3 physiological entropy, energy, information jobs of the system and the central mind-number in contact with them all coordinates its functions.

So we also talk of 10 inner dimensions or ‘sub-systems’, represented by a tetraktys:

In the graph, each 3 corners are sub-systems of ‘Information, entropy/motion and Energy/reproduction’ put together by a central 10th dimension (the black ball/hole/point/knot that messes with all of them). Indeed the central point of the ideal tetraktys communicates with all the other parts and embodies the whole that ’emerges’ as a point in a higher ∆+1 world. Thus we talk of the ‘subsystems’ of a being.

For example, a human being is defined in medicine as a system of cells, attached by 10 sub-systems:

In the graph upper left 3 ST-ructural human body sub-systems are its membrane, sustain and motion.

Bottom 3 ‘chemical ðƒ’ systems or hormonal brain (creative, distributive and reproductive). And the nervous system singularity-mind… of the whole. Those graphs show that as we grow in planes, the ideal geometry of the lower ‘atomic planes’ disappear, as long as the ‘logic concepts behind it’ – hyperbolic fractal body-waves branching, bilateral symmetry, etc., remain. Since function is more information than form.

So the convoluted bilateral networks that connect the singularity brain with all its antipodal points of elliptic geometry have the same function that an antipodal ‘representation of a non-Euclidean’ sphere.

Yet in the human organism, the lines that connect them is not made of straight lines but it does work because what matters here is the symmetric territorial order of the singularity which constructs its membrane with opposite ‘rays’/nervous lines and will constantly balance and hence act as a leverage with its ±inverse directions for its antipodal elements, two hands, two kidneys, and so on.

Morphology then starts with simple laws of Non-Euclidean topology, which become disguised by the adaptation of each function to the available space within a membrane, as the T.œ adapts to its outer world.

The 3×3+1 physiological systems of the human being, lost its ‘ideal’ mathematical symmetry, translate to ‘existential algebra’ the concepts of decametric scales. They can be easily subdivided in 3×3+1 Nervous integrative mind system, but also in the fractal entangled Universe, into 5 x 2, positive and negative dimotions. For example, the entropic system branches in 3 the digestive, respiratory and excretory systems, which feed the Dimotion of locomotion, sustained by the Muscular and skeletal system; the reproductive ST system branches in 3, the blood, circulatory, reproductive and excretory systems; and the informative system, in 3 Nervous, lymphatic and endocrine systems, whereas the Nervous system doubles as the one site of consciousness, origin of the Dimotion of perception, and social evolution of the whole.

Physiological networks then belong to a different branch of geometry, fractal networks, which are more efficient in the distribution of the 5 dimotions of exist¡ence between parts and wholes of the ∆±1 scales and so we introduce the concept of a number *as part of a social *network by merging Non-E topology (2^{nd} postulate) and Number theory.

In that regard, a complex analysis of the simplest numbers shows that the more perfect form is the 10-cellular system or tetrarkys, in which 3 x 3 triangular corners act as organs of energy, information and reproduction with a 10^{th} central element that communicates all others and acts as the one of the higher scale, representing the entire organism.

Thus as the number of cells grows, the topology of the system will grow in degrees of freedom and complexity till resembling more and more the repetitive, geometrical forms of social organisms. Topologies become thus at the end, complex networks, adapted to different functions of complex organisms.

As abstract as all this might seem, when observing nature we shall see how those type of events, waves and social planes happen in all the scales of the Universe, from atoms which form crystal networks based in the equality of the same atoms or at best in the existence of a ‘body-mass’ of equal atoms intersected by a few ‘stronger’ atoms that form a complementary network of higher resistance, to the body rejection of cells with different DNA.

What things we can do with numbers can reflect then many of the actions of its networks. For example:

– We can study how social groups organize themselves or fluctuate between states=functions. This is the study of the internal point of view of networks as a collection of self-similar points. Those changes of states are often defined by a differential equation as informative systems have less spatial extension/motion but are more complex networks with more bits of information=points. Thus differential equations, most of them of the type Y (ti) = aX^{3}±bX^{2} ± cX ±D, express ∑Se<=>∏Ti transformations, where Ti is a network in 2 or 3 dimensions of time bits, bits of information and Se is a network with one (same organism) or 2 (Darwinian feeding) scales of lesser complexity than Y, such as f(x)=Y^{n.} It follows from the Fermat Theorem that there is a restriction to the number of solutions a system can find, which is n=3, the maximal number of dimensions an informative sphere can have as it displaces itself over a plane of energy.

The relationships between limbs and heads that exchange in a 3^{rd} region called body, form and motion, such as the head designs the motions of the limbs, which move the head, and both exchange in an intermediate region of elliptic nature called body more subtle types of form and motion to create more complex cycles that will in fact reproduce both systems can be mathematized in infinite different ways, using matrix, combinatory theory, differential equations, polynomials, Riemann surfaces, etc.

– We can study how networks grow and multiply creating new species and we can add them and observe how they reorganize creating curves which are differentiable to obtain the rate of grown and diminution of the organic population. The study of herds of energy and networks of information in its life cycle is one of the key disciplines of all sciences specially physics and ecology.

– We can study them as networks with form through its geometrical ways of exchanging energy and form, from the simplest point to the line of 2, the triangle without a central focus, the structure of energy, which can however turn into a pi-cycle, the 3, the 4 with its zigzag, solid quadrangle and cross structures, the 5 and first 3 dimensional structure, and so on.

Each number will increase the possibilities of the game, yet when we reach 10 we play a perfect game with 3 triangles that act as organs of energy, information and reproduction, and a central point both in a 2-dimensional or 3-dimensional geometry, acting as the collective action/will/intersection/knot of all cycles – the first clear, complete ego structure in 3 dimensions with perfect form and complementarity. Thus beyond 10, while some numbers might bring slight improvements to the cell, most forms are just growths of the primary numbers in multiple associations.

*Recap*. All the structures of mathematics, regarding of the notation we use, reflect events and forms of knots of time arrows (st-points or numbers), as mathematics is a language whose grammar derives from the Universal grammar of spacetime. Numbers are thus formal networks that try to achieve the essential arrows of time. And so certain numbers (1, 2, 4, 5, 7, 10) deploy better those arrows and are the commonest on nature.

– We can study the evolution and reproductive creation of new networks with successions and combinatory is important in multiple time-spaces since we find always complementary systems of reproductive energy and information, each one with a ternary choice of evolving differentiation (energetic, informative and balanced species). So especially in the classification of species of different sciences we shall find simple combinatory laws that explains the differentiation in 3, 6, 8 and 10 elements depending on the triads and dualities of multiple space-time systems.

– We can study a key antisymmetry of time and space expressed with the language of probabilities: Sequential events are studied with probabilities in time, whose symmetry in space are the study of percentages of populations in space, such as if each event in time is the birth of an individual of a population both probabilities and percentages are the same.

This confused physicists in some cases, as in an electronic nebulae, which is a population of fractal electrons in space, but it is studied as time probabilities, and created the bizarre theory of multiple universes (multiple, probable electrons) instead of a fractal Universe (fractal self-similar micro-electrons, which are bundles of ultra-dense light forming a nebulae which also acts as a ‘whole’ electron, self-similar to its parts). Thus the study of probabilities in time events and growing populations of a wave of space-time cycles is an essential tool: we can study the proportions, herds, groups and networks of self-similar st-points in its evolution either with probabilities or differential equations.

**The Generator’s Ternary Symmetries and Its S=T 1, 2, 3 Dimensional Analysis**

There are 3 relationships in space-time between entities, which are part in non-Æ of the laws of the fourth postulate of similarity, that we* relate to the 3 elements of the fractal generator:*

ST: Complementary adjacency, in which in a single plane, *membranes of parts fusion into wholes*, and in multiple scales, parts become enclosed by an ‘envelope’ curve that becomes its membrain. Its main sub postulates being the realm of topology proper.

$t: Darwinian perpendicularity, in which a membrain/enclosure is ‘torn’, and punctured by a penetrating perpendicular, causing its disrupter of organic structure. Its main postulates being the realm of Non-Euclidean geometries.

- ð: Parallelism, in which two systems remain different without fusioning its membrains, but maintain a distance to allow communication and social evolution into herds and network supœrganisms. Its main postulates being the realm of Affine geometry.

The correspondence of those relationships with the 3 elements of the generator, $<ST>ð§ are immediate:

– ST-Adjacency allow to peg parts into present space-time complex dualities.

-$-Perpendicularity simplifies the broken being into its minimalist ‘lineal forms’, $t.

-§-Parallelism allows the social evolution of entities into larger §ocial scales.

They will define ‘ternary organisms, in which the 3 topologies in 1, 2 or 3 s=t dimensions of a single space-time plane, can be studied in ceteris paribus analysis or together, *but no more, as all other attempts to include more dimensions in a single plane are ‘inflationary fictions caused by the error of continuity’ *

Dimensions thus must also be considered besides *the 3 logic relationships.* And there are 3 levels of complexity in dimensions, lineal, 2-manifolds and 3-D volumes that express also the ternary generator:

So for the 3 lineal coordinates, the equivalencies are immediate:

1D¡ Γ: $t: length/motion <ST width/reproduction> §ð: height/information.

As lineal length is the shortest distance between two points, height the projective geometry of perception from antennae to heads, and its product mixes them to reproduce in the width dimension where you store your fat…

**Change of symbols in algebra from 4D to 5D: the new equality-feed back equation. Correspondence.**

It is important before we go further into algebra, to understand that the r=evolution of non-AElgebra steams from the fact we are in a pentalogic convergent manner to create forms, hence = is substitutes by <=>. And so every equation of science must be reassessed not as an equality but a relationship of equivalence that through a ‘dimotion of exist¡ence’ can be transformed into each other.

This also means that ultimately = is equivalent to an action, and so we should write 5 different ‘stages’ of equivalence, one for each dimotion of exist¡ence.

For example, e=Mc2 doesn’t mean that energy and mass is the same; but that through an « entropic process mass becomes energy. Often this kind of relationships then require a ‘ratio’ or ‘universal constant that explain us how one variable becomes the other and this is the role of C2, so in fact we have to write: M < c x c < E.

So 5D non-Euclidean algebra introduces changes in the basic interpretation of the equations of mathematics, but beyond those principles it *closely embodies *the principle of equivalence, as the new ‘operandi’ of non-AElgebra enclose as a ceteris paribus analysis the simple equality symbol.

Equality however is not real, as absolute congruence does not happen often. So we have to couple the 4^{th} postulate of non-Euclidean congruence, experimental facts and rules of non-AElgebra to fully grasp the diversity of variations of the ó symbols of feed-back equations. This can be done exhaustively and systematic as we shall try to do in the paper on the stientific method and also here to bring meaning to the abstractions of algebra. And we shall also introduce those concepts on the papers of mathematical physics to better understand it, as Universal constants and ratios are closely related to those feed-back equations.

The Universality and lack of generality of the symbol = therefore disappears now by the fact we try to operate always in more detail equations with the necessary < > ≈ «» 5 symbols for the 5 Dimotions of existence.

= thus come in 5 favors and with them come different operandi to signify, SS, St, ST, Ts, TT elements. So knowledge of an equation comes also from a deep reflection of its structure to observe in more detail and translate it to GST equations.

If we consider GST, the ‘Generator of SpaceTime’, S≤≥T, we can then consider a first differentiation into 5 subspecies of St dimotions. Then we evolve them into chains and so the first question we wonder is how many ‘events’ of two chains of spacetime are possible and what they signify.

It is then when the concept of a function of existence of 5 Dimotions of space-time moving in 5 Dimensional space acquires all its meaning for complex formalisms of mathematical stiences with non-AElgebra. How can we represent that 5 Dimotional space? This is the subject of Ænalytic Non-E geometry, (æ¬E). Enough to say a good model departs from the complex plane in 4 Dimensions where the I is the S coordinates and the real the Time coordinates..

**4 ^{TH} AXIOM=POSTULATE OF RELATIVE CONGRUENCE: SIMILARITY, ¡NDIFFERENCE, TRANS-FORM-ATIONS.**

*4 ^{th} Postulate:* Equality is no longer only external, shown in the spatial perimeter of any geometrical form (Euclidean congruence) but also internal and further on it is never absolute but relative, since we cannot perceive the entire inner form of a point – hence the strategies of behavior such as camouflage. Forms are self-similar to each other, which defines different relationships between organic points, according to their degree of self-similarity. The

*4*postulate is thus the key to explain the behavior of particles as the degree of self-similarity increases the degree of communication between beings. Some of the most common behaviors and ‘events derived from this postulate are:

^{th}1) Reproductive functions in case of maximal self-similarity or complementarity in energy and form; ei->Sei or Max E x min. I (male)= Min. e x Max I (female).

2) Social evolution, when points share a common language of information, i=i -> 2i.

3) Darwinian devolution when forms are so different they can’t understand each other’s information so instead they feed into each other: i ¹ i. In such cases if those 2 entities meet they will start a process of ‘struggle for existence’, trying to absorb each other’s energy (when E=E) or simply will not communicate (when E¹E, since then there is neither a common information to evolve socially nor a common energy to feed on). Yet because any point absorbs only a relative quantity of information from reality, similarity is relative, faked for purposes of hunting with biologic games such as camouflage or sociological memes that invent racial differences, allowing the exploitation of a group by another.

The 4^{th} postulate defines systems as identical when they are equal in its 3 ternary parts, the outer angular momentum or ‘membrane’, its central Active magnitude or singularity, focus of the forces and the vital energy*, enclosed within them, and all others as similar with different ‘angle of congruence’. We* distinguish 2 different interactions according to the degree of equality of its ternary parts, as systems can be symbiotic, if *their individual, cellular or atomic ‘fractal points=parts’ are similar enough, interacting through *its 3 physiological ‘lines=networks’ evolving in parallel creating an organic plane, as those described in the next graph for each scale=science, or they can be entropic, destructive, predatory, when they are dissimilar and don’t speak a common language of information to coordinate its actions, whereas the stronger system will perpendicularly break and feed on the weaker one.

The geometric complexity of the 4^{th} Postulate is caused by the topological forms created by any event that entangles Multiple Spaces-Times. Since it describes the paths and forms of dual systems, which connect points: Self-similarity implies parallel motions in herds; since equal entities will maintain a parallel distance to allow informative communication *without interfering with the reproductive body of each point*. Darwinian behavior implies perpendicular confrontations, *to penetrate and absorb the energy of the other point*. Finally, *absolute, inner and outer self-similarity* brings boson states, which happen more often to simpler species like quarks and particles that can form a boson condensate as they do in black holes, *where the proximity of the points is maximized. And indeed, the same phenomenon between cells with the same inner information /DNA originates the ‘collapse’ of waves into tighter organisms.*

Finally if there is no similarity neither in body or mind, its existence as ‘cat alleys’, that never cross (relative invisibility). We talk then of Skew T.œ.s.:

The 4^{th} Non-Euclidean postulate is implicit in the work of Lobachevski and Riemann who defined spaces with the properties of self-similarity (Riemann’s homogeneity), which determines its closeness (Lobachevski’s adjacency).

*4 ^{th} postulate of relative congruence & angle of parallelism as a mirror of its 5 pentalogic dimotions and variations of angle define Darwinian or social, reproductive outcomes to communicative events between fractal points.*

Thus in praxis we assess similarity by an ‘angle of parallelism’ that increases social evolution into herds and supœrganisms, or perpendicularity that ‘scatters’ systems into entropic destruction – elements those of an entire fascinating new field of 5D topo-biologic studies that analyze in geometric terms, the vital topology and relationships between form and function in all systems of Nature from particles to organisms.

This simple geometrical truth however is essential to all systems of nature, whose angles of connection determine the functions and symbiosis between parts.

The Universe always starts with an asymmetric being, which can go both ways: towards a social evolutionary symmetry that lasts in time and implies a mirror parallelism, or an antisymmetric destructive, perpendicular event in which one part punctures and absorbs the energy of the other. It is the topo-biological ternary principle of non-Euclidean, Non-Aristotelian I-logic geometry that puts together both the biological and mathematical properties of reality. The concepts of symmetry=parallelism, antisymmetry=perpendicularity and asymmetry are mirrored by the 4th Non-E Postulate of similarity. But we can extend the concept of asymmetries also to asymmetries of time, between the young age of locomotion and the old age of information, of actions=Dimotions between the step and stop similar actions, and the entropy and social evolution actions, which bring us the final asymmetry of scales between the upper arrow of whole with more spatial size and the lower arrow of parts with more information. When those dualities: step-motion/stop-perception and scale up (5D: social evolution), scale down (4D: entropic dissolution) are put together we obtain the most complex balancing dimotion, reproduction, and when they are all added up in the existence of a being, we get its world cycle.

In the graph we can assess the different 5 mirrors in which mathematical Space and logic Time reflects the game of 5 Dimotions=actions of existence, which then expressed by territorial monads GENERATES its logic REALITY. In Geometry fractal points=monads will other through waves of communication of energy and information that grow into reproductive networks a territorial plane, creating a super organism, which will related to the external world according to its relative similarity=congruence, assessed by its angle of parallelism or perpendicularity.

In logic terms, a super organism, by breaking its formless asymmetry into different spatial configurations according to congruence (social parallel systems, complementary gender-mirror systems, Darwinian perpendicular systems, or systems that are dissymmetric and do not share any reality) builds a casual pyramid of growth from a fractal point through waves of communication into social networks that become ready to act – move, feed, perceive and evolve socially.

Since we must add to the mathematical and logic languages-properties of reality the 5 actions, or organic properties of the scalar Universe as essential to the game as they are its logic and mathematical more abstract laws – a fact the egocy of æntropic men of course reject, as it must remain in its monad-subjective monologic the only claimant to life properties.

Thus the pentalogic of generational space-time is established by its Non-Euclidean fractal points, its ¡logic congruence with reality in which it will order a territory to perform its 5 vital actions=Dimotions of existence, and the mathematical, logic and organic laws of those 3 languages will be therefore the bottom line of the ‘Creative process’ of the Universe – nothing chaotic except the entropic Dimotion, which conforms the monologic of huminds.

**The Generator’s Ternary Symmetries and Its S=T 1, 2, 3 Dimensional Analysis**

There are 3 relationships in space-time between entities, which are part in non-Æ of the laws of the fourth postulate of similarity, that we* relate to the 3 elements of the fractal generator:*

ST: Complementary adjacency, in which in a single plane, *membranes of parts fusion into wholes*, and in multiple scales, parts become enclosed by an ‘envelope’ curve that becomes its membrain. Its main sub postulates being the realm of topology proper.

$t: Darwinian perpendicularity, in which a membrain/enclosure is ‘torn’, and punctured by a penetrating perpendicular, causing its disrupter of organic structure. Its main postulates being the realm of Non-Euclidean geometries.

- ð: Parallelism, in which two systems remain different without fusioning its membrains, but maintain a distance to allow communication and social evolution into herds and network supœrganisms. Its main postulates being the realm of Affine geometry.

The correspondence of those relationships with the 3 elements of the generator, $<ST>ð§ are immediate:

– ST-Adjacency allow to peg parts into present space-time complex dualities.

-$-Perpendicularity simplifies the broken being into its minimalist ‘lineal forms’, $t.

-§-Parallelism allows the social evolution of entities into larger §ocial scales.

They will define ‘ternary organisms, in which the 3 topologies in 1, 2 or 3 s=t dimensions of a single space-time plane, can be studied in ceteris paribus analysis or together, *but no more, as all other attempts to include more dimensions in a single plane are ‘inflationary fictions caused by the error of continuity’ *

Dimensions thus must also be considered besides *the 3 logic relationships.* And there are 3 levels of complexity in dimensions, lineal, 2-manifolds and 3-D volumes that express also the ternary generator:

So for the 3 lineal coordinates, the equivalencies are immediate:

1D¡ Γ: $t: length/motion <ST width/reproduction> §ð: height/information.

As lineal length is the shortest distance between two points, height the projective geometry of perception from antennae to heads, and its product mixes them to reproduce in the width dimension where you store your fat…

**External dimensions/networks of organisms: territories. **

*The key element of Non-E geometry are the 3 topologic regions of all systems, as that is the underlying structure that evolutionary topology develops, with a singularity, @, dominating a vital territory enclosed by a membrane.*

It is the mixture of function in time through actions of survival that dominates the spatial ternary structure of those T.œs, which guides the understanding of vital geometry.

I.e. “Though most arachnids are solitary animals, some spiders live in enormous communal webs housing males, females, and spiderlings. Most of the individuals live in the central part of the web, with the outer part providing snare space for prey shared by all the inhabitants”. *Britannica*

Indeed, regardless of the vital topology of the point, all ‘build ups’ of new geometric scales start with the simplest form, a ‘bidimensional territory’ with a membrane, a central singularity and the vital energy between them. So geometry not only evolves in the humind in complexity, it does so in the evolution of the vital topologies of new forms in each single plane:

In the graph, all systems regarding of its ‘perfect geometry’, have the same ternary structure, to which vital geometry adds motion, in a Klein-like Non-E structure, where borders of entropy=death can’t be crossed; so they are relative infinities – military borders, balls of fire, membranes. On the left we study in more detail a mammal territory. Any animal territory is an i-logic space-time with 3 zones:

An informative central territory (1) or den, where animals reproduce and 2 secondary homes where the herd performs secondary organic cycles (2,3).

An energetic membrane (M, 5) – an invisible limit that provokes a confrontation if a stranger crosses it and where most energetic preys ‘flee’ away from the den of the predator.

An intermediate zone with cyclical paths of absorption of Entropy and information; where we find a hunting territory, places to drink (E), to bath (B), socialize (A), defecate (D), etc.

In organic terms, a dimension is a network. So a living organism can be considered a sum of cellular quanta united by 3 basic space/time discreet dimensional networks, which are its physiological systems: the digestive/energetic network, informative/ nervous network and reproductive/blood networks around which cells teem, creating a stable, organic st-point. In other words the Entropy and informative networks of a living being are its internal, diffeomorphic dimensions (of relative length and height), to which the organic system adds a 3rd, reproductive dimension that combines both elements and represents the width or ‘volume of cellular quanta’ of the system.

Finally its movement in the external world becomes its 4th temporal dimension. Yet that 4th dimension of external activity can also be considered a network territory in itself, sum of the 3±i cycles of existence of the being, creating a bigger vital space that will become the basic unit of an ecosystem or social organism made of individuals of the same species. In the figure we draw the vital territory of a minimal social pair of mammals, differentiated in 3 clear sub-sectors:

__Max.Information: Informative den or central territory (1,2,3):__

It is the territory of reproduction used to copulate and store basic food and Entropy to raise the young. It is a forbidden zone where not even hunting is allowed (4). In social species of great mobility, aerial or marine, where borders are much more extensive, this territory is very ample and tends to be located in warm latitudes.

Entropy=Information: Dual Territory of Entropy hunting and informative socialization (5).

It is the feeding, social and hunting territory, on which the central informative being feeds itself. It is outside the zone of reproduction. It is the winter territory of many migratory birds.

Given the relativism of all movement, in biological territories the informative singularity moves to hunt its Entropy quanta, as opposed to galaxies where stars and space-time dust moves towards the central worm hole.

Within those limits there are also neutral territories of communication, courtship reproduction and free Entropy, like water troughs. So the intermediate territory works both, as an informative and energetic territory where different victims and predators trace parallel cycles and come together around meeting points (E, B, R).

__Max. Entropy: Borders that limit the territory.__

Membranes are dangerous topological zones of dual osmosis. So the informative center watches to control any invasion of its hunting/social territory. Membranes fluctuate according to neighbors’ power. For example, the vital space of a fish increases during mating, since the couple is more powerful than a single individual. Marks (M points) fix those limits and reduce combats. They are often invisible, as most territories are defended against competitors of the same species, who understand the informative code of those marks; but rarely against members of other species. So we find all kind of linguistic marks:

*Smells* (common in mammals, like foxes, rhinos, antelopes), excrements (in canines and felids) or other glandular secretions.

*Optical marks* often connected to scents: The brown bear creates marks in trees, rubbing them with his head, warning adversaries of his great Max.SxT size and force. In human empires (nations can also be treated as biological territories) visual marks correspond to armies displayed in the borders. In human homes those marks used to be shields with weapons; now they are cars and other proofs of money, the new language of social power.

*High pitch, acoustic marks,* proper of birds, which are triggered when a rival enters the territory.

**Recap**. Vital territories of animals and human nations can be explained with the 3 topological regions of st-points.

The structure of §ð<TS<$t, territorial spaces with a central point of view, developing its particular worldview, trying to reach infinity with his distorted geometry, affine to a projective geometry where far away means small, defines each world of a Universe, which is objective when ‘clashing’ each form with all others – so only eusocial love, and emergence through the scales of the 5^{th} dimension make survival possible. Geometry is then the study of the spatial form that the functions, which dominate the vital, sentient Universe, adopt in their existential actions.

And as such is the best method to visualize the ‘meaning’ of algebraic and analytic equations both in abstract and mathematical physics. Each of the different laws of bidimensional plane geometry then can be studied as a reflection of the efficiency of vital Dimotions in a simplified geometric form, where curved paths become perfect lines, distances are measured without error and angles have no geodesic distortion. But still *vital geometry will add to those structures* *its vital interpretation, besides the ab*stract knowledge introduced by the Greeks, on the lines of the previous graph (which become lineal forms in other systems such as ‘polygonal molecules’ or human artificial constructions).

In Geometry thus we can also use the ‘pentalogic points of view’ on each theorem and also specific of geometry the vital meaning of the *5 angles of congruence (4 non-E Postulate), from Darwinian perpendicularity to social parallelism.*

**5 ^{TH} POSTULATE – DEFINITION OF A MIND-POINT OF A NEW SCALE.**

The fifth and only standing non-E postulate is equivalent to the first, defining a fractal point but from the p.o.v. of its inner mind-center of reference as the Universe has infinite of them: The graph shows the difference between the Aristotelian, self-centered, Euclidean=light humind and the Universal mind: ¬-Aristotelian, Non-Euclidean:

__The human mental light-Euclidean space one of many multiple spaces made with different force pixels.__

Euclidean geometry is the specific mind-mirror of light space-time and its 3 perpendicular Dimensions.

The Universe has ∞ mind-mappings made with different pixels that mirror for each singularity its territory of order (bodywave) and world beyond. The human ‘visual mind’ made of light is NOT the only mind-mapping. In the graph, on the left the ‘physicist’ view of a single continuum light spacetime for the whole Universe. In the right side the multiple p.o.v.s

Descartes did understand this multiplicity; so he published his mapping of the humind in a book called the ‘World’ to differentiate it from the ‘Universe’ with infinite monads, each one holding an entire world in itself (Leibniz) the very essence of the definition of a fractal time§pace organism. Yet Humans lost this earlier understanding – as we noted on the introduction – when Galileo didn’t argue the fact that the Earth moves but our mind creates a still space, a mirage of the senses. And physicists followed suit, creating its philosophy of reality called ‘naïve realism’.

Space as a mirage of the mind would be then understood in philosophy, both in the Eastern tradition (Buddhist Maya) and the western tradition influenced by them (Soviet, German schools starting with Leibniz, followed by Kant who noted Euclidean geometry was the geometry of the human mind, and Schopenhauer, who saw it as a representation.)

*– The mind or 0-point is, the relative frame of reference that mapped the ∞ cycles of time of the Universe, reducing them to a ‘World’,* to fit them into the infinitesimal volume of the brain.

*º-mind x ∞-cycles = World: equation of the Linguistic Monad: ∞ Mind-Worlds in 1 Universe.*

The mind though believes to be the center of the Universe in the ‘ego paradox’ as he sees every thing turning around its infinitesimal point, which hosts inside all the linguistic perception of reality, or ‘world’ it confuses with the universe. So the mind is a fractal point •, but believers to be it all.

The paradox of the Ego – who make each mind to feel so important – is then rooted in the self-centered structure of the mind, which selects information from its point of view, creating, an infinitesimal linguistic mind-mapping of reality – which then it confuses with the whole universe:

*0-linguistic mind x ∞ Universal space-time cycles= Constant World mapping of reality, with @mind at its center.*

The ∞ information of the Universe is reduced into the relative infinitesimal volume of our mind gives us a constant mapping, from where we expel all the properties that are not interesting to us and our self-centered view.

The mind is a singularity or infinitesimal 0-point, a relative frame of reference that maps the ∞ cycles of space-Time of the Universe, reducing them to a World to fit selected useful information into the finitesimal volume of the brain.

In mathematics the mind equation is 0 x ∞ = C; ∞^{0}=c; that is, the ∞ time cycles of reality become in the ∆º self-centered scale of a mind, a constant world that mirrors all what exists in the Universe both in time-motion (∞) and spatial form (º).

Because of such synoptic capacity to ‘mirror’ the laws of space-time in minimal size through the concept of number, which excels the previous synoptic language of verbal phonemes, mathematics soon became the most efficient language we know, but it does NOT create reality. It is just the best mirror of reality. Languages/mirrors occupy an infinitesimal part of the whole, yet paradoxically hold the maximal information of the Universe according to 5D metrics:* Space size x Time speed of information = Constant.*

Thus the mind of the most efficient survival species of reality, particles, ‘atoms’ and ‘galaxies’ (black hole atoms of the top quark decuplet) should be mathematical and imprint a local order which multiplied by ∞ of such species makes mathematics the dominant mind-species but not the only one and still a mirror of the true reality which is ‘scalar space’ and ‘cyclical time’, dimensional form and motion – the 5Dimotions of reality.

A language *is first a reflection of the laws of ¬∆@st, Dust of space-time* and its ‘Universal Grammar’ and Fractal Generator equations, without which they cannot order≈recreate locally the Universe. So 5D mathematics advances the discipline by focusing better the mirror to include the bio-topo-logic properties of scalar space and cyclic ¡logic time; and by putting in relationship maths and ‘existential algebra’ (ab.¬Æ), the a priori Disomorphic=equal laws of 5D, making it an experimental science; connecting its laws with the laws of fractal spacetime.

**Metric equation of mathematics: 0x ∞ = K search for ‘wholeness’ in a single equation. **

Thus we define the Generator equation of algebra, which as a mind language derives from the mind 5D metric:

*0-finitesimal spatial mind x ∞ time cycles = Constant mind-world: §@(mind space)<≈>∆ð (universal cycles)*

So t*he ‘Generator Equation’ of all digital numbers is, 0 x ∞=K and its algebraic expression, Sx<≈>ƒð.*

Both equations together allow to represent all possible metric equations of all supœrganisms of the Universe. As such neither 0 is an absolute zero, nor ∞ is an infinity, concepts relative in a scalar Universe, where the limits of scales in size and time speed and perception in each self-centered p.o.v. only allow to talk of relative infinitesimals, we call finitesimals, 0%, and relative infinities, whose symbol is µ. 2 evident truths are then the existence of an experimental finitesimal, h=m l² ƒ, the minimal unit of angular momentum or Planckton of the galatom, and a relative infinite, =c, its maximal distanced-speed, which conforms the most extensive metric we know of with precision, beyond which dark space and time is unperceivable to mankind, so we write 0s (h)x (c)=K

It follows also that mathematical infinities are inflationary mirrors whose contradictions (Cantor’s Paradoxes) are NOT solvable (the Zermelo axiom being an ad hoc addition, against the logic of truth), because precisely they limits of reality enter in paradoxical contradiction with the infinite inflationary mirrors of languages.

The study of those 4 elements of all realities, its actions and ternary operandi, structures *the dynamic ‘Generator Equation’ of all Space-time Systems, written in its simplest form as a singularity-mind equation:* *O x ∞ = K = ∞º=1*

So the *5 ^{th} postulate *defines points as informative knots or linguistic eyes – minds of information that absorb a flux of forces used by the point to perceive a relative world. A non-Euclidean point corresponds then to our concept of a relative mind that gauges the information reality with a certain force, similar to the concept of a monad in Leibniz relativistic space-time. In words of Einstein: a point of space is a fixed frame of reference.

Thus, Non-Euclidean mathematics fuses the logic and geometry of the fractal Universe, greatly improving our understanding of Reality even in terms of mechanist measure. Since mathematical solutions to problems with several points of view are impossible to find in continuous space (i.e. 3-body problem in gravitation), given the fact that a network of infinite points of view is local and relative and each point is a focal knot that acts from its perspective. Thus, the absolute truth of a system is the sum of all its points of view, which influence each other. Yet even if we cannot calculate precisely with mathematics, systems with more than 2 bodies, since those systems are organic, hierarchical, made of networks with attractor points, fractal structures and self-similar paths, the new mathematics of attractors, fractals, scales and Non-Euclidean systems, refine greatly our analysis. In essence, indeed, we observe that ‘networks’ integrate parts/points into wholes, which then ‘act’ as a single point. So in the complex models of i-logic geometry we can tackle many problems by defining sets of points as ‘wholes’ of a ‘higher space-time’.

For example, the previous principle of local measure, where each point is a relative center of the Universe, is called in relativity the diffeomorphic principle, which now becomes explained as a partial case of the wider law we called the ‘Galilean paradox’; the duality particle of information/wave of energy becomes a specific case of the application to physics of the duality of energy and information found in all systems.

The expansion of the laws of quantum theory (complementarity principle) and Relativity (relativity of scale, local measure, etc.) to other sciences and the organic principles of the 3^{rd} and 4^{th} postulate to physical particles is therefore the consequence of those postulates. Yet it requires the understanding of the new, i-logic, organic laws of the Universe and its networks, because E-mathematics has clear limits to extract all the information of the Universe, given the fact that it syntax includes a priori errors and simplifying postulates (single space-time continuum of points without parts, etc.).

Thus, when the event described is complex, performed by a great number of points/variables you enter into non-lineal systems, which require topological descriptions (chaotic attractors, fractal non-differentiable equations and Non-Euclidean mathematics), and the i-logic laws of organic networks and systems – a better syntax in which to fit experimental evidence, especially in phenomena of informative nature (since only formless energy is continuous and resembles the models built with a single arrow of energy and a single plane of space-time).

So while classic physical systems calculate accurately the energetic, continuous properties of the Universe an overview on how multiple points of view emerge into wholes requires organic laws. This search of whys also applies to the understanding of mathematics. For example, the previous postulates resolve the long-standing question of what is the nature of the 1^{st} and 5^{th} postulates that seemed redundant (as the 5^{th} describes also properties of a point like the first does, and the 1^{st} seem to describe a non-geometrical property). They are no longer redundant, but they are more concerned with causal logic and time than spatial geometry in its purest forms (points, lines and planes.)

There is also self-similarity between the fractal postulates of i-logic geometry (since the 5^{th} is geometrically self-similar to the 1^{st}, as both are concerned with points) and the 4 dimensional time paths/arrows of the universe. This is not casual since all languages of space-times depict in self-similar ways the 4D Universe. Thus if the 1^{st} and 5^{th} postulates define *a gauging point of information* as the fundamental unit of the Universe, the 2^{nd} postulate defines a line or flow of communication of energy and information between 2 points, which *reproduces* part of the information of the ‘generator’ point across a surface of space; the 3^{rd} postulate defines those points, which are not similar as *energetic* substances that will be absorbed by the points. Yet if those points are self-similar they will gather through the arrow of eusocial love, creating according to the 4^{th} postulate a network of space/time, a new organic plane of existence. So the 4 arrows of space-time are explained by the 4 postulates of i-logic geometry.

** **

**III. GEOMETRIC NUMBERS**

**S-Numbers: Polygons as vital organisms**

Numbers are social gathering of indistinguishable forms. When studied in space thus numbers must have regular efficient configurations. So ∆§ numbers are 2D polygons or 3D Platonic solids.

Its importance in vital topology lays on the fact that polygons start the creation of superorganisms, with a membrane – the polygon proper, which closes a vital space and can by connection of points through lines-waves of communication, create a central singularity. Thus numbers as fractal points grow organisms. And this is self-evident in the study of Nature. We can then study together with entangled pentalogic both ‘scales’ of social numbers as ‘growth in time’ of polygons, efficient configurations in ‘surface’ worlds (land, water surfaces), and polyhedrons, which will be efficient configurations in a larger ∆+1 3D world (water depth, atmosphere, vacuum for atomic and molecular scales).

**POLIHEDRONS**

Polygonal numbers are best to represent the social evolution of forms into new scales that emerge as envelopes into wholes.

We shall conclude the geometric analysis with the 5 regular solids whose obvious symmetry with the 5 Dimotions of existence, means each of them specializes in 1,2,3,4,5D, in parallel to the simplex forms and mirror symmetries that compose them:

3D platonic solids are 4, 6, 8, the only decametric regular numbers in 3 dimensions. Then we have only two more in the 20 scale, 12 and 20 vertices, and that’s all!!!

They play each one a role as actors of one of the 5th dimotions – the decametric 3 completing a 4,6,8 Fractal generator:

$t-tetrahedron (legs)<S=T-Octahedron (body) > §ð-Cube (head)

Indeed, the tetrahedron is basically a triangle in 3D, acting as the spearhead of a locomotion; which in timespace is traced by lineal members moving on steps as your legs do. Hence its 2D¡=$t function. The Octahedron has a clear mirror symmetry as all reproductive dual genders do – already posted in academia.edu its paper; and the cube, is the form of the 3 closer to a filling sphere with a well-connected singularity of maximal symmetry with the outer world.

So this leaves the two complex dimotions of social evolution and entropy for the triangular icosahedrons, *with maximal number of social triangular units (20 faces of the simplest 2D surface)*. And the 5^{th} dimotion of entropy for the pentagonal dodecahedron… Alas again the pentagonal evil Death symbol again… Why? That I won’t tell…

Just quote the master of ‘transcendental philosophical geometry’, Mr. Plato who affirmed that, “The Universe is the body of an organism whose logic mind we call God.’ & ‘God used a dodecahedron to design the Universe’ *Plato*

Ok, I’ll tell. Think in vital terms. Imagine a Universe with those 5 species, no curves. The 3 first are the smallish ones. Cubes perceive, and keep quiet, hidden, filling easily empty space; its survival strategy – social growth, with its identical forms, repeating by the mere motion as lines first, flat squares and giant 3D cubes. It is a single gender ‘feminine’, informative reproduction. The other series does the same but with mirror gender symmetry, as the triangles of tetrahedrons expand from 3 to 4 and peg inversely into octahedrons and then grow into icosahedrons.

But the dodecahedron is NOT concerned with the survival strategy of reproduction. He rather eats them all!!

How is this? Easy, the dodecahedron has the largest mouth and the largest volume, even larger than the icosahedron, so it swallows them, in an action of vital geometric feeding; and once caged inside, remember pentagons are the ‘first’ polygons, which can do a smaller replica of itself, NOT a perceptive singularity but a *crunching stomach*.

Each point is a cell of a dodecahedron. As it happens there are as many faces in the icosahedron to eat as vertices in the dodecahedron that feeds on it – both connected through the golden ratio of feeding and reproductive events, a+b (feeding)/a = a/b; a key relation of top predator/prey forms. And so it opens its mouth, swallows it, cages it, crunches it and burps (; But the pentalogic Universe allows different dimotions. So now imagine instead a reproductive event. It is not far fetched. Biologists consider that ovules first ate sperm cells and latter they learned to merge their genes.

So once a ‘male’ entropic dodecahedron has eaten the ‘female’, reproductive icosahedron it can actually merge into a mixed form, by merely attaching to each dodecahedron point a triangular face, becoming the icosidodecahedron (ab.‘dodecosahedron’) – a new species which is more perfect than its parental ones, as it shares the information of both. It is a first gender case in ideal geometry, since *Gender reproduction happens in all scales of stience,* from the simplest circular ovum swallowing a lineal seminal seed – a cyclical and lineal form, merging into an |xO=ø topology, to the previous example of ilogic mathematics.

The dodecahedron social points, aka numbers feed on the icosahedron faces, reducing them either by swallowing them, which will result in a growth of size of the dodecahedron or by merging into a reproductive evolutionary process far more interesting, as any merged system is more efficient than its 2 separated parts-species, a more evolved form, which survives better in a process similar to all genetic-memetic-reproductive processes, that create in any stientific scale more efficient systems by communication.

Indeed, the dodecosahedron is the top predator polyhedron form, as it has ALL the ilogic elements and symmetries that enhance survival in the fractal Timespace Universe: individual ego-centered points, shaping inner trilogic systems in yellow, tetralogic dual triangles married to dual pentagons, decalogic reproductive bilateral social parts, plus all the magic stable numbers of vertices, edges, faces & diagonals, despite its external ‘soft membrane’ –unlike its ‘dual figures’ the Catalan solids that *show a defensive external membrane*, in which the swallowed form emerges into convex ‘teeth’, but has a less efficient internal organic structure:

In geometry, we say that any polyhedron is associated with a second **dual** figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. The graph is an example from the other dual top predator/prey series, a cube of larger ‘mouths’ and bigger volume-spatial efficiency swallowing an smaller octahedron of lesser volume.

What differs now is an added property – a topologic ‘defensive’ concave and convex dual form which can establish as Vaughan found in defensive flat geometries for the French army, entropic expansive concave points and smaller implosive, informative sinks, which maximize the entropic reach of cannonballs with minimal surface exposure to the external army. Indeed, a ‘mathematical’ ideal duality, we shall find latter on in the stience of biology between top predators of soft membrane but much more efficient inner structure and defensive armored animals with less evolved internal networks (squids, with eye-perception and evolved nervous systems vs. armored trilobites, simpler organisms; Tyrannosaur Rex vs. armored triceratops, Lions vs. armored, skin thick, horned herbivores) and social systems (defensive walled cities vs. nomadic horse armies).

We are all VITAL geometries and so the rules of geometry carry from ¡logic spacetime to ilogic mathematics, to the different stiences of vital reality where those forms acquire topological motion. The specific case of polyhedrons being of special interest to molecular chemistry, which we shall study in detail when upgrading to 5D the discipline (as those are processes that do happen in solid crystals).

**Vital transformations.**

Though polytopes are all essentially in topologic terms spherical membrains, its study brings topology down to the details of its scalar social numbers or parts of a network of points; allowing its dynamical transformations.

A key sub-field of 5D vital geometry is the analysis of the Disomorphisms and symmetries between polytopes to understand how they transform into each other through 5D entropic, 3D reproductive & 4D evolutionary dimotions.

**4D: social evolution.** All solids with a center singularity evolve their surface increasing their vertex, and growing their volume/body ratio towards the perfect relationship of the sphere – which we consider *the 6th perfect solid of ∞ vertex. *Among the Platonic solids, either the dodecahedron or the icosahedron may be seen as the best approximation to the sphere. The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest to that of a sphere of the same size (i.e. either the same surface area or the same volume.) The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most.

**5D: Entropic devolutions **mediated mostly by the 5D dodecahedron. Feeding though requires a degree of similarity, reached ad maximal for the dodecahedron and icosahedron. What makes them *identical to switch from 5D mind state to 4D entropy state is A*pollonius’ finding that the ratio of its surface is the same that the ratio of its volumes:

*So i*ts content of ST vital energy is the same; but as the dodecahedron dies, it changes first its tight surface, which as we know in death processes grows in excess detaching itself from its volume – from 12 to 20 elements (another key number, with many vital interpretations, being indeed the number of amino acid variations on the protein surface of living solids, and so on)… and then reached its maximal form on the stable surface, it can only switch arrow of time.

As death is exactly that process: when all the vital energy is consumed and the skin is fractured ad maximal and no more motion can be extracted from the body, the system changes arrow and decomposes back to the past.

Other feeding case is the parasitic covering of one form by another. So the dodecahedron is generated (Euclid) from the cube, *by covering it as a parasitic membrane, unlike the 3 other species, which can be considered to reproduce the same form. *All this might seem trivial to mathematicians, but they would miss the point of what 5D mathematics tries to show – the vital nature of space… which must be explained as the ultimate formal substance of reality, having organic living properties, simplified into a still mirror, NOT the other way around.

1D. Perception. If the sphere is the perfect platonic solid with absolute symmetry to perceive an ‘equal distant’ territorial larger sphere focused in the central point without deformation (Poincare Conjecture), platonic solids are next in regular symmetries, hence as they are ‘economic’ in points compared to spheres their efficiency makes them to repeat constantly in the Universe on the simplest forms of chemical, atomic existence (simplex platonic solids), NOT in the planetary orbits, where Mr. Kepler thought God drew them (:

We see then once more at play the scalar laws of inversion of form, from the ‘galatom’, or spherical atom, given the ginormous number of ‘static photonic points’ trapped in its electronic nebulae/star plane (quantum/cosmological scales) to the molecule or lineal polytopes of minimal quanta. Molecular surfaces tend to be as the graph shows, perfectly regular polytopes for the singularity to apperceive through the Van der waals forces of electronic perception the gravitational and electromagnetic forces of the world – reason why crystals, regular atomic systems which can ‘scan’ reality in a near-spherical are formed as units basically with those forms and the strongest ‘pre-cyclical’ pi=3 hexagonal system (which also can form the ultimate platonic dual systems, a pentagonal, hexagonal cover, the strongest ‘fuller dome’.

In the graph we see the all pervading forms of maximal resistance in the membranes of physical T.œs. In crystals the cubic system is overwhelming. In metals only cubic and hexagonal systems exist. In architecture the only systems which can be grown in size without external reinforcements maintaining its stability ad infinitum are the Fuller Domes, made with triangular, hexagonal ‘pi=3’ and hexagonal & pentagonal combinations, whose form grows ‘ad infinitum’ towards the perfect platonic solid – the sphere.

**The cube, a 3D st solid with informative roles. **

In that regard, another ‘inverse duality’ in 3D happens between the sphere and the cube, similar to that between the triangle and the circle in 2D, with inverse properties:

The cube is excellent for social evolution into larger networks, while the sphere tends to be lonely. The cube is the preferred crystal form for matter to reproduce filling ad maximal space and socially evolve, from 3D to its social larger 5D whole grouping.

How then one transforms into the other? If the triangle achieves the feat by rotation or social evolution into hexagonal π=3 circles, the cube is the closest form to become by elliptic deformation the sphere, in another st beat, as it bloats feeding on energy into spheres, it deflates back into a cube. So the triangle becomes a circle by adding a 4D rotary motion and the cube becomes the sphere by adding a 5D feeding motion.

Since the key function of reality is reproduction and all is motion’ the survival of the cube resides in its simplex reproduction by mere translation in space: a line that moves grows a dimension into a plane that moves and grows into a cube that moves and grows into a line restarting the game, and adding a new 5D scale. Yet each growth changes the function of the form according to the generator equation that *changes S-topology & T- age, hence function as a system grows in ∆-scale, such as:* **∆-1: $< ∆º ST > ∆+1>ð§**

The partial equation of the fractal generator that encodes within it all laws of the universe means in topology the transformation of topologic varieties, as we grow in scale, from lineal limbs/potentials at ∆-1, into Ø-ST-iterative space-time into ∆+1 O-spherical particles-heads of information.

The cube is the ð§tate, excellent for social evolution into larger networks, the preferred crystal form for matter to socially evolve, from 3D to its social 5D grouping into minds, the closest form to become by elliptic deformation the sphere, in another st beat, as it bloat feeding on energy into sphere, it depleats into cube.

And then again as the form shows, the cube displaces to form a line on the ∆+2 plane, NOT a fourth-dimensional spatial being, which does NOT exist.

So if we start the reproduction of cubes in the cubic ∆-1 line its function is 2D lineal motion; then the ∆º plane is an ST reproductive body-wave form and the ∆+1 cube the §ð informative function (with maximal volume and minimal perimeter compared to the line and the plane)

And then again, the cube displaces to form a larger line on the ∆+2 plane, NOT a fourth-dimensional tesseract, which does NOT exist in reality regardless of its usefulness to prove and model S=T dimotions.

So given the *reproductive nature of motion*, we talk of distance as the sum of adjacent ternary ‘stœps’: the cube state of a ‘squared’ line stops and chooses any of its 6 directions of motion; then it reproduces in present sliced planes on the chosen direction and moves as a zig-zag line, stopping, iterating and stepping as all reproductive motions do. Beyond 3 dimensions there are no more dimensions in a single plane, so the cube generates then a line of the larger scale, transposing its function again, completing a full zero-sum cycle in scales, creating a replica of itself in the larger ∆+1 world.

So we observe in those reproductive translations two symmetries *at work together:*

*$≈t≈$… Translation* = reproduction of motion through the $-length dimension, causing….

*∆o-> ∆+1: ST+ST+ST: Reproduction of form through the width dimension, *which makes the being, grow in scales of the fifth dimension, symmetric to the change of position in timespace causing…

∆-1: $< ∆º ST > ∆+1>ð§ a change in its topological functions=forms.

The next form of reproduction of the 3 varieties of topology is the rotary circle, which becomes a sphere with one more dimension, the strongest membrane that encloses and captures the vital, hyperbolic energy of the being. And then it becomes a toroid, which appears as a larger circle.

And it is worth to notice the circle has only 2 states as circle and sphere, unlike the cube which has 3 Dimotional states as cube, line and square; a fact of paramount importance when considering the different speeds of motion, reproduction, explosion and implosion along the 5Dimotional events of lineal v. circular forms that paradoxically favors the circle over the line and explains why 1D vortices accelerate faster than expansive lineal big-bangs.

*But on the negative side, the ‘cube is a solid form’, but the reproduced cycle becomes an* ’empty sphere’. It is not a ball, only the surface that encloses a volume of empty space – the simplest, strongest topology normally made at the ∆-1 scale of strong triangles of entropic nature. So the circle becomes the membrane that acts as the entropic envelope that closes the ‘empty vacuum’, becoming a non-Euclidean Klein disk whose inner vital space can never reach because the dense membrane will kill its lighter internal particles – and indeed, we shall see how atoms and galaxies – the two limits of our scales of form, follow such pattern. Since even if the form is a disk, while it can create in the first rotation a dense sphere, in the second rotation it can only form a hollow toroid.

But the most efficient form of reproduction is that of waves and bodies, with its ‘networks’ that branch and hence reproduce in an exponential growth a single point; hence the hyperbolic form of the ‘growth’ phase of waves; and the enormous multiplication of ‘surface’ of a fractal networks.

Finally all those forms can transform into each other: cubes can feed and curve elliptically into cubes and lines are small stœps of curved geodesics, both type of iteration can merge, so lineal limbs/potentials at ∆-1, Ø-ST-iterative waves and ∆+1 O-spherical particles-heads of information can in rare occasions transform into each other, but specially *can transfer flows of energy and information among them, which at ∆-1 level will acquire the shape of each of the 3 adjacent topologies of the supœrganism.*

So functions and forms do change but all possible events are encoded in the generator and so if we understand the underlying ∆st basic laws of the Universe encoded on it, everything keeps falling into place. And those laws are vital laws that have a survival purpose. For example, the hollow sphere KILLS by enclosing and trapping as lionesses do hunting zebras by enclosing them or dogs shepherding and enclosing sheeple, moving around in fast circles as ‘angular momentum’ that becomes by virtue of T=S a vital membrain in motion. Then the membrane will sharply penetrate perpendicularly the zebra herd and kill; the military border of a human social territory will give a coup d’etat and collapse into the capital and conquer… the battle will be lost once in Cannae, Hannibal had encircled the prey (graph, where the red color of entropy represents the dimotion of the horse that ‘closes the roman pray’ as a moving membrain like the dog does).

So ∆st laws might seem abstract to you but are the stuff of which experimental survival and existence is made.

**Hyper-dimensional polygons.**

As S=T 5D metrics implies a dimension of space is another form to interpret a motion in time, an important field of 5D geometry is the study of 4D (2S=2T) regular polytopes, which close the entangled scales of regular social numbers as T-Dimotions or S-till elements that ‘construct’ the real structures of the Universe, to extract 5D¡somorphic laws; even if we do NOT see static fixed 4D forms in our Universe (as 5D includes entropy, which is the negative destruction of all others, 4D is the limit of constructive engagement, often of 2 still polygons with 2 motions=degrees of freedom). We just make some key statements on this discipline:

With the simplification of the inner volume of fractal points, we can match polytope laws with the symmetries of Spacetime beings. So 1D polygon of points with no parts -open lines equivalent to 1D cylindric limbs/fields are ∞ (Natural Numbers); 2D polygons akin to cyclical, O-particles/heads are ∞ in flat space, *& 3 hyperbolic polygons, equivalent to Ø-body-wave reproductive systems, intersection of 2D lineal & cyclic forms are ∞ in 3D space.*

*So there are ∞ growing ensembles of 1D limbs/fields + 2D particle/heads=3D hyperbolic body-waves.*

Growth of complex finite polytopes peaks in 4D, as further increases of dimensions reduce them to the ternary ‘single plane’ structure of a triangular, square and pentagonal ternary ensemble; *which means there are no more dimensions beyond the 4D 2S=2T systems, and all further ‘Dimensional growth’ is a dimension on new scales of 5D which are memoriless, erasing the previous emergent complexity of the fractal point, which stays on its inner parts. (Because the hyperbolic body-system is more complex being its minimal form 3D, its ternary growth peaks at 5D).*

In the side the existence of a regular 4-polytope is constrained by the existence of the regular polyhedral which form its cells and a dihedral angle constraint sin π /p sin π / r < cos π /q to ensure that the cells meet to form a closed 3-surface. And yet despite those restrictions we have 6 polytopes. And this matter because our world as in relativity can be analyzed in a single plane as a 4D ensemble of its 3 classic dimensions and a time motion of social organization, as *the 2 inverse dimotions of entropy and information do NOT happen within the same event simultaneously in ‘regular entagnelments’. ***So pentagonal polytopes make no sense, being the 5 ^{th} dimotion entropy the destruction of the regular in-form-ation shared by them all.**

*Thus the most complex 4D structure of regular points are the 120-600 dual pentagonal mirror ‘gender symmetry’,*reached through

**combinations of 5 Dimotions, 6 ‘triangular radians’ of a π sphere (**

**µ**

**polytope) & its 120 angles. The result are the final stable 120 dodecaplex and 600 tetraplex maximal figures of 4Dimensional geometries. And we shall find this 120 magic final number in the few cases the system is perfect enough to transcend the octaplex to be the real final ‘island of stability’ in atomic systems or oldest possible age in the very few cases the system transcends the 80 years ‘radioactive barrier’. And we can imagine the 120-600 vertex entanglement the limit of ¡logic evolution of a complex social system in the Universe.**

**But all those polytopes must be understood as ternary systems which reproduce in motion forming a larger 5D scale; so 4D is a motion, and that is indeed how geometers construct them, bringing to each 3D point an internal volume; that is, making each vortex of a 3D polygon a new ‘square’ to form the tesseract which is equivalent to ‘enlarge’ its inner form. While another 4D polytope will be merely the motion of the square that becomes a larger ‘1D’ line in a larger world, as we make a 2d plane by moving across a 1D line.** What about the sphere? It is the ultimate regular polytope and so we wonder how a 4D sphere looks like. The answer is also telling of the 5D S=T structure of the Universe, where time motions and spatial dimensions are equivalent – but only motions truly exist. So we have to consider beyond 3D, new spatial dimensions to be beyond algebraic manipulation time motions. This is also our experimental evidience, when we generalize what we see of a 3D sphere as it moves through a 3D plane: a series of growing and diminishing cuts, passing through a 2D world.

And so a 4D sphere moving, passing through a 3D world appears as a growing and diminishing sphere, *indistinguishable from a *5D Universe. *Thus we postulate an Einsteinian “Principle of Equivalence’* as a proof of the existence of a 5^{th} dimension of parts growing into wholes:

‘A 4D spatial system moving with a 5^{th} dimension of time, IS (equivalent to) a 3D spatial system with two scalar dimensions of growth and diminution in inverse ∆±1 scales.”

So beyond our perceivable entangled 3D holographic parts, all further dimensions are scalar motions of time (4D, 5D inverse, SS-formal, implosive or TT-entropic expansive dimotions). Specifically we differentiate dimensional growth

**S=T.** In a single plane, beyond 3D spatial still forms, new dimensions are temporal, moving ones, which only through the ‘persistence’ of a memory create new dimensions of space, (as explained on our analysis of fractal dimensions). Motion then become reproduction of information, and ‘social forms of growth’ – a cube moving to in-form a squared line, sphere turning into a dual spiral diminishing and growing to form a larger flat disk, etc.)

**∆±¡.** In scalar planes, when the form is ‘static’ in the same position. Then the being experiences a ‘scalar travel through the fifth dimension’ growing its inner parts from points into ‘cells’. This in ideal polytopes means to draw in each vertex a new tetrahedron, cube or dodecahedron). Or as a whole growing or diminishing in size-scale.

So that is the dynamic reality of 4D spheres passing through our world, which do NOT exist as such, but as S=T for any mental spatial mirror that trans-forms T-motions into S-till dimensions, 4D polytopes are useful to model and solve mathematical problems of real physics, by the algebraic method of converting a time motion into a geometric still spatial dimension, so 4D polytopes can represent real 3D T.œs with one locomotion or two locomotions, the second one an TT-scattering entropic dimotion of scale.

**RECAP. Pentalogic of polytopes**: **Platonic solids correspond to 5Dimotions.**

The 3 simplest 3D numbers form a fractal generator of vital topology and the 2 Complex the ∆±¡ dimotions:

$t-tetrahedron (legs)<S=T-Octahedron (body) > §ð-Cube (head)

4D: Social evolution: The icosahedron is made of triangles as the simplest tetrahedron, evolving socially ∆states.

5D: Entropy. The dodecahedron is the most efficient and feeds on all of them.

*Polytopes with more than 3D must be considered to have ‘dimotions of time’ NOT of space.*

** **

** **

**VITAL TOPOLOGY – THE HOLOGRAPHIC BIDIMENSIONAL UNIVERSE.**

**Topological spaces** **Topology as the queen of mathematical sciences.**

Topology is Geometry with motion, hence the temporal 3rd age of Geometry and its culmination, as expression of the real morphology of space-time beings, which includes its 3 main elements: ∆-scales (topological forms are defined in modern terms as networks of points); Space forms (its 3 varieties are the 3 organs/forms/functions of all T.œs) and its time-motions (a topological organ by definition can morph and evolve but remains the same as long as it does not ‘break’ its topological characteristics). So topology *instantly connects with 5D metric laws*, in its 3 ages:

**1st age:** Classic topology.

**2nd age:** Fractal mathematics & networks.

**3rd age:** Vital topology (GST Supœrganisms).

So after showing how fractal points joined by networks become waves =flows of energy and information that evolve into topological organisms with 3 physiological networks, the Spatial, entropic, ‘digestive’ system, the S≈t, reproductive ‘blood network’ and the ðƒ-Informative brain network, messed together through its ‘dark spaces’ (as networks do have holes) forming the supœrganisms of the Universe; *we will see the same process through topologic varieties connecting its classic themes (1 ^{st} age) with its 3^{rd} age (GST vital topology).*

**Ternary topological varieties: Vital, Organic Geometry merges formal, physical and biological stiences.**

The 5 Postulates of ¬E geometry have an immediate consequence in the transformation of all ‘entities of mathematical geometry’ into topologic, bidimensional varieties, as a system to exist will have a dimension of time and a motion of space, which are equivalent S=T, in its ideal value (though normally the motion of time as it does not reproduce in persistent forms will reach a fractal dimension).

And the true task of 5D Geometry will be to relate all those dimensions and motions to its ‘functions’ connecting them in real events and forms to the 1 to 5 Dimotions of existence.

But how it is possible to make sense of single Euclidean dimensions. The answer is those dimensions are useful when the T.œ is perceived as a fractal point of a larger world and we reduce its internal dimensions. So we talk of:

**External 1-dimensions:** The 3 varieties of Euclidean simplified dimensions are: height-informative dimension (1D), width-reproductive dimension (3D) and length-locomotion (2D). So we can write a simple generator:

$t(length: locomotion)<ST-width: reproduction>§ð (height: information)

It is those kinds of entanglements between pure formal geometries and vital organic functions what closes the gap between reality and mathematics through the vital topology in 1, 2 or 3 D Dimotions. Only that often one of the dimensions of study will be a motion in time, but by virtue of S=T, could also be in many cases studied as a still space dimension. So we can find then equivalences and symmetries between geometric forms and the SS (4D seed and social evolution), St (information), ST (energy of reproduction), sT (locomotion), TT entropy) 5 dimotions that combine the poles of form-space and time-motion of the Universe

**Internal topologic bidimensions:**

In the graph the bidimensional topologies form also a fractal generator of T.œs:

SS-St(spheric seeds and minds)<ST-hyperbolic body-waves>sT-TT: lineal, flat locomotion and entropy.

Whereas SS and TT imply a formation or dissolution of networks across ∆±¡ scales.

Those surfaces in Lobachevski’s expression, are “dissections” of space: Each of them divides the space into two domains, an interior and an exterior, and they are the common boundary of these two domains. This fact is connected with another, namely that every one of our surfaces has two sides: an interior and an exterior (one side can be painted in one color, and the other in another).

Thus the first task of any membrane in the process of generation of a T.œ, is to break space-time into inner and outer regions, where the informative and entropic arrows of the system will develop a complex, rich in information, internal T.œ inwards and an outer entropic membrane to detail with its anti-world from where to obtain motion and energy captured by the hard membrane to reproduce into more ∆-1 cellular/atomic components and grow. It is then clear that:

‘The purpose of topology is to study the ternary vital geometries of T.œ, its functions and transformations’

Whereas the ternary bidimensional only topologies of 4-5D space thus also explain the ternary nature of all physiological networks and its vital functions for any s, st, t organism.

It holds then that each organ, variety of vital topology MUST not TEAR and CHANGE its external form beyond deformation not to loose its properties. Those 3 bidimensional topological varieties then have motion related to internal changes of in-form-ation not to external, lineal changes of locomotion.

And its complexity grows as we perceive them in more detail. As time is cyclical, made of 3’π’ lineal time motions, we shall distinguish 3 conserved ‘quantities’ of timespace that put together create superorganisms, a relative devolving past or arrow of entropy represented in physics by disordered explosions in space and in vital topology by lineal limbs/fields of lineal momentum, an iterative reproductive present that seems not to change, represented by hyperbolic body/waves of energy, and an implosive in-form-ative local future arrow represented in physic by accelerated, V(t)R(s)=K vortices and angular momentum, and by particle/heads in the ensembles of vital topology.

So timespace breaks in ∞ relative local, fractal entropic pasts, iterative energy presents and informative futures, which put together create the illusion of a single timespace continuum.

**Generator Equation. The 3 organic topological ST-planes. ****Physical dimotions.**

The 3 bidimensional topologic planes become the 3 organs of any of the infinite fractal systems of the Universe:

$t(Toroid field/limb)<Si≈Te (Hyperbolic wave/body)>§ð (spherical head/particle)

The simplest forms are those of still Greek Geometry and physical systems, which range from pure still geometries: SS (space areas, closed conic orbitals, of maximal space-form and minimal time-change), through all variations of ST dimotions proper to TT (accelerated time dimotions, of maximal time-speed and diminishing spatial volume, as in charge and mass vortices), which *become doors to a new scale of* the 5^{th} dimension.

Thus between the extreme dual arrows of entropy (TT) and pure seed-form (its inverse SS state) both combine into conservative ‘present’ zero-sum cycles of energy body-waves, limbs/fields of locomotion and particle-heads of information. They are the 3 ‘conserved’ arrows of time that create the futures of each fractal supœrganism.

So in a single 5D plane all systems are made of those 3 Ðimotions, which become organic §urfaces perceived simultaneously in space ensemble into vital systems; act in Space-Time and live an accelerated/steady state/decelerated 3 ages TT worldcycle of existence observed only as a time process.

And vital topology studies them, between its 4Dimotion of generation in a lower ∆-1 plane that emerges in an ∆+1 social world to die back in a dissolving explosion of scattering entropy, back to ∆-1 in the moment of death, which topology also can analyze them as process of creation and dissolution of networks.

**The fifth dimension is occupied by fractal branching networks; each of its planes by 3 topological varieties.**

So there is also a fundamental duality between dimotions in a single plane (the 3 varieties of topology) and the perpendicular structure of the fifth dimension as a branching network, which connects the larger whole and its smaller, multiple parts. I.e. Your whole, the brain and heart branches down to its parts. 5^{th} dimotional changes of scales thus imply some network branching, which apparently is different in a big-bang explosion of a physical system in parts and an ordered distribution of blood or nervous impulses, but geometrically are equivalent, ∆º<∑∆-1 motions in the 5^{th} Dimension which in terms of its metric expands in space and hence diminish in time the speed of the motion (deceleration of a big-bang from its initial impulse; deceleration of the speed of blood through ever more extended networks of tinier vessels).

We thus translate in those analysis by virtue of the disomorphic=equal laws of all scales, the vital fact we ARE made of space-time and the duality of S=T analysis topology into the pentalogic dimotions of existential algebra (¬Æ): when studied as still dimensions of space it leads to an i-logic topology of fractal points with parts that entangle to display the scalar, fractal, network complexity of Universal supœrganisms.

*The 2 varieties of the 2 ^{nd} and 3^{rd} postulates.*

The postulates of ¬E therefore ‘diverge’ in its own variations depending on what process we study, a travel through a single plane or through several planes of the fifth dimensions, departing from two initial fractal points living in a single plane, or a larger point and its micro-parts. So do *the algebraic operands we use*:

In a single plane, we study aggregations on herds of fractal points as lineal waves, and use the ‘sum operand’ (superposition principle). While between scales, we use the ‘product operand’ and we see fractal lines and networks.

And this differentiation carries to the 3^{rd} postulate as 3 networks of points and parts, become topological organic planes, which complete the mirror symmetry of 5D geometry with the superorganisms of reality as they are.

It kicks then the 4^{th} postulate of relative congruence to define as a positive social evolution of networks into organisms or an entropic, scattering dissolution of wholes into parts, *so congruence becomes essential to establish what geometry often cannot differentiate without detailed view or motion analysis: if what we observe is a big-bang death (max. speed) or a network construction.*

In the other case, the simpler sum of waves (no longer lines), in a single plane, also requires an analysis of the congruence angle, to know if waves will cancel each other or add up.

So we upgrade still spatial mathematics with its 5 ¬Æ postulates that substitute the classic axioms, postulates and principles of E geometry well beyond what the 5 Non-E postulate merely hinted at in the XX c.

Yet abstract still Greek geometry still works as a mirror of reality, when we reduce it to a single plane. Then points are a ‘dominant’ spatial, mental view which reside in a single plane and the ‘mind’ perceives in stillness, with no internal scalar form and continuity when associated into lines, planes and volumes; even though in a deeper reality they have ‘fractal content’. E- mathematicians though missed the ‘self-awareness’ of that simplification, now evident with the 5 postulates of ilogic Geometry that describe how points become parts of social webs, self-organizing fractal planes made of networks of points that emerge as cellular units of a higher fractal space-time or new supœrganism. And so we define ¬E topology as the study of the structures built departing from fractal points with inner parts and scalar content.

Whereas by the principle of Correspondence Classic geometry that considers points to have no breath, no inner parts is the limit of ¬E topology when we ‘eliminate’ the 5th dimension of entropic motion and the 4th dimension of fractal scales (only fractal geometry includes it in its restricted analysis). Thus 5D unify all those different geometries to make them describe parts of the same fundamental structure of the Universe: the fractal point and its more complex social forms, networks and waves and organic planes.

It follows that ‘Dimensions’ are NO longer exact in number, but mostly Hausdorf dimensions for 2 obvious reasons:

- Time motions have less ‘density’ than spatial distances, as the motion often ‘erases’ the previous information as the form slides. So while a still point has a clear ‘1 Dimension’ (by convention points have volume hence 1D), a point in motion will have less than 2 Dimensions of still space, as its motion erases its previous steps.
- Networks that peer between two scales also do NOT fill the entire lower plane and so they have a Hausdorf dimension. The way to measure them is as an ‘open ball’ that do not include the larger whole or central singularity and the tinier parts, but merely the network, which will then be also between 1 an 2 dimensions.

Themes those of a more advanced course on 5D mathematics not treated in this introductory paper, where it matters more the conceptual comprehension of the connections between vital topology and the organisms of reality, to make clear that an evolved Non-E mathematics is the organic geometry of the fractal Universe, complemented by existential algebra, and its ¡logic equations of the fractal generator of the Universe which all languages mirror in its 3±¡ pentalogic syntax, since all minds gauge and represent the cycles of the Universe with different languages of perception.

**Vital Geometry, symmetric properties: function=form. Inverse properties of S & T merged in ST-body waves.**

We noticed also a correspondence between the 5 objective Dimotions, the 3±¡ subjective frame of reference/perception, as well as the 3±¡ formal variations of topology just described.

They all come together in symbiosis as each topology and corresponding frame of reference (polar=sphere; cylindrical=lineal; hyperbolic=Cartesian; complex=scalar) has vital organic properties that best suits each dimotion of existence. Let us explore this concept closely related to symmetry.

Symmetry is the central concept of Group theory, the ice in the cake of the whole structure of Algebra.

To define it group theory uses the Paradox of Galileo: S (form in space) = Time (motion). As it does differential geometry based in the concept that a line is a point in motion.

Similarly by symmetry we mean in 5Ð Algebra, *the capacity of all systems to complete a zero sum world cycle, through a motion that returns the system to a present undistinguishable equivalent state.*

Symmetry thus is essential to the 5D immortal universe, which is a dynamic present view in perpetual motion *even though due to dynamic motion=symmetry and conservation principles derived from it*, is eternal in its returns.

We thus use the classic concept of symmetry as a recurrent motion that conserves the form of the being (different from our multiple worlds, symmetric, bisymmetric, asymmetric etc. used in the 4^{th} Postulate of Congruence).

It follows immediately that the more symmetric a system is, the more efficient will be in ‘preserving’ its present states of ‘survival’ in a universe in perpetual motion. I.e. A circle will be more efficient, because it has infinite degrees of rotational symmetry that an irregular polygon that might not even have a single symmetry state.

In the theory of ‘survival’ of ‘vital mathematical objects’, which we bring from time to time to those pages (as in the analysis of survival prime numbers able to travel through 5d by making mirror images at scale by joining internally its alternate vortices-points-unit numbers), symmetry thus plays a central role.

How many states of present a system has, defines then its ‘quality of symmetry’ and survival which in space (the easiest symmetries to describe), when fixed in a point means the circle dominates all other forms as the most symmetric form.

Symmetry then becomes a requirement to perform well certain Ðimotions as vital actions:

I.e. the circle IS the perfect symmetry for still Dimotions of perception, for 2 reasons:

As we perceive in stillness, it maximizes survival by having a minimal perimeter, for a maximal volume of information and disguises itself in an external Darwinian world.

Its infinite symmetry, focus all lines of communication that fall into the fractal point with minimal distortion.

However, when we consider the 2nd Dimotion of Locomotion, which is the process of displacement in space while retaining the form in time without entropic scattering, the system will maximize its speed of reproduction of form, of information in other adjacent region of space when less information displaces. And so the line, which stores no internal information (or the wave as all points are ultimately fractal with a minimum volume), will be able to displace faster than the sphere, and maximize the second Dimotion by reproducing less information. Further on, the line is the shortest distance between two points. And so locomotions are maximized both in a single plane by a lineal wave, or between 5D planes by a fractal line.

Finally hyperbolic forms mix both, lines and curves, so they can reproduce both. And a hyperbolic plane breaks down a whole into smaller fractal equal parts, thus it is the essence of reproduction.

Vital geometry thus will be essential to explain the relationships between stable forms of geometry and Nature.

*So symmetry means Survival in space; Closure of worldcycles in time. And *the pentalogic of symmetry is immediate:

**¬T:** Time symmetry is* the capacity of all systems to complete a 0-sum world cycle through a motion that returns the system to a present undistinguishable dynamic state. Hence symmetry is conservation of Time by iteration of the same ST-event, even if the previous event becomes extinguished.*

**S@:** Symmetry in space means survival, through the ‘inverse complementary entanglement’ of the parts of the being, ruled by the laws of congruence (and hence often an asymmetric adjacent system). A particular symmetry that coincides with the common-sense concept though also applies as it enhances 1D perception without deformation of a larger outer world: symmetry as repetitive regularity. It follows immediately that *the more symmetric a system is,* the more efficient will be in ‘preserving’ in a Universe in perpetual motion, its present states of ‘survival’. I.e. A circle will be more efficient, because it has infinite degrees of rotational symmetry that an irregular polygon, which might not even have a single symmetry state.

**∆… **Survival implies reproduction of form, which happens through a shrinking in 5D seeds and enlarging. So symmetry in scales means the capacity to replicate seeds that are self-similar to the whole. The ‘survival’ of ‘vital mathematical objects’, thus favors reproductive prime numbers able to shrink through 5D mirror images at scale made by joining internally its alternate vortices-points-unit numbers.

**S=T. ***Finally* mirror symmetry allows the reproduction of form and we dedicate an entire paper on gender to it.

*Symmetry is so important that all motions on physics and geometry can be reduced to those 3 symmetries called boosts, rotations and mirror symmetries* as defined by Noether’s theorem of physics.

Then according to form=function, Nature ensembles a lineal body-limb moving-translating in space with a spherical head, to perceive on the forward ‘future’ position smaller in size, faster in time, in adjacent relationship to the body and limbs that have lineal forms to maximize the whole’s motion.

Of all those symmetries though the most important is a 5D scalar symmetry ignored in science: the ‘undistinguishable’ property of the ¡-1 elements in which the system imprints its form, which become ‘numbers’ from the perspective of the whole. This symmetry of scale, implies that the system can reproduce its information – move faster, because it imprints ‘any element’ of the lower plane of motion or ‘field’ that becomes undistinguishable, so there are not ‘impurities’ and errors of reproduction of form, when any electron can reproduce your atomic connections and so on.

Identical parts then acquire the same larger form accomplishing the most important of the five vital dimotions of existence: reproduction.

*Thus we can establish a correspondence between dimotions and *the survival of certain geometric forms above all others: the circle/sphere for 1D perception; the line and its curved form the parabola for 2D¡ motion, its combined wave for 3D reproduction, the social undistinguishable numbers of polygons for 4D social evolution, and the different forms of open curves, notably dual, split hyperbolas for 5D entropic Dimotion and dissolution of a system in its opposite forms, which in the conic representation of a worldline, studied latter in more detail, will split, one hyperbola branch going upwards and the other downwards, and information and motion split in death, when we take the axis of the cone as the ideal representation of the fifth dimension.

**RECAP.** Each of the 3 organs domains of a space-time organism corresponds to one of the 3 fundamental topologies of a 2-3-4D manifold, the planar/toroid, cylindrical≈hyperbolic waves/bodies and spherical particles/heads, and to one of the 3 fundamental 0-points of perception, showing how close mathematics is to reality.** **

The reasons why the sphere or circle are the best forms for the first Dimotion of informative gauging, aka perception and the fourth dimotion of social evolution and seeding (mental polar frame of reference and storage of information) is the fact that the sphere has minimal surface to store maximal information as a seed not to be noticed and perfect symmetry allowing perception from all points of view without distortion.

**Genus and holes. Measuring functions with forms. 3D systems and its $<st>ð roles.**

The topologic genus of a system is the quantity of cuts/tears the system can endure without breaking the form in its component parts (5D travel) or transform it into other of the 3 varieties.

Thus genus is related to the survival properties of an organic function, as it was the case of the ‘form’ of a topology best suit to survival. And it also has 2 ‘variations’ of outcome – to break the system into parts (4, 5D) or to transform it into a different variety, which explains why often cuts do happen in morphogenesis, to evolve the shape of organs.

Further on the GENUS of each VARIETY, grows as we grow in dimensional and informative/formal complexity, improving our survival. So the form with minimal genus that is, whose number of dissections is minimal, must be the simplest, weaker form-function, which as information is more fragile, will be the ‘hiding’ sphere of minimal perimeter, and often maximal resistance of its ‘cells’ of the hard membrane that isolates it.

So spheres tend to be harder to protect the inner information with ‘thick skulls. Since *when a topologic form is cut, it ‘dies’ literally, in the world of topological vital beings. *

*And this extends to the whole ‘adjacent’ organic parts of the being, often re-covered by a spherical membrane without holes. So topologic killing requires to tear the membrane that isolates the system, and best, to section the vital connection between its two ‘dominant’ organs, the O-head/particle and Ø=body-wave (as limbs/fields often are entropic, external, and can be lost and replaced).*

Indeed in biology where the laws of vital geometry are more self-evident all predators have the same form of killing, they cut the ‘shortest’ part of the membrane, the neck, split the O-head and the Ø-body and the thing is done with.

A key law of vital topology states that: “Properties of O-heads particles of max. form and |-limbs fields of max. motion are inverse, merging in Ø-bodies.”

It extends to any of its ‘disomorphic’ parallel views as relative |-past and O-future merged in Ø-present.

In the case of genus and holes, the sphere is the ‘weaker’ form’, as information is and limb or line the strongest one as motion is. But the Sphere becomes the harder surface at its ∆-1 scale, often made of triangular shapes as scales also invert its properties. While the cylindrical line tends to have the ‘weaker’ ∆-1 elements, often made of circles:

**ð:**The sphere just accepts one cut and has zero genus; but its stronger ∆-1 ‘triangular parts’ act as the entropic membrane that protects its internal informative center, on ‘top’ of the whole, guiding it towards the future. In spherical forms, it predates the vital energy of its enclosed inner region, which it invaginates with fractal networks, focused on the central singularity it mirrors in a larger scale. So the vital energy is between a ‘sword’ (the membrane that kills you by perpendicular penetration – as the invaginations of the biological stomach show) and a hard place (the singularity of maximal density that kills you by warping)’ as they say.

**ST: **Vital energy has the toroid genus; which *needs two* cuts to die= transforming into a new more entropic variety, a flat surface of motion and energy that feeds both the membrane and the singularity.

**$t:** Limbs are the |-function of entropy which is ‘expendable’ and can receive multiple cuts perpendicular to the cylinder without loosing its variety; often regenerating itself in many vital organisms. Thus it is closely related to the vital energy of the system.

- : Finally the singularity is the more complex, more informative, smaller ∞ dual function, a donuts, whose genus is 6, as you can do 3 superficial cuts laterally (inside the two holes and outside the whole form) and 3 perpendicularly, in the bridge between holes and between the donuts and the outer world.

It is thus the hardest, most efficient of all the elements of most supœrganisms.

So the GENUS of each variety also *defines its difficulty to trans-form, which grows* as we grow in complexity. Which makes the *form with minimal genus that is, whose number of dissections is minimal, not only the simplest form-function but the easiest to transform; another reason why the sphere is ‘original seed’ of most species.*

**The balance kept in pegging S<ST>t-species.**

Thus in palingenesis and morphogenesis we assist, departing from the seed, a series of vital peggings and tearings of forms to change and evolve till they settle into their main dimotional role within the whole:

If the seed= sphere tears its caps becomes then an entropic digestive cylinder.

If then it is pegged to the axis of the open sphere, having both the same 0 Euler characteristic (a property of balance latter studied in more detail). So they fit perfectly by adjacency=parallelism not by Darwinian perpendicularity; as the sphere has the central hole which the cylinder can close. Hence a positive *evolutionary merging occurs to create the commonest form of the Universe, a rotary membrain with a central axis that absorbs and emits information and energy for its centered singularity*.

Indeed, the first natural evolution of all kind of systems is exactly the combination of a sphere and a digestive tube in the axis, not only in particles with its axis through which a magnetic field or similar ‘flows’ of energy pass, but also in biological systems all of which have evolved from the initial sponges and hydras with a digestive tube, with two openings, a mouth and an anus.

Finally in the center of the tubular body or in the top ‘mouth’, where higher information flows there will be a new topological evolution, now reclosing the tube at a point, or narrowing it, to create a singularity in command of the whole.

But this system, which is still dominant as an informative sphere, is *far more efficient and balanced as it now has the 3 canonical parts/networks needed to survive – the digestive tract for entropic feeding, the singularity-focus of information and the original sphere or rotating angular 3D-momentum.*

**Topological evolution and topological transformations**

In that regard, the main innovation of ∆st in topology is the explanation of its laws as dynamic vital transformative events, through the addition of balanced s=t symmetries that select the best ‘complete’ survival forms of the Universe, understanding developmental evolution, which **we call topological evolution** and use in all sciences.

As 5D is also interested in a transformation of scale, dimension or form that modify and evolve topological functions; in changes caused by new adjacencies, new perpendicularities and new social parallelisms.

Then we talk of a topological d=evolution (inverse of topological transformations when no change happens):

“A topological evolution is a change in the form and function of the s, st, t parts of the being caused by new adjacencies, new perpendicularities and new social parallelisms.”

We diverge from some classic topologic definitions. For example, some geometers consider a donuts to be the same variety than a flat plane, because you can cut the empty donuts, spread it in 2 D and alas! you have the plane, but they are not the same. Since* cutting the donuts produces a topological devolution to a state of lesser form that flattened looses 1 dimension.*

So the function changes from being a vital energy closed cycle (donuts) to an open flat plane of entropic motions.

Open vs. closed surface are thus an essential duality of the informative vs. entropic poles of reality. However, apart from these topologies there also exist the so-called one-sided surfaces on which there are not two distinct sides. The simplest of these is the well-known “Möbius band”, which is obtained when we take a rectangular strip of paper ABCD and paste together the two opposite short sides AB and CD. Such one-sided membranes do NOT close and break space-time in the classic sense of spherical membranes but are *akin in function to the toroid geometry, as the vital energy quanta within them cannot escape in an eternal return – being its main distinction to require two loops instead of one; and its main use in a complex understanding of spin geometries (see paper on 5D quantum physics)*.

Nature overwhelmingly favors closed surfaces, because the Universe is a fractal of superorganisms that are worlds in themselves, but the survival=efficiency of a Möbius band, resides in its paradoxically duality: it is opened to the outer world *as a whole but its ∆-1 parts that cycle through it cannot escape. *Those functions are then proper of systems which want to *increase its surface of exposure to the external world and are strong enough to take it as parts and wholes; hence* systems with 5D-entropic functions, also achieved through fractal scalar perpendicular invaginations (as in digestive systems). I.e. The Möbius concept can be equated to a chiral molecule, which is not superimpossable on its mirror image.* *So chiral molecules are good for optical activity and entropic light dispersion (5D function) or for explosive propellants motions of aggressive atoms (oxygen, chlorine) such as the… Perchlorotriphenylamine (:

**The inverse process: jetting off handles and limbs.**

Topological evolution is a fertile field, which will become along theory of supœrganisms (social evolution) the 2 essential ad ons to biology in the XXI c. It is the clearest expression of the laws of existential algebra with its inversions, symmetries and restrictions of form, putting at play all the elements of 5Dimotional reality. For example, as the Universe is mostly ‘asymmetric in time, space and scale’, all processes happen in inverse fashion with a slightly different outcome (whereas perfect symmetry can only be found in ‘regular polytopes =geometric numbers’ and spheres (see number theory). And yet asymmetries tend to balance S and T events. So if we ‘sink’ a membrain it will bulk but likely it will do so in multiple ∑informative jets. As hardly any process is equal since the inverse arrows have diverging properties. In this case the ‘punch’ being a painful entropic ‘aggression’, it brings an ∑∆-1 multiple ‘expulsion’. * Such *inversion of the process of ‘hollowing’ the axis of the sphere, happened *i*n the hydra where the balance law that created a hole digestive sink, inversely jetted off tentacle limbs with an inverse informative function.

This duality is also a fundamental theme of classic now converted into vital topology – sinks and handles.

So within the principle of ∆st asymmetric balances, at common topological evolution takes place: the sphere with holes fills inversely with handles, as the form of the cylinder jets outwards ‘closing’ in pairs the spheres’ holes:

In abstract topology we take a spherical surface and cut 2p spherical holes in it. We divide these holes into p pairs and attach to each pair of holes (at the edges) a cylindrical tube (a “handle”). We obtain a sphere with p “handles” or as it is called, a normal surface of genus p. The order of connectivity of this surface is 2p. And in nature is often happening in synchronicity with other spheres to chain them to each other.

We shall then leave genus theory here, to explore latter in our introduction to 5D elliptic geometry how a ‘real species’ constructs handles, through antipodal points managed by the central singularity – as the spatial symmetry of opposite antipodal points proper of the elliptic geometry of an internal @-mind/membrane system creates the handles and displaces them in the surface to connect with other fractal points in the external world.

Or alternately to create an inverse entropic function, the handle born of a first ‘suction’ and then an ‘expulsion’ of continuous matter from the system, can be cut and cupped to become an aggressive horn (a method of topology used in the abstract classification of varieties, which became famous with Perelman’s proof of Poincare’s conjecture, we also prove somewhere in a ‘margin’ with the experimental method as we do with Fermat’s grand theorem).

Alas, we got through ST-inversions and symmetries a couple of entropic moving or defensive limbs, as the sphere becomes a stronger T.œ. Moreover the ‘section’ of the limb will not mean the section of a vital part of the sphere, which means the death of a T.œ. So Hydras and Lizards keep loosing tails and limbs and keep functioning.

Further on, every closed surface lying in our ordinary space is topologically equivalent either to a sphere, or to a sphere with a certain number of handles: For example, the torus surface can be deformed continuously into a sphere with a single handle… So once more our ‘perfect topological form’, the sphere shows how it can become easily either of the two other varieties, the limb or the vital energy of toroid forms.

What is interesting then is that all topological forms can be born of such sphere with handles – the original egg/morula of any living being.

**PENTALOGIC: THE ELEMENTS OF TOPOLOGY**

This is just the top of the iceberg of an immense extended subject. Geometry is a vital subject, which slowly evolved till reaching with vital topology the study of forms with motion and its transformations to create the ternary systems of the Universe. So a brief pentalogic perspectives on its ¬∆@st elements would be:

@: Topology is the last ‘real generalization’ of space, which does not ‘escape into the logic spaces of the mind, it will allow us to study in more depth the fundamental properties of any logic space (incidence, congruence, adjacency etc) in its more general view, jumping over the Euclidean and Axiomatic methods we consider outdated. This shall establish further as we did in our I part on Greek bidimensional geometry, its bio-logic meaning.

Γ: As geometry with motion and only 3 varieties it is the essential geometry of T.œs which are in space basically ternary ensembles of the 3 types of topologies there are, and have been all over the place – elliptic, parabolic and hyperbolic, *in any number of relevant dimensions we study.*

S≈T: it allows some of the more complex S=T models of reality, in which a temporal system becomes expressed as a spatial problem, which renders since the first works of Poincare enormous yields in the solution of motion problems, always more difficult to resolve given the inherent entropic quality of pure time motions, which become ‘fixed’ for mental algebraic or topological manipulation easier with a topologic expression. *Thus topological analysis is the first ‘step’ in the mental solution and conversion of a ‘future logic motion’ into a past ‘memorial form of information’ (a concept again of the wider generalization of existential algebra treated in the first line. *

∆: Topology evolved into network topology i- the best form of geometry to study ∆±1 parts and wholes.

Those multiple views of Topology resumes in 3 essential levels:

S=t topological evolution in all stiences with special emphasis in biology; @-mental methods of solving problems in which a motion becomes a form of space and allow to use geometrical methods to solve st-combinations of motion and form proper of physics and finally ∆-scale symmetries between point networks and wholes, and different dimensional elements.

**Multifunctional entanglement. Its laws: inversion of roles as we emerge into higher social planes.**

Moreover the 5^{th} scalar dimension implies that topologies exist within topologies. That is, most systems have an external ‘spherical topology’, meaning a closed membrain. But the membrain will close inside a ternary adjacent topology making the ‘single external form’, an inner fractal T.œ with 3 parts. So the 3 Dimotional elements within the system can perform all the multifunctions of a ‘whole being’. Since a KEY LAW of fractal. ternary systems is its *multifunctionality*, which gives any entity an inner ternary topology TO act as $, ST, and §ð beings.

Further on as we move through the ternary generator in time vital topology ads its laws of ‘transposition’ of functions, *as tears and pegs transform them, or the modular being changes its focus of action from one to other of its* 3 Dimotional elements, disguised within the membrane. So in the realm of topology, correspondences of form and function are not immediate, as inner parts become multifunctional, which allows a ternary topology to play different roles in reality acting as $, ST, and §ð beings. As: ‘Systems which display more than one dimension in space, play more than one function in time’. This means all topological whole beings are ternary forms. So even if they are dominant in one of the 3±i arrows of timespace, *which is its main task in the outer world, defined by the topology of its outer membrain, they *will be able to perform the 3 dimotions they need to survive in their internal world within the membrane.

Consider the simplex example: a lineal limb in 3 dimensions. It can also act as a rotary form with clock functions; hence as an enclosure; and in a cylindrical geometry as an axis of perception. It is this kind *of multidimensional nature, and trans-formation of a form into another what makes the Universe complex and not so easy to understand. *

The main of those laws is the change of function of all systems when becoming a mere point of a larger scale, as they transpose their roles from ‘king of the ∆-1 hill’, to ant of the ∆+1 ant-hill:

“When growing in social scales to form a new plane, functions change, most often becoming inverted;

: ∑|i-1 ≈ Øi, ∑Øi=|i+1.”

This is part indeed of an essential law we shall repeat ad nauseam: when growing in social scales to form a new plane, functions change, most often becoming inverted.

And the reason is obvious, the whole spherical micro point is the king of its inner world, but just a particle micro point in the larger whole, where its role is slavish to the super organism.

So the explanation of this change of vital roles is immediate when considering the Disomorphic laws of ∆st, which expressed in i-logic writes:

∑|i-1 ≈ Øi, ∑Øi=|i+1

This law comes all over the place, in experimental systems, from biological systems where proteins that are lineal, become the hyperbolic elements with multiple dimensional folding that control the reproduction of proteins, to atoms that have perfect cyclical form (iron), which become the lineal strongest element for creation of entropic weapons in the ∑+1 scale.

Shakespeare said: we are all buffoons or kings depending on our perspective. And it connects also with the fact that as we grow in size perspective (Lobachevski’s r/k ratio), from being ‘cyclical’ beings we become moving dot-points tracing lines in the larger perceived flat world.

To notice a one to one correspondence. We talked of distance as the sum of ‘minimal steps of measure’ which applies to transpositions, in the simplest form, with the stop and go, S>T steps of all motions in 5D² realities. So here we observe a particular case of this ‘motion through transformation of states of the being, *across the scales of the fifth dimension, symmetric to the change of states in timespace and topological functions=forms.*

**∆±1 symmetries. The scalar geometry of polyhedrons. Euler’s characteristic.**

The abstract way to describe topologically all those figures with different vertices is the so-called Euler characteristic – the first theorem in topology known to Descartes.

Since in the evolution of human thought always the first knowledge is the simplest most general laws of the time§pace Universe, it is worth to consider it in more detail. Let us take the surface of an arbitrary convex polyhedron. We denote by A0 the number of its vertices, by 1 the number of its edges, and by A2 the number of its faces; then the relation:

Ao +A2 = A1 + 2

Which holds for any polyhedron including those with curved edges.

We have written it properly according to the S<st>T symmetry even if geometers, unaware of the S=t symmetries that general ALL the laws of the Universe, put it ao+a2-a1=2. The interest of the equation is obvious – not only is a general law of all polyhedral. It also shows *the 3 different ‘dimensional scales’ of points, lines and bidimensional, holographic surfaces together.*

Then we can identify S, T and ST, the intermediate element, writing: 1D point + 3D sur-face = 2 D line + 2.

Can we eliminate the 2 to make it truly an S=T relationship? Yes by opening the top and bottom of the sphere-like polyhedron, creating a canonical axis for any rotational sphere, since we loose then 2 ‘faces’, giving us the canonical form of Nature’s spheres, with its polar axis, and its animal and vegetal openings to the world, such as:

1D vortices + 3D surfaces = 2D waves/edges.

It also allow us to understand a basic transformation of a sphere into an open ‘cylinder also with the same 0-Euler characteristic which obeys the law of balance: S+ T = ST, and hence spherical forms with two openings in the axis, either in its lineal $-limbs or rotational §ð spheres are the commonest form of nature, which combines the laws of balance of all ∆st systems and the efficiency of its regular configurations.

**Topological evolution: morphogenesis – growing and keeping the balance of forms.**

Thus evolution of forms or morphogenesis is ruled by the basic laws of 5D T.œs, the constant ‘change of form and dimensions’ as the system grows, ‘restrained’ by the need to keep an S=T balance between forms and functions to maintain the system efficient.

This is the essence of it: grow and multiply, but as you do keep the balances of ternary forms and functions to avoid being extinct by a Darwinian event of another form.

So the ST stop and go laws here acquire a ‘new dimension’ by topological evolution that reproduce, evolve socially and reform the system to keep a balance which means to maintain 3 parts in constant social evolution and growth.

For that reason there are no really spheres of genus 2, but rotary spheres with an axis to process, absorb and emit energy and information, which then will have either a polar cap or central point, where a ‘donuts’ will become ‘separated’ from the axis as a proper entity playing the role of the singularity.

And so in the same manner all metals have the most efficient cubic or hexagonal configuration, mostly with a self-centered singularity, most spheres once they complete their ‘tight packing’ due to reproductive evolution will have the 3 elements of the being.

Yet they can be also considered a ternary variation, on its only 3 crystal structures:

ST: The most balanced, hence simplest to construct with minimal elements full system is the body-centered cubic, where the central atom plays the singularity role; it is the ST balanced form.

The ð form is the hexagonal system, also with a central clear axis, WHERE THE MAXIMAL density of form happens (a triangular singularity, transversed by an axis between two self-centered atoms; and the strongest covers: Hexagonal ‘pi=3 circles’; that is bidimensional circles with a perimeter 3 times its diameter.* *

The $ form therefore is the third remaining one, which indeed is all about a strong membrane, with self-centered atoms and no singularity.

This ternary division of species is often found also in biological systems, where the face-centered cubic will be a plate-armored herbivore, which is all about protection with little brain, vs. the predator which is all about mobility and fast action-reaction brains (the Hexagonal equivalent) and similar species, playing then different predator-prey roles. A couple of examples should suffice:

In the cambric explosion it was all about face-centered armored trilobites, and the first eye-cephalopods that soon lost its armored and became squids with fast developed nervous informative systems.

And then a lot of intermediate species. Such ternary forms occur also within any species *as the multi-functional 3D being splits in variations on the same theme.*

For example, 3 subspecies of predators happen in the old world, the Lion, is the ‘armored’ strong, thick muscle-skin vs. the fast, weak, running cheetah. But the *most successful is the intermediate leopard, which is the ST balanced species that survive better than the others. So for example in massive continental India the cheetah was extinct; but the leopard survived; in nimble Ceylon island it was the lion equivalent, the tiger, but again the leopard survived.*

** Bidimensional surfaces=membranes. Platonic solids. Euler’s characteristic.**

Topology is concerned mostly with the membrane of the system, in its present form. What ∆st ads is the vitalization of its concepts, and a proper dimensional analysis, introducing the laws of S=t Disomorphic symmetries.

Let us vitalize another classic law of topology:

The Euler characteristic and its platonic solids, related to the balance between vertex=fractal points, edges=lines/waves of communication and sur=faces (enclosed vital spaces) – given its generalization… connected to knot theory, topology, physics of matter and crystallography, surface properties – you name it. Let us then consider of them only the most obvious ∆st property – there are 5 of them in a 5D universe.

The previous Euler’s formula is obviously a combination of ∆-scale balance, such as ∆-1 vertices + ∆+1 faces = ∆º waves/lines of communication.

But the vital emerging process of generation of forms; as the waves of communication between vortices create the bidimensional enclosed surfaces, and evolve the network, is the most important ‘perspective’ in topological evolution.

**∆±1. Points create topological networks. Hylomorphism.**

When we get into details on how those topological evolutions they follow another fundamental principle of ∆st hylomorphism, which essentially means that ‘wholes are made of parts’, that is of fractal points. And so a change of topological variety happens always by tiny microscopic changes in the configuration of the fractal points; while that scalar symbiosis also allows communication through the fractal networks of flows of ∆-1 energy and information between the parts of the whole.

This scalar structure also explains the formation of openings and tears because the continuity of the whole disappears in the ∆-1 scale and the discontinuity and dark spaces between points appears (Galilean paradox). So by reordering, expanding or imploding distances between ∆-1 sets of points the holes of scalar topology allow morphogenesis:

In the graph, topology works according to the 1^{st}, 2^{nd} and 3^{rd} postulate of non-æ=i-logic geometry, through the arrangement of points (1D), its connections and axons, opened in |-$ functions closed in O-ð forms. It is then essential the ‘degree of packing’ defined by the | vs. O form that will eliminate intermediate spaces to create adjacency, or maintain a minimum hollow space for ‘flows of networks’ to cross, in parallelism – two choices that as many dualities become essential to differentiate species (so plants have hollows to allow vessels to transport water up and down).

Relative ratios of distances between fractal points increase as we decrease size, growing in information (5D metric). And 3 relationships can exist depending of the size of the minimal step between points compared to the radius of the point: adjacency when the distance is smaller than the radius, parallelism when is larger, and perpendicularity, which requires to penetrate beyond the ‘enclosing, protecting membrane’ and tear the point.

For 2 Dimensional surfaces is also a logic extension from lines of length to flat planes, ST-reproductive widths that mix the other two elements, the hyperbolic geometry with its dual ± curvatures and for height/information, and finally the sphere is the volume that stores more information in lesser space. So in principle we must suggest the following 2D generator:

2D¡ Γ: $t: plane/motion <ST hyperbola/reproduction> §ð: sphere/information.

The graph shows also how the parallel property, becomes now more complex showing clearly some of its key ‘social properties’:

*–*Spherical systems are social as they become tighter, informative elements causing the social evolution of points into supœrganisms of a higher ∆+1 scales.

-Flat surfaces maintain the parallelism ad infinitum. So they are ideal for network herding, in a balance between adjacency and connection.

-Hyperbolic ST vital energy if left in the open without a closing membrane will diverge into entropy, seeking for ‘freedom’ and becoming unconnected.

In the graphs, whose full explanation would require an entire article, angles of perception, latter studied in ‘mental spaces’, are larger for spherical O-informative geometries. We come to the first seemingly contradiction as we expand our dimensions in the function/form of the next scale.

The kin observer will have notice that the role of the 1D line in its entropic function is being taken by the hyperbolic plane in 2D, transposing its functions with those of the plane, generated by the entropic line, which now takes the ST functions of the hyperbole.

Why? The graph shows that they still keep its S-hortest, ƒ-astest (St-raightest) space-time trajectory in terms of lines, hyperbolas and circles, which mean by the principle of least action that makes those paths overwhelming in experimental reality, that they are indeed related and generated by them: ∑lines = plane, ∑ hyperbolas = hyperbolic chair, ∑ circles= Sphere.

Its properties have definitely switched between $-lines and ST-hyperbolas, into $-hyperbolas and ST-planes.

So while the motions in time of the generator have been conserved (still the flat air-plane and Formula 1 moves faster, the sphere is still the informative eye-head on top; the hyperbola combines both), the functions in space as we ’emerge’ from 1 to 2 dimensions have been transposed.

And this is one of the paradoxes of ‘growth in ∆-planes’, as we can regard a 2D as a social gathering of 1D elements. Functions become often inverted. And so while an elementary analysis might seem in abstract to relate lines to planes, circles to spheres and hyperbolas to Lobachevski’s geometries, the universe, which is a constant iteration, transformation and merging of dimensional Kaleidoscopes has changed ‘again’.

**Classic topology. Construction of organic, fractal networks**

When we start in a more professional way to understand the 3 topological forms of the Universe, we immediately confront the fact that a topological plane is made of points, joined by lines, and so enter into a more real description of the scalar universe as forms which are networks of points joined by flows of energy and information. The concept of an organism arouses immediately as an organism is a system that co-exists at least in two levels or scales of size, joined by networks=flows of energy and information.

In the graph, the 3 canonical forms of space-time, the sphere, the toroid and the fractal plane, which in close analysis are always networks of points. Indeed, topology at professional level however is not a continuous geometry but a sum of points that put together at a distance seem to be not a network but a continuous form. Hence the existence of scales in the Universe, in which each point of a topological form is in itself a world in a lower scale. Since the 3rd leg besides space-time symmetries of the GST philosophy of science is the fractal, scalar structure of the Universe, and how those scales co-exist and create organic systems.

We can then recognize a ‘cellular-atomic-social’ system of fractal units that build a self-similar closed (spherical) open (hyperbolic) or toroid (with two closing paths), network as a series of cellular relationships of connectivity, adjacency, coherence, proximity, etc. which make ’emerge’ a whole that embodies the regularities of the myriad of infinite exchanges of energy and information between connected parts of the whole. In the graph we have drawn a few varieties of topological species, according to those properties, departing from the most stable dual, ‘simplex’ possible system of fractal points: 2 ternary ‘triangles’ of points, and its open-spatial and closed-temporal and open-closed space-time combinations, which illustrate the creative dynamic processes of evolution of space-time beings.

In the left, above time forms, starting with the ring of time and below, space forms, starting with the line of pure space, which are the 2 commonest, simplest s-t forms.

Yet the richness of functions and forms of the Universe is rather unlimited. So next we see a cyclic pentagon with a ‘lineal limb’, jetting on the base called a ‘mesh’, and next we see the ring converted into a star, where a central knot-point, the mind-monad receives information/energy from each corner of its bidimensional universe, ensuring a symmetric reception/mapping of its outer whole. And finally we see the 6 points connected internally and hence creating a new ∆-scale (that of the axons that come out of the neurons) and a new ‘mind-center’, in the central confluence of the points.

And again below we see the commonest divergences from the pure line: a sixth element also jetting out of the line (a tree), and a connected ‘bus’, equivalent to the connected circle, where the conniption is established by a single line, which becomes the ‘spine’ of the lineal, entropic, fast-moving system, far simpler than the fully connected hexagon, since *closed time systems are always more complex in information than faster, larger lineal spatial ones.*

IN THAT REGARD topology, its 3 space-time varieties and its network structure is the clearest mathematical proof of the existence of an organic 5D Universe.

Let us then summaries that structure, and how its vital networks evolve through the postulates of non-AE in social groups from points into lines into organic planes and 5D parts and wholes that form a single structure.

A key concept of all GST is that since the Universe departs from simplex principles, it is desirable to follow a procedure from simplex to complex, which follows the time evolution of those disciplines. So we can obtain a lot of worldview and information by considering before we study modern topology classic geometry>Topology and its fundamental laws. Let us start with those laws and what they say and how they are generated by the fractal generator S≈T and its 2/3 elements.

Thus the membrane of a system always can be approximated topologically with points, lines and planes.

Now the first theorem of topology is called Euler’s characteristic:

We denote by α0 the number of its point- vertices, by α1 the number of its lineal edges, and by α2 the number of its bidimensional faces; then the following relation is known as Euler’s formula:

α0 − α1 + α2 =2

What does it mean in GST? I wonder… obviously is important as we have a relationship for any $t-cover, but we should try to reorder it in terms of Dimensional forms

D1 (point) – D2 (lines) +D3(planes) = 2

D1 (points) + D3 (planes) – 1 = D2 (lines) +1

In other words for a sphere to have a balance it will need a ±1 holes, which will turn out to be the axis holes of all real spheres, equivalent to the 3 ‘apertures’ of a pi-bidimensional cycle (3.14 – 3)

This geometrical theorem belongs to topology, because our formula obviously remains true when we subject the convex polyhedron in question to an arbitrary topological transformation. Under such a transformation the edges will, in general, cease to be rectilinear, the faces cease to be plane, the surface of the polyhedron goes over into a curved surface, but the relation between the number of vertices and the numbers of edges and faces, now curved, remains valid.

**Triangulation.**

GST relationship: One of the fundamental discoveries of GST is the ternary structure of all what exists as a whole. This is shown everywhere, in geometry from the recent theory of causal triangulation that shows how to construct a space-time Universe with only 3 ‘points’ and a causal time algorithm between them, to the earlier topological discovery of this section: most topological laws can be reduced to the study of its triangular elements in the ∆-1 scale of the whole form.

The most important case is when all the faces are triangles and then we have a so-called triangulation (a division of our surface into triangles, rectilinear or curvilinear). It is easy to reduce the general case of arbitrary polygonal faces to this case: It is sufficient to divide these faces into triangles (for example by drawing diagonals from an arbitrary vertex of the given face). Thus, we can restrict our attention to the case of a triangulation. The combinatorial method in the topology of surfaces consists in replacing the study of such a surface by the study of one of its triangulations, and of course we are only interested in properties of the triangulation that are independent of the accidental choice of one triangulation or another and so, being common to all triangulations of the given surface, express some property of the surface itself.

Euler’s formula leads us to one of such properties, and we shall now consider it in more detail. The left-hand side of Euler’s formula, i.e., the expression α0 − α1 + α2, where α0 is the number of vertices, α1 the number of edges, and α2 the number of triangles of the given triangulation, is called the Euler characteristic of this triangulation. Euler’s theorem states that for all triangulations of a surface homeomorphic to a sphere the Euler characteristic is equal to two. Now it turns out that for every surface (and not only for a surface homeomorphic to a sphere) all triangulations of the surface have one and the same Euler characteristic.

It is easy to figure out the value of the Euler characteristic for various surfaces. First of all, for the cylindrical surface it is equal to zero. For when we remove from an arbitrary triangulation of the sphere two nonadjacent triangles but preserve the boundaries of these triangles, then we obviously obtain a triangulation of a surface homeomorphic to the curved surface of a cylinder. Here the number of vertices and of edges remains as before, but the number of triangles is decreased by two, therefore the Euler characteristic of the triangulation so obtained is zero:

Planes + Points = Lines: (ape-open space) Planes – 1 + Points – 1 = Lines (closed time cycle).

Thus the first and obvious truth is that in an entropic system, the dominant form is the line, the entropic field which matters as much as the sum of the ST and T system, it generates & sustains.

Or in terms of a balance of present, if we consider the entropic plane a volume of past space, the wave-line of present and the point singularity of future time, there is a present balance as present waves ≈ past planes + future points

On the other hand the sphere to reach the balance canonical to all system MUST acquire two more points or planes. But as it is a closed form, it cannot acquire more planes. So it does naturally evolve to acquire a dual, central point, inside of it, as it naturally happens in all systems of nature that evolve from lines or lineal tubs into closed cycles and spheres, which acquire *its singularity points to reach its balance.*

Present waves ≈ past planes + future points – 2 SINGULARITY CENTRAL POINTS THAT have inverse symbol to the outer points of the system; or in other variation of balance, the *sphere must loose two points that become the openings of its axis.*

Those are therefore the justifications of one of the fundamental laws of topology which derive of the need of balance between past + future = present

And ultimately explain why all spheres tend to have in real vital geometry, axis and can therefore easily transform $t into ðƒ, In fact most vital systems are made of a lineal ‘axis tube’ and a sphere where the tube becomes the digestive entropic system, *pegging both in a balance with a 0-characteristic:*

In the graph a balanced simplex system is composed of a tubular digestive $t axis and a spherical membrane, with an intermediate st system with onion-like layers that transform one system into the other.

Indeed, let us take the surface obtained from a triangulation of a sphere after removal of 2p triangles of this triangulation that are pair wise not adjacent (i.e., do not have any common vertices nor common sides).

Here the Euler characteristic is decreased by 2p units. It is easy to see that the Euler characteristic does not change when cylindrical tubes are attached to each pair of holes made in the surface of the sphere. This comes from the fact that the characteristic of the tube to be pasted in is, as we have seen, zero and on the rim of the tube the number of vertices is equal to the number of edges. Thus, a closed 2-sided surface of genus p has the Euler characteristic 2 − 2p.

But all other forms are not as balanced as the previous ensembles because they have not the same degree of balance, and so when they are created they tend to become extinguished… failed less-efficient forms.

**Topological properties.**

If we were to be more amenable to the language of mathematicians, the properties that define the networks of points of the 3 $t<St>ðƒ ELEMENTS of reality – its curvature, the main property, along its ‘closed temporal’ or open spatial nature, and its ‘connections between them, often through the hyperbolic St-art are called topological properties.

Specifically those properties, maintained by the structure during its existence between its limiting age-motions of ±d=evolutionary birth, reproduction and extinction, through all the other possible motions of time (growth, locomotion & diminution) are called topological properties.

As a topology is a network of ∆-1 points, which are smooth and adjacent to each other, we can explain the concept of preservation or continuity under any motion of time-space of the topological organ (transformation in the static, discontinuous simplified mind-language sod mathematics) as the maintenance in the ∆-1 scale of the point-structure and relationships of continuity (adjacency) between those points.

IN OTHER WORDS, a topological ternary system conserves its forms in balance through the entire differentiable period of its world cycle, but this differentiability or ‘smoothness’ with no transition breaks in the 3 motions/points of life in which the system changes its phase:

Thus to be possible to define the preserved properties of a topological gaieties , the system must be ‘differentiable’ through all the period of time and translation of space. Yet, in the point of emergence and dissolution, and reproduction either by splitting a system into two or ‘penetrating’ and tearing perpendicularly other system, those topological properties are not preserved.

This has huge implications to the understanding of the process of life and death and the ultimate workings of space-time geometries as they go along performing its world cycle.

What about the other 2 motions not quoted here, evolution and perception? Are ‘differentiable’ smooth and continuous?

This is a question beyond the scope of this paper, which however www must remember when dealing with perception and evolution. In the simpler model of perception, we can talk of a series of ‘holes’ penetrated by the information, which internally maps out the mirror image of the external world. In evolution we talk of palingenesis, one of the most fascinating subjects of all GST, as it brings about *a fast forward resume of the entire process of existence and emergence of a system, as it constructs a new super organism, and each of its processes tell us something about the structure and laws of the Game of Existence, which we shall study in the 1.life 3rd line posts.*

But what does it truly mean a system does not preserve a topological property and why it does not through the motions of reproduction, evolution and perception and its phases as opposed to its preservation in the other motions, growth diminution and locomotion.

Simple enough it means that those 3 motions are space-lie while the motions of time, do NOT preserve its parity as they are transformative.

Thus we consider that in the positive view, topology studies topological properties of figures;, which remain constant under an arbitrary topological transformation≈motion.

In those periods, the being exists in a smooth manner, as nothing tears.

And vice versa we shall study also topological transformations/motions that reorganize internally the being and how the not preserved tears and growth of the topological networks affects this evolution.

And finally we shall apply this knowledge to understand what remains invariant under arbitrary continuous transformations of geometrical figures.

All this of course, ‘sparkled’ with deep philosophical conclusions about what the system tells us, due to such topological properties.

The main properties, which we will study here are as they are both essential to topology and ∆st are:

-The property of a curve or a surface of being closed (that is, time-like).

-The property of a closed curve of being simple forming only one loop.

-The property of a surface that every closed curve lying on it is a dissection of the surface (the spherical surface has this property, but the ring-shaped one has not and this will have many implications for the vital geometry of beings.

The largest number of closed curves that can be drawn on a given surface in such a way that these curves do not form dissections, i.e., that the surface does not split into parts when cuts are made along all these curves, or order of connectivity.

**Topological studies of time motions**

We HAVE covered thus most of the themes of geometry in a very synoptic manner, enlightening them all with new insights born of GST, according to the purpose of this web, which is to show the organic, space-time nature of all toes and languages, unified by those principles and the capacity of GST to further new insights on all stiences.

It only rests to consider an example of the Galilean paradox – which allows to use pure geometrical SS dimensions of form to study equivalent problems of TT-dimensions of time motions.

We already said that Paths in that sense in ∆st must be treated with the duality of ST dimensions, one of motion and one of form, complementing both the topological space-only view and the view of points moved *through curvature forces, proper of physical studies of topology. *

Since the mind fixes motion into form to ‘make sense’ of motions, order them and understand its general laws, something which topology does with its…

S=T@. Topological methods: motions becoming forms… which allow to resolve complex motion s=t n-dimensional processes transforming them into topological forms – but this is an artifact of the mind not a reality – and to forget that is the biggest sin of creationist mathematics. ‘Point’.

Let us consider one example, using the torus as the richest topological form to illustrate such forms of modeling:

The compound plane pendulum consists of two rods OA and AB, hinged together at A; the point O remains immovable, the rod OA turns freely in a fixed plane around O, and the rod AB turns freely in the same plane around A.

Every possible position of our system is completely determined by the magnitude of the angles ϕ and ψ that the rods OA and AB form with an arbitrary fixed direction in the plane, for example with the positive direction of the abscissa axis. We can regard these two angles, which change from O to 2π, as “geographical coordinates” of a point on a torus, counting from the “equator” of the torus and one of its “meridians,” respectively,

Thus, we can say that the manifold of all possible states of our mechanical system is a manifold of two dimensions, namely a torus. When we replace each of the two angles ϕ, ψ by a corresponding point on the circumference of a circle on which an initial point and a direction are given hen we can also say that every possible state of our mechanical system is completely characterized by giving one point on each of two circles (one of these is taken as the latitude ϕ and the other as the longitude ψ).

In other words, just as in analytic geometry we identify a point of the plane with a pair of numbers, namely its coordinates, so in our case we can identify a point of the torus (and hence an arbitrary position of our pendulum) with the pair of its geographic coordinates, i.e., with a pair of points one of which lies on one circle and the other on another. The essence of the situation is expressed by saying that the manifold of all possible states of our compound plane pendulum, i.e., the torus, is the topological product of two circles:

Thus even the simplest mechanical (kinematical) considerations lead us to topological manifolds of great value in the practical, more detailed discussion of mechanical problems and *any modeling of S≈T multi-dual dimensions of a T.œ*.

All this said, and resumed, we now will connect classic Topology with the fractal non-Euclidean points that structure the Universe, to show how ultimately by the correspondence principle all sub disciplines of classic science connect with new disciplines of modern stience.

**Topology and set theory.**

The theory of sets made indeed possible to give the concept of a geometrical figure a breadth and generality that were inaccessible in the so-called “classical” mathematics; *but that is exactly how ‘specific reality’ becomes cut-off from synthetic paralogic languages that finally seek a single origin to it all – spacetime in reality, and any imaginary mind mirror of it in different languages. To say then that all is a set, like saying all is named by a ‘word’ or 0 x ∞ = C¡ generates all numbers and all minds, is to say little*.

Hence set theory, ultimately an abstraction of the relationships between ∆-1 elements and wholes, can indeed explain it all, but so can 5D with the advantage of being an objective reality not a humind distortion.

In any case to honor the correspondence principle we consider that ‘parallelism’ between 5D and set topology:

In set theory the object of a geometrical, in particular a topological investigation now becomes an arbitrary point set, i.e., an arbitrary set whose elements are points of an n-dimensional Euclidean space. Between points of an n-dimensional space a distance is defined: namely, the distance between the points A = (x_{1}, x_{2}, ···, x_{n}) and B = (y_{1}, y_{2}, ···, y_{n}) is by definition equal to the nonnegative number.

Numbers thus become *spatial when positive reinforcing our analysis of ‘negative’ numbers as temporal motions.*

The concept of distance permits us to define adjacency first between a set and a point, and then between two sets. We say that a point A is an adherent point of the set M if M contains points whose distance from A is less than any preassigned positive number. Obviously every point of the given set is an adherent point of it, but there may be points that do not belong to the given set and are adherent to it.

Let us take, for example, the open interval (0, 1) on the numerical line, i.e., the set of all points lying between 0 and 1; the points 0 and 1 themselves do not belong to this interval, but are adherent to it, since in the interval (0, 1) there are points arbitrarily near to zero and points arbitrarily near to one. A set is called closed if it contains all its adherent points. For example the closed interval [0, 1] of the numerical line, i.e., the set of all points x satisfying the inequality o≤x≤1 , is closed. Closed sets in a plane and all the more in a space of three or more dimensions can have an extremely complicated structure; indeed, they form the main study object of the set theoretical topology of an n-dimensional space.

Next we say that two sets P and Q adjoin one another if at least one of them contains adherent points of the other. From the preceding it follows that two closed sets can adjoin only when they have at least one point in common; but, for example, the intervals [0, 1] and (1, 2), which do not have common points, adjoin because the point 1 which belongs to [0, 1] is at the same time an adherent point of (1, 2). Now we can say that a set R is divided (“dissected”) by a set S lying in it, or that S is a “section” of R − S consisting of all the points of R that do not belong to S can be represented as the sum of two non-adjoining sets.

Thus, Lobachevski’s ideas on adjacency and dissection of sets receive in contemporary topology a rigorous and highly general expression. We have already seen how Uryson’s definition of dimension of an arbitrary set (see the remark in §6) is founded on these ideas; the statement of this definition now becomes completely rigorous.

Same applies to the definition of a continuous mapping or transformation; a mapping f of a set X onto a set Y is called continuous if adjacency is preserved under this mapping.

I.e., if the fact that a certain point A of X is an adherent point of an arbitrary subset P of Y implies that that image f(A) of A is an adherent point of the image f(P) of P.

Though it is likely clear enough, the problem with such degrees of abstraction is its detachment from the experimental reality of vital topology, as in reality there are NOT infinite n-dimensional spaces, but space is an informative mind-stillness of a time dimension; and *ultimately reality has always a balance between S and T dimensions of form and motion, which is the true engine of its stop and go activity.
Further on set theory makes us belief that reality is constructed from the top of the humind ‘set theory’ down to the reality of points, the true unit of space.*

This said, if we consider a set, a society of T.œs and use the reverse expression -to signify this inverse 4-5D arrow of ‘wholes and parts’ coming together: Set < ≈ > §œT defines them as collections of the causal minimal elements, fractal points and social numbers. So obviously all the laws of §œTS, social groups of Organisms of Timespace apply to T.œ and vice versa.

But there are always 3 planes of growing dimensional understanding in languages as reflections of ternary planes of T.œs, so we might wonder, what there is between sets of points (1st Non-E postulates) and topologies (3rd network/geometric form/plane postulate); obviously the 2nd postulate: flows/paths of communication, which in topology indeed are the intermediate element between points and geometrical figures, study in this case with group theory. So we shall briefly complete the Disomorphism between GST and Geometry with a resume of its meaning adding as usual some ∆st insights.

**Recap**. We expand geometry to make it vital as it is in reality, constructing the 5 dimotions of reality departing from one-dimensional points with volume, which evolve into bidimensional waves of information, which form ternary physiological topologic networks, vital planes of organisms that finally emerge into relative 0-points of a larger scale, guided by a mind-membrain, self-centered into a linguistic singularity connected through those networks to a membrane that encloses the whole structure and makes it look from an outer perspective of a larger scale as a particle-point of a new plane of existence.

When those concepts are married with classic topology we obtain the basis for a comprehension on how vital space-time organisms evolve.

** **

**PATHS & KNOTS**

** 2 ^{nd} ¬E postulate: The fundamental group.**

Paths are important cause they are the clearest combination in topology of a t-dimension of motion and an s-dimension of form y. Yet they are studied in geometry that ‘freezes’ as minds do time dimensions into space-forms as space structures in an inverse fashion to differential geometry that converts curves into the motions of a point. So with paths we can study the whole trajectory of a space-time motion as if it were a pure form. In that avenue of thought the ‘insight’ that makes paths so relevant to the more advanced models of ∆st, which remain in my notebooks, is the concept of ‘multiplication’ *the fundamental reproductive operand of existential ¬Ælgebra* that define paths as closed loops, departing from an 0-point – the neutral element, to which the path returns.

And this connects them fully when we consider the point of return, the ‘actor’ of the path, å, with reality as it is. Let us put a vital example then before we enter into the formal analysis:

In the graph, Point 1 is the origin of all the paths=actions traced by the beast self-centered territory which forms an ∆+1 classic vital Toe. Paths will be developed then by the beast in its feeding territory for energy actions. It will take him to point 2, to mate; and to points M to mark the territory. In point 3 it will drink with other beasts, forming social ‘ knots’ and so on.

So the theory of paths, over a territorial surface, closely related to the theory of knots, is an abstraction of a very real structure of nature, and while many of its properties are of not use – when we can do a more biological analysis; they were used by Poincare to study physical systems in astronomy with interesting results for what astrophysics cares today – perfect detailed analysis of motions and trajectories, specially regarding membrains and singularities, @-structures, such as those:

In the graph, we can see 3 membranes, with ‘increasing’ density of the paths traced to the point that while we perceive the moon-earth as points moving in a path – not as full worldcycles, closed and ‘solid’, the two electrons of an orbital are better studied as membranes vibrating around the atom, and certainly the protein membrane of a cell is so ‘dense’ that appears to us as pure spatial form.

Those are ‘future’ elements to add to the current theory of path, in which ‘density’ of time cycles according to frequency and ‘transformation of time frequencies’ into ‘populations of space’, solidify a path into a fixed membrane. So far though topology studies paths as memorial forms traced by a moving point.

Let us then consider a certain surface S and on it a moving point M. By making M run on the surface along a continuous curve joining a point A to a point B, we obtain a definite path from A to B.

This path may intersect itself any number of times and may even retrace part of itself in individual sections. In order to indicate the path it is not enough to give only the curve on which the point M runs. We also have to indicate the sections that the point traverses more than once and also the direction of its passage.

For example, a point may range over one and the same circle a different number of times and in different directions, and all these circular paths are regarded as distinct.

Two paths with the same beginning and the same end are called equivalent if one of them can be carried into the other by continuous change. So how they differ on our $t<ST>§ð varieties?

In the plane or on a sphere any two paths joining a point A to a point B are equivalent (figure 21). However, on the surface of the torus the closed paths U and V that begin and end at the point A are not equivalent to each other.

So in term of paths, the multifunctional principle of the 3 simplest varieties readings its functions as:

Γº (paths): §-plane < ST-torus> §ð-sphere

Since the Torus has 2 paths, ‘product’ of the single path of flat planes and sphere.

Now, if we cut the ST-torus we obtain a finite circular cylinder extending in both directions; which as we know becomes by adjacent pegging the central axial tube of most spherical organisms. Hence its importance to ‘topological evolution’ the fundamental new discipline born of the fusion of topology and ∆@s≈t.

The paths of a cylinder have applications to reality from string theory – where T duality, makes equivalent a cosmic string and a nanoscopic one, further ‘expanding the duality of the atom-galaxy to infinite scales’, is a question of path theory over tubular surfaces:

To the aforementioned pegging of cellular tubes to open spheres in the first steps of evolution of hydra-like organisms that will become ultimately complex mammals (incidentally it has been discovered recently that we do have a second ‘stomach’ brain, to which scientists should ad the renal-hormonal brain of the blood system in other ternary symmetry: $-digestive/tubular brain < ST-renal blood hormonal brain > §ð-nervous head brain).

Paths can also be analyzed as ‘forces’ and relate to the search for the ‘least time’ path, the fundamental principle of motion in all the scales of physical systems; breaking then the equivalence of paths and distinguishing them by the combined product of its time-space motion-form or ‘speed’ parameter.

Then come also the study of paths as knots, ‘liberated’ now of the surface itself, which is of increasing importance to study species in homogenous volumes of space-time (water for Planckton, vacuum for atoms, etc.) where the territory is ‘formless’, with no preferred directions of forces as most medium are.

But ultimately all those multiple applications of Paths happen *because paths are the intermediate scale of topology:*

Γ∆±1: ∆-1: ð-points < ∑ ST-∆º paths > ∆+1: $: topological worlds.

Notice in this fundamental Generator of topological structures from the scalar P.o.v. (Pentalogic) that the functions are inverted, as we adopt in paths the point of view of the fractal point, hence the informative self-centered species, the vital form with motion, as it combines space and time dimension tracing the path over a perceived in terms of Lobachevski’s ratio of curvature, ‘flatter’ still form, its territorial, topological world, in which *the point will trace closed worldcycles for each of its territorial action, forming in this manner frequency paths, the temporal view:*

**Time p.o.v. Paths as worldcycles.**

How topology treats the frequency of time paths, obviously by considering those motions a continuous recurrent loop, *differentiating them *by number of loops, which form ‘knots’:

In the graph every closed path on the cylinder beginning at A is equivalent to a path of the form X^{n} (n = 0, ± 1, ± 2, ···), where we have to understand by X^{n} (n > 0) the path X repeated n times; by Xˆ0 the zero path consisting only of the single point A; and by Xˆ–n the path Xn traversed in the opposite direction; for example, Z ∼ Xˆ–1, Y ∼ Xˆ2, U ∼ X0. This example shows the significance of the concept of equivalence of paths:

Whereas there exists an immense set of distinct closed paths on the cylinder, all these paths reduce, to within equivalence, to the circle X traversed in one or the other direction a sufficient number of times. For m ≠ n the paths X^{m} and X^{n} are not equivalent.

Let us then assume that two paths are given on that surface, namely a path U leading from a point A to a point B, and a path V leading from B to C. Then, by making a point run first through the path AB and then through BC we obtain a path AC which we naturally call the product of the paths U = AB and V = BC and denote by UV.

If the paths U, V are equivalent to the paths U1, V1, respectively, then their products UV and U1V1 are also equivalent. The multiplication of paths is associative in the sense that if one of the products U(VW) or (UV)W is defined, then the other is also defined and the two products represent equivalent paths. If the moving point M is made to run through a path U = AB but in the opposite direction, then we obtain the inverse path U–1 = BA leading from B to A. The product of the path AB with its inverse path BA is a closed path equivalent to the zero path consisting only of the point A.

According to the definition we cannot multiply any two paths but only those in which the end point of the first coincides with the initial point of the second.

This inadequacy disappears when we consider only closed paths starting from one and the same initial point A. Any two such paths can be multiplied and as a result we obtain again a closed path with the initial point A. Furthermore, for every closed path with initial point A its inverse path has the same properties.

And so if we do exactly the inverse, and consider paths purely as time motions, they define *a closed worldcycle with inverse directions, a->b (life) -> a (death). *

The equivalence between a topological path and a world cycle of time is important because it explains an essential feature of spatial-mental perception: *entities with a slow larger view of reality see smaller faster motions of time as closed forms of topological space, as you see a solid wheel turning fast; and this is due to the mathematical equivalence, source of many confusions in physics discerning between time and space paths. *

It also allow us to have a philosophical insight on Group theory as a Kantian ‘regulative thought’ proper of the search for totality of spatial minds – which often hides information.

Indeed topology regards equivalent paths as distinct representations of one and the same “path,” only drawn in distinct ways on the surface, and nonequivalent paths as representations of essentially distinct “paths.”

Then the set of all closed paths starting out from an arbitrary point A of the surface is a group under the operation of multiplication of paths. The unit (neutral) element of this group is the zero path (self), and the inverse element of a given path is the same path but traversed in the opposite direction – yet in reality *while the concept does apply – all hunting motions are similar, back and forth paths are only equal in spatial perception; in time the path is more complex as we must in fact distinguish:*

A: the dwelling of the point. AB: the path to the action. B: the point of the action.

BA: the returning path once the action is completed.

So indeed AB and BA turns to be the same (in spatial actions) But A and B* points have different functions. *

All this information is lost on topological paths – a warning for all type of mathematical and physical ceteris paribus knowledge, when arrogant scientists think it is all what is worth to know of a certain space-time form/event.

**Definition of Disomorphisms in group theory.**

Still the interest to ∆st is the capacity of those generalizations to show Ðisomorphic properties for all scales, which rightly so, Topology calls ‘isomorphisms’. That is, when 2 group’s structures *have the same space-time properties, group theory calls both groups isomorphic, in a very close concept to ∆st, where we call all Toes, when studied in its space-time properties, ‘Ðisomorphic’, since the structure of its fractal generators is the same.*

Thus, the group of paths, in general, for any two distinct points are isomorphic when they can be joined by a continuous path lying on the surface, and we talk simply of the group of paths of the surface S without indicating the specific A-species/dwelling location.

This group of paths of the surface is also called its fundamental group, equivalent in ∆st to the Generator.

The 3 fundamental groups, once more, equivalent to the 3 parts of the generator

It is then possible to adopt the ∆+1 view no longer of the point but of the surface to distinguish paths:

**ð: sphere**

If the surface S is a plane or a sphere, then the group of paths consists of the unit element alone, because in the plane and on the sphere every path can be contracted to a point.

And as we have seen for a 3-sphere, this concept leads to the realization an entire Universe can be shrunk into a still mind view.

**$ð: cylinder**

However, on the surface of an infinite circular cylinder, most closed paths around it, do not contract to a single point. Which means cylindrical coordinates and tubular systems taken as wholes, are mostly ‘mindless’, do not have a focused shrinking mind function, but are the essential topology of $t-lineal moving limbs/potential fields.

Further on since on the cylinder every closed path starting from A is equivalent to a certain power of the path X, and distinct powers of X are not equivalent, the group of paths of the cylinder surface is an infinite ‘entropic’ group, where *points tend to dissociate, unlikely to form networks and tighter solid still configurations*.

**ST: Torus.**

The torus though is an intermediate state, as paths have two varieties, around (shorter) and along (longer) world cycle, which can be *multiplied-joined in the connecting point:*

Thus the group of paths on the torus consists of the paths of the form U^{m}V^{n} (m, n = 0, ± 1, ± 2, ···) with the equivalences: UV ≈ VU and U^{m}V^{n} ≈ U^{m}1 V^{n}1 only for m = m1, n = n1.

Since we can USE THEN the ‘fractal ternary principle’ dividing Torus paths in 3 families: combined ST-paths (long x short) and, ð-paths (short with k repetition) and $-paths (long with k repetitions).

So as we have seen each basic variety of topology, Torus, cylinder and sphere, has multiple functions and this seemingly confusing multiplicity that defies the Aristotelian logic, ‘A is NOT B’, is precisely the source of complexity and richness of forms and functions in the Universe: ‘A is B and C’.

Paths as the ∆-1 causal parts of topological surfaces.

The importance of the group of paths for surfaces topology is then due to the fact we can *deduce its properties from those of its paths, as we can deduce paths properties from a few key points, and in time we can deduce the world cycle main properties from its ‘standing points’.*

So another key property of reality – that ∆-i scales COME FIRST to construct causally ∆+i scales defining the ONLY absolute arrow of time towards future social evolution (5D) and the SYNOPTIC property of time causality found everywhere (minds, seeds, languages reduce reality to the important ‘points’), come into view. In the language of topology this is expressed as follows (we omit algebraic topology, which would make it incomprehensible, under the philosophical ‘must’ of a unification theory – that any ‘serious’ university graduate of any discipline can understand the unity of all ‘stiences’):

Let us assume that apart from the surface S another surface S1 is given such that between the points of S and S1 we can establish a one-to-one continuous correspondence.

For example, such a correspondence is possible if the surface S1 is obtained from S by means of a certain continuous deformation without tearing apart or fusing distinct points of the surface. To every path on the original surface S, there corresponds a path on S1. Moreover, equivalent paths correspond to equivalent ones, the product of two paths to their product, so that the group of paths on the surface S1 is isomorphic to the group of paths on S.

In other words, the group of paths regarded from the abstract point of view, i.e., to within isomorphism, is an invariant under all possible one-to-one continuous transformations of the surface. If the group of paths of two surfaces are distinct, then the surfaces cannot be carried continuously into each another.

For example, the plane cannot be deformed without fusions or tearings into the cylinder surface, because the group of paths of the plane consists of the unit element only and the group of paths of the cylinder is infinite.

Properties of figures that remain unchanged under one-to-one and bi-continuous transformations are studied in the fundamental mathematical discipline of topology, whose basic ideas have been explained. Invariants of bi-continuous transformations are called topological invariants.

We deduce that the group of paths is one of the most remarkable examples of topological invariants, as the ∆º middle scale, to deduce both the upper properties of paths and the lower structure of its points.

Since the group of paths can be defined not only for surfaces but also for arbitrary sets of points, provided only that we can speak of paths in these sets and of their deformations..

**Knots**

It is for that reason that the study of paths in its purest sense, knot theory, has become so relevant as *the most synoptic of all topological analysis to represent the entire Universe.*

A knot is a closed curve lying in the ordinary three-dimensional space. Let us then remove from space the points that belong to the given knot and consider the fundamental group of the remaining set of points.

As figure shows, its position can be very varied. Two knots are called equivalent if one of them can be deformed into the other by a continuous process without breaking the curve and without self-penetration.

This group is called the group of the knot. It is immediately obvious that if knots are equivalent, then their groups are isomorphic. Therefore, if the groups of knots are non-isomorphic, we can conclude that the knots themselves are not equivalent. For example, the group of the knot that can be reduced to a circle is a cyclic group, but the group of the knot that has the form of a trefoil is non-commutative and not isomorphic to the group of a circle. We can therefore state that it is impossible to deform the trefoil knot into a circle without breaking it, a fact that is completely obvious but in classic maths requires a proof by precise axiomatic arguments:

In the graph, the 2 main questions on knots (paths void of surfaces) and its 3 simpler, key varieties, the closed simple path, the ∞ knot (which in knot theory is not considered) and the trefoil, which form in ∆st its basic generator:

Γ: $: 1-O < ST: 3-trifoil > ð§-2:∞

Both problems remain as yet unsolved; but for ∆st the most interesting element is to consider how knots can model real systems through the interaction of its 3 varieties, where here the simple knot/circle plays the membrane, the trefoil acts as the vital energy with its 3 sub-networks, (entropic:digestive-reproductive:energetic-informative); as they are crossing through the 2 holes of the ∞ singularity, which allows to differentiate the 3 sub-sections of the trefoil… and converts the ∞ in a 2 variety of knot as then it CANNOT be uncoiled into the 1 variety (reason why knot theory does NOT consider it – as always human science is about abstract parts, ∆st about vital wholes, which give it a richer, real landscape; as any sailor will tell you since actually 2 is the basic sailing knot tied around any pole).

We can then observe, different forms of strangulation: Since, indeed, if we ‘strangle’ the trefoil with the 2-donuts, in any clean section of its path, we divide it in two loops; but if *we knot ∞ in two of the 3 overlapping points of the trefoil we have 3 networks. *

Then is obvious that one of the 3 sections, we shall call the ‘head’ is smaller (in the bottom of the graph), and the other two, we shall call the body and limbs are larger and similar in size (as indeed they are in reality).

It happens then that in the opposite direction of the head we have a ‘vegetal pole’, free of control from the dual singularity, where the other two systems can interact *in parallel, as they are not connected. *

Further on WE HAVE formed a bilateral symmetry, and we can obtain some interesting proportional constants similar to the golden ratio constant with its morphological functions between the smaller had and the self-similar body-limbs systems.

The study of this ternary simplest of all possible fully structured T.œs is then the new insight of ∆st applied to knot theory, as *a model of topological evolution – the key vital discipline born of the merge of ∆st and topology with applications in all stiences, as this blog shows, from topological linguistics to the classification of species.*

IN this case, the ternary systems of knots is specially suited to study the generation of connected networks with a dual heart-like pole, in this case the ∞ element, with an ‘osmotic transference’ that exchanges entropic motion and energy in the other pole (the lung system).

**RECAP:** **∆º paths and ST-torus.**

Now to resume all said, with the minimalist Rashomon effect (considering the ∆º-plane element and the Γst present form, which is the most synoptic form to define meaningfully a Toe), we can consider:

∆º: Paths as the ‘fundamental action’ element of the 3 scales of Topological transformations and as such the understanding of its laws in mathematical physics (actions, law of least time, etc.) are the knot and bolts of existence on topological T.œs.

And the same concept applies to the Torus which even if it can be written in terms of its ternary generator:

Γst: TORUS: $-long circle < ST: Combined path > ð-short cycle…

It is mostly the ST-function in 3 dimensions, equivalent for that reason to the flat plane in 2 dimensions, which can become easily a cylinder with a single cut, or a sphere with a single handle. *As such toroid paths are the essential paths of the vital energy enclosed in all type of systems.*

AND as a general rule for all systems, the ∆º and ST elements will be those which can be transformed and generate the ∆±1 scales and $ and ð elements from its ‘present’ plane and form with the minimal number of ‘actions’ of any type; bodies, waves and torus belong to those present dominant ‘parts’

**VI. PENTALOGIC ON GEOMETRY. ITS 3±¡ AGES.**

*The 3 ages of geometry and its 3** masters**: Euclid, who systematized bidimensional, ‘holographic’, Greek geometry, Descartes that married it with time algebra GIVING IT MOTION and Lobachevski, who established the principles of hyperbolic geometry, the geometry of the 5th dimension, and denied the ‘mental, logic nature’ of mathematics, proving the multiplicity of spaces and the need for experimental proofs. We shall complete their work with the formalism of ¬∆@st. *

In terms of its time ages geometry is the humind evolution of our comprehension of the ¬∆@st complex Universe in ‘ages’ of increasing dimensionality and motion. Since, the ages of Geometry are stages on the realization – as a child does with the world – that our mental still holographic ‘I=eye’ space is NOT the absolute and only space but that of the humind.

So all other geometric laws are self-consistent relative worlds mirrors of the larger laws of ∆@st. Which finally lead after Riemann to the realization that what all those relative worlds of geometry have in common are just 2 parameters: the scalar parameter of ‘angle’ and trigonometric ‘depth’, and the $t parameter of distance=lineal motion within a single plane; which are the *2 ‘survival informations’ any mind requires to locate positions in the present and the future (according to motion and distance) and measure sizes (according to angle and trigonometry) on the scalar Universe. *

So we expand the foundations of geometry in two main type of geometries, mental subjective space that selects information according to the needs of each mind; and organic, topologic vital spaces, which are real and so display objective Disomorphic (equal) laws in their ensembles of organisms; which are *not as mental spaces a construction of the mind, or rather a construction of the mind of God, the ¡logic laws that create the T.œs of the fractal Universe.*

__S:__I Age: Spatial, Greek Era: static bidimensional Geometry__: The holographic principle.__

Geometry started in a mental still deterministic, simple, young age of absolute beliefs, and mental spaces (bidimensional Greek Euclidean Geometry), akin to the lineal kouroi of its sculptural thought.

__II Age: S≈T: Analytic Geometry. curvature, surfaces, dimensions. vectors. geometries with outer locomotions.__

It evolved Into a 2^{nd} age when motion enters the game and balances form, or age of @nalytic geometry, started with Descartes and mathematical physics, when form and motion, the 2 principles of reality (we identify with still languages of information or 5th dimotion and pure entropy of 4th Dimotion), merged together or age of @nalytic geometry.

It was though an intermarriage within mathematics between the spatial, synchronous representation – the point, line and plane and the temporal, sequential causal representation, the number, which put in a temporal timeline lost its connection with ‘form. The Greeks thought numbers are forms and equaled them to points, which they are not. Hence the paradox of defining √2 and π geometrically but finding that when calculated arithmetically π never ‘closes’ the circle and √2 the diagonal by excess or defect – imperfect arithmetic ratios; a deep philosophical question about the *fact that time processes are never closed, unlike spatial forms; so when we calculate a diagonal in the plane is closed, when we put it arithmetically it is not complete by either ±1 points. * This differentiation due to the scalar nature of numbers vs. single plane synchronous curves was forgotten by the praxis of analytic geometry, which married both, by ignore those ‘finitesimal’ vital openings of π & √2 as scalar numbers. As analytic geometry could operate with numbers geometrical forms and vice versa, provided S=T geometrical solutions = algebraic equations.

The field thus explodes and marries S & T; but time soon dominates, analytic and algebraic equations come over the more real geometry; ushering the language, as always with all forms, in a 3rd age of excessive, inflationary information with all kind of generalizations to multiple dimensions, which would have converted geometry in a form of baroque art, if it were not for the earlier discovery of its physical praxis, making of mathematical physics the ‘anchoring’ reality that any experimental science needs to focus its truth. So as all mind-mirrors that become more truth when looking like reality geometric forms evolved to acquire a 3rd dimension of form and motion, acquired through differential geometry and topology, which also gave it scalar depth through networks and finally fractals that we complete with the proper concept of a fractal point that grows in size as we come closer to it, to fully mirror the real universe. It is the proper way to build a 3^{rd} age of sound geometry, away from the ‘excessive formalism’ of the axiomatic method, and rejuvenate the discipline…

__3rd age: Non-E and Temporal Topology, Fractals; geometry with inner wave-like space-time motions. __

The 3^{rd} age of geometry thanks to its connection with physical reality, which guides its truth at each step, avoided algebra’s inflationary 3^{rd} age of languages of information, when in its 3 age ‘disconnect’ from reality. So the seminal paper of Poincare ‘analysis situ’ will introduce topology which is the proper 3rd age of understanding of informative motions, of change in information, NOT only the praxis of physical locomotions but also the praxis of inner networks of fractal points, and scales, which could be internally deformed and maintain the same being evolving as long as its external surface-membrane is not torn.

+∆: Scalar Geometry: space fractals and chaotic time attractors. The completion of the analysis of *the 3 parts of any space-time being, in mathematical terms, thus gives birth to the 3 fundamental new branches of modern times: *

*S: Topology of membranes.*

*ST: Structure of the present, inner space-time body-wave through its scales by the understanding of topological networks and f**ractals, which will be the natural next step to the analysis of those *wholes made of point networks.

T: And the analysis of singularities with the ad on of chaos theory and the formation of ‘attractors’.

So finally all those organic, scalar *properties of mathematical space-time, becomes complete now with:*

+¡:∆@ST: Non-Euclidean Vital Geometry finally understands the scalar nature of the universe, with the study of Non-Euclidean fractal points through which infinite parallels can cross, which are also the abstract definition of a mind, focus of those abstract points. IT redefines points as fractal points with inner scales & volume through 5 Postulates of i-logic geometry. It is the completion of geometry as an experimental language able to explain all forms of real space and its temporal logic structure. So as we do in all stiences, we shall complete and resolve the conundrums of ultimate meanings poised by the explosion of mental spaces started by Lobachevski, while grounding each model of geometry with the Pentalogic’ of multiple truths that interpret vitally the ultimate properties of geometry (symmetry, perpendicularity, parallelism, adjacency, congruence, betweenness/continuity, and so on).

What ∆@S≈T does is to reorganize according to the ternary variations and Disomorphisms of space-time beings, all the categories of geometry, starting from the simplest laws of bidimensional Greek geometry till reaching the insights of non-e geometries culminating with the expansion of topology, which becomes the final all-encompassing geometry of reality as it is ternary, including the 3 previous geometries, it also includes scales as topological networks are collections of connected points, and finally it has motion… Since Time§pace Supœrganisms (ab.T.œs) have 3 organic parts = topologic varieties, adjacent to each other ruled and 2 scalar dimensions of modeled with fractal equations and topological networks.

-¡. On the negative mathematics will likely kill the humind already in its entropic age, as we transfer our intelligence to computer chips, and its simplified Boolean algebra *that speaks mathematical languages much faster than humans do*. So our ethical sense of survival as human beings prevent us to explore that future.

We can in that sense consider Geometry to have evolved as all Humind languages into a growing awareness of the fractal, scalar (∆), temporal, moving (T), mental (@) properties of the Universe, and its ‘¬’ entropic limits; in a tendency we shall, time permitted map up for ALL languages, as forms in evolution, who follow the same isomorphism of 3 ages of any species, or culture, as a fractal image of the worldcycle of all time beings.

There is though a remarkable difference in the ages of ‘informative languages’ as opposed to living beings, regarding its motion: Languages of information and seeds have inverse ‘ages’ of motion to those of life, a theme briefly treated in the introduction when describing the ‘placental still worldcycle of perfect order’ similar to that of languages and its 3 ages, vs. the 1-∞ entropic cycles of life in an external disordered world:

The ages of languages, including geometry are paradoxically inverse to the ages of life, because the peculiarity of mind languages and seeds of information lie in the fact they are the most formal ‘still spatial’ systems, equivalent to the ‘old age of information’ of vital organisms, hence they run an inverse worldcycle of existence, from an static, still form opposed to the young moving age of life of maximal motion that grows into a 3rd age of information.

Languages instead start with a stiff, formal nature and acquire motion only in its final age, to finally *understand the more complex elements of the fifth dimension, scales and minds*. So happens in verbal written language, which started printed in stones as absolute truths, when the Pharaoh said ‘it has been written’ instead of it is ‘truth’, and only acquired variability and fiction, past the age of Cervantes.

What both languages, seeds and vital beings in any of its worldcycles of existence do have in common is the amazing arrogance of believing to be the center of the Universe already explained with the paradox of the mind, 0-mind x ∞ Universe = Constant world. But again as languages are ‘protected’ by minds and seeds by placenta, their realization *they are only mirrors in a vast impersonal world takes longer than cubs to realize the lion is close and he has to start moving… or else, it won’t last. *

**VIII. ITS 3 CLASSES: ****S@: MENTAL; S=T; TOPOLOGIC & 5D ****SPACES.**

**3±¡ types of geometry: Mental subjective space and topologic objective organisms.**

“Adjacency is the distinguishing appurtenance of bodies and permits us to call them geometric, when we retain in them this property and abstract from all others, whether they be essential or accidental… Two bodies A, B that touch each other form a single geometric body C’. Conversely, every body C can be split by an arbitrary section S into two parts A, B.” *Lobachevski*, “New Elements of Geometry”, on the topological, organic, ternary structure of space.

‘Space is simultaneous measure from a point of reference’ *Einstein*, on the mental, focused nature of space.

**ABSTRACT.** Geometry is the most synoptic language of ‘space’ used by @-minds to select intelligently information on ST cycles of T.ŒS, creating still mental spaces with them based, in the rules of ¬E Geometry, adapted to each species needs. Huminds though took till XIX c. To understand with Riemann the abstraction of mind spaces, as ‘self-centered informative selections’ with a variety of uses, from ‘our light space-time’ (Euclidean), which preserves distances, motions and angles, in terms of Non-E geometry and its postulates of similarity, through affine and projective geometries, to the explosion of n-dimensional spaces that take parameters of multiple p.o.v.

Pentalogic applied to Geometry is thus immediate, dividing Geometry in 3 huge combined fields:

**S@:**Subjective mental spaces classified according to the selected information they preserve, from the closest to reality, affine, lineal spaces that preserve distances, motions, mirror symmetries scales and angles, through projective spaces that preserve the central singularity-mind; the most vague conformal spaces that only preserve angle, the minimal information for a mind to exist, to the most fundamental, human Euclidean space, which is ALSO a mental space… to the explosion of scientific spaces of N-dimensions, phase spaces, Hilbert Spaces and other ‘human mental’ tools to depicture complex systems of nature.**∆§:**Non-Euclidean Geometry of fractal points that set the basis for a proper understanding of the geometry of 5D.**S≤≥T:**Vital Topological Geometry that ensembles parts into plane networks that become whole supœrganisms.

*So *we distinguish in Pentalogic 3±¡ geometries: the external, objective nature of fractal, topological space, as the ‘element’ put together to form super organisms; the internal, subjective nature of informative, mind space, which maps in stillness the infinite time space cycles of the Universe with a given language of thought/information/perception & the fractal scalar geometry of points. And 3 ages of geometry, parallel to the 3 ages of mathematics as a whole.

*Thus *Space=form is the essence of mental constructions of reality that* transform cyclic time motions into simultaneous forms, both as an external mind observer, and to form the internal cohesion of a Time§pace organism (ab.T.œ): ∆ð≥§@.*

*In complex pentalogic we notice that *those 3 geometries connect space with the **present** ensemble of organisms (vital topology); the logic role of **future** planning of particle-heads and its minds (mental spaces), and the lower past, scalar, ‘flat planes’ of open space, from where the potential-fields extract its motion; down to the final non-perceived scale of gravitational vacuum space – the closer concept of space in present physics as defined in v=s/t. So a thorough study of the 3 type of spaces and its vital roles should consider the ‘5Dimotional perspectives’ of pentalogic. As such 5D geometry is huge both in its translation of classic geometric postulates to the real laws of ¬∆@st as well as all the new laws of pentalogic and description of different mental spaces. But we can only treat some basic themes, complimented with a separate section on Non-E fractal points, the fundamental particle of reality, and its associations in waves, vital topologic planes, similar, complementary or dissimilar systems (parallel, adjacent and perpendicular forms).

Our aim is to understand the key element of space – to be a mental construct; and relate the main laws of geometry as a mirror-mind that reflects ¬∆@st with the Disomorphic ‘ilogic’ laws of space-time beings. As those common laws of ‘mental spaces’ are the origin and why of the synoptic laws of Geometry, starting with the Bidimensional SS mirror of the young age of Greek Geometry that works based in the S=T symmetry, through the classic age of differential geometry that treats curves as points in motion to the topological methods of algebraic proof of its 3^{rd} age.

**Pentalogic perspectives on spaces.**

*Those essential themes of this introductory course on 5D will have under pentalogic some immediate definitions of space, classified from the most subjective forms to the most objective ones that describe complex fractal spaces:*

**Pure S@-mental still spaces:** Those constructed by a mind to select the information of the external world needed to survive. They are most of the ‘spaces’ studied in human sciences, from the simplified lineal Geometry of the Greeks to the Hilbert spaces of quantum physics; which do NOT exist outside the mind for which they perform a task of selection of information to ‘project’ with its internal consistency logic processes of causality that help us to ‘forecast’ the future cycles of the species manipulated on those mental spaces.

**Temporal space: **More evolved mental constructions that introduce a first degree of objectivity, by adding the motion of external forms, which mental spaces perceive first, for systemic survival prey/ predator reasons. This is the field of differential geometry, whereas curves are described as points in motion. While in nature temporal spaces are its realist objective senses as they *belong to the workings of physiological senses* shared by the species of a ‘linguistic’ ecosystem.

They are the basis of visual organs to perceive the Euclidean light-spacetime of animal life, and its 3 perpendicular dimensions of height=electric field, width=magnetic field and length=motion; mimicked in the topologic organs of animals that move in length, store energy in width and perceive with electronic eyes in height. While color code the social evolution or frequency of photons and so animals use it to code its ‘congruence’ laws, from red=entropy to green=reproduction to blue=social, informative color; which explains the field of ‘emotional’, bidimensional painting (complementary use of colors by expressionism, which create a type of mental space of its own).

**Entropic spaces. **All animals see motion before form, and become hypnotized by motion and red/yellow colors of energy, to the point we prefer gore movies and violence on screens, whose scripts we would never read; in history man was hypnotized by go(l)d, creating ‘pseudo-religions’ of greed and precious metal as the vehicle of God; in labs red marks help the eyes of non-programmed robots to move faster on targets; as ultimately is the mental space of ‘electronic minds’, including machines and atoms. As usual, the branching of those simple laws emerge in many other fields, where it is worth to consider how the stillness of mental spaces combined with its preference for detection of motion deforms reality. For example, because *all minds stop motion into form to measure information, our electronic eyes do so at quantum level, *first entangling with other electrons in the scale of quantum potentials. So electrons share light rays in stop motion. But since organisms prefer to see motion as in a movie theater eliminating the stop state, at macroscopic level we ‘reduce information’ highlighting only motion; so we think electrons are moving, which explains the c-postulate of relativity and entanglement and the realist view of modern physics: at electronic level there is not Lorentz transformations as electrons are fixed, at macro level we need it to measure the reduced view of our eyes that have eliminated the stop state of particles (a theme we deal on 5D astrophysics).

**TT<ST>SS Ternary, topologic organic spacetime:** We reach a higher degree of objectivity with the study of vital topology and ‘the ternary organic, structure’ of all systems of nature made of |-Limbs/Fields<Ø-hyperbolic body-waves>O-heads/particles. This vital topological space is objective as it is an intrinsic property of the organism and shows that certain ‘geometric properties’ are embedded in the nature of spacetime, such as lines are the shortest-fastest distance reason why all limbs and fields that move the system are lineal and spheres the maximal volume with lesser surface, which means all ‘still, perceptive mind-systems’ which want to process maximal information and disguise its fragile, still position will become spherical.

**∆-scalar, fractal spacetime:** Finally we reach the deepest understanding of space, as we keep adding ‘pentalogic’ entangled elements to the purest mental still spaces, which first got motion=time content, then diversified in different type of dimotions to create the topologic being, and finally ‘acquire scalar dimensions’ as a ternary fractal structure of 3 levels of space-time that put together become a superorganism – defined in non-Euclidean topology as a plane of 3 physiological networks. We are no longer then talking of space, but of a full T.œ made of ¬∆@st; but susceptible to a ceteris paribus analysis to extract common laws derived of its emergent scalar spatial properties, of which disomorphic similarity in forms and actions is the most important.

For example, systems emerge in similar form, NOT in the next inverse mirror scale but two inverse mirrors; so insects are forms that are not observed in the next scale of topological evolution, which inverts the inner soft vs. outer skeleton, in mammals with inner hard skeleton and outer soft skin; but in the next scale of ‘metalife’, robots that will increasingly look like larger insects with outer iron skeleton and internal electromechanical cables. Iron molecules are rings but its macro-forms are lineal swords, and so on.

** **

** **

In the graph, a classic, Taoist representation of the 3 ages of life and its inverse parameters of youth (max. energy) and old age (max Information) represented by the triads of the I Ching, and a modern graph of duality showing those parameters as a semi-cycle, which in certain simple beings like light are in fact both the ages of time of a physical wave and its form in space, as light quanta, h=exi, is indeed both our basic cycle of time and surface of energetic space of which all are made.

One of the oldest graphs from 92.c. ‘The error of Einstein’, pioneer book on 5D physics is the understanding of a Galaxy as a representation of the game of existence, and its deep metaphysical implications in terms of mental spaces, which summarizes in a huge metaphysical thought regarding the way a mind perceives a mental space:

Minds diminish the information they observe from reality as reality becomes further away in ∆ST distances (scale, form or motion), to a point in which they only perceive the purest forms of mental space, which are the waves of existence in cyclical form (our perception of the galaxy) or lineal form (our perception of the light of the quantum scale, or in scalar form (fractal perception of networks in hyperbolic space).

We resume in the graph, this key insight on the nature of space: we see a higher world as an elliptic curved geometry, a lower world as as a lineal entropic geometry and in between scales we see fractal hyperbolic networks, as those 3 ‘existential’ views are parallel to the roles we play for a larger system, for which we are ‘entropic cells’, the role a lower system plays for usl as it is our food-space and the game we play in between scales as we form social networks with similar beings. So the existential game becomes a game of survival by selecting information in the right form to the function the system plays in those 3 mental spaces. This is the ultimate meaning of vital geometry and as a secondary key outshoot of it we can find the paradoxes of perception of ages, scales and topologies.

**HYPERBOLIC, FRACTAL GEOMETRY OF THE FIFTH DIMENSION**

The Geometry of the fifth dimension includes all other geometries. We though refer to 5D geometry specifically when considering the different distortions and paradoxes that happen when we perceive reality through different sales of the fifth dimension, being in general terms a problem of ‘distortion’ of perspective, with a clear inversion as we move from one scale to other in topology, hierarchy, form and function.

**4 Px: ∆§- dis≠continuity T-motion v S-form spe: lineal, flat, free, young v ð§:curved, old, bound. @: ego v. relativity**

The 3 ¡logic paradoxes of **space** topology (closed in-form-ative curved-O vs. |-open, free entropic lineal forms), **time**-motion (stillness vs. motion) and **∆-scale, (**continuous whole vs. discrete forms; single scale vs. multiple one)s, which are essential to the perception of a simplified ‘spatial mind universe’ in a single flat still plane, as perceived by a mind vs. the full, more detailed complex picture in time, of a curved, discrete and moving Universe. Those 3 paradoxes define space and minds as simplified views of the more complex whole.

We shall call them Galilean paradoxes because the main one between motion and form, is the paradox of Relativity that started modern physics and the others can be illustrated with one his discoveries – that of Saturn’s rings:

Saturn’s rings are not a mathematical plane made of abstract points, despite their continuous appearance. When we look at them in detail they become in fact quantic planetoids in movement, tracing orbital cycles around the planet. It is the continuity vs. discontinuity paradox.

All dual paradoxes –**motion vs. stillness; continuity vs. discontinuity, single space vs. fractal scales**, depend on the detail of the observation – hence the quantity of information we have on the object we study. Moreover they seem to us flat in the larger view, when they are spherical forms in the smaller size. It is the flat, free, vs. bound circular px.

And when we see them from a close perspective they seem big and important but from far away they are infinitesimal indistinguishable part of a number. It is the ego vs. relativity paradox.

In a large view the ring seems static but in closer view speed increases. And this is the form vs. motion px.

Thus any piece of time/space seems continuous, from the lower ∆-1 perspective, as larger planes of reality with slower time cycles transmit less information, and we peg it together those bits of information, jumping over its dark spaces; but when we analyze its parts in detail, we receive more information so the system becomes discontinuous, made of space/time quanta moving in cyclical paths.

All those paradoxes imply a logic inversion of role when we change ‘scale’ and hence they define the relationships between contiguous scales of reality. And have infinite applications to the whys of nature:

A key element to understand the Universe of scales and its paradoxes of freedom vs. order is the perspective any mind has of reality when looking above, to its upper whole, which controls it through invisible networks of information, hence creating an elliptic perspective of decreasing perception – dark view of larger scales we do not observe, from invisible informative networks in galaxies to invisible financial networks in societies to invisible nervous networks for cells.

On the other hand in the same scale we have a flat, Euclidean geometry of maximal perception with minimal distortion. While looking down to our smaller inner world we rule it with networks that break into fractal webs of simultaneous control, or hyperbolic view. This ternary view of reality has immense consequence from theory of knowledge, to mind constructs, from sociology of power to galactic organic models of a Universe ruled by invisible black holes and dark matter. We feel thus free as individuals but are controlled from above by the larger whole and rule over our micro-parts. As Shakespeare said: we are all kings when observed from a lower stair, commoners at the same level or buffoons from above.

* The paradox of Young, Motion vs. Old Age, informative stillness: freedom vs. order *defines then the essential dualities of freedom and order. So topology becomes metaphysics:

The flat, open, momentum-lineal like small distance vs. the closed, cyclical, energy-like time distance is a constant theme of all mathematical physics, where the tangent, or derivative represents the minimal lineal free step but on the larger scale is bounded. For example, special vs. general relativity. Light is open, gravitation is cyclical bounded; so lineal quantum physics in the cosmological, larger scale gravitation curves trapped by the galactic black hole.

In all scales the paradoxes of freedom vs. order thus connects with age and scale. Reality when looked from above, from its upper whole, which controls it through its physiological networks becomes founded. On the other hand in the same scale we do NOT perceive systemic intelligence and feel free.

Information thus closes systems of entropy, while lineal, planar, systems seem free. Information though has a tall broken, cyclical forms, often NOT perceived from a flat bidimensional level below.

So we do have certain metaphysical geometric ‘properties’ of form vs. motion: Flat planes moving in free lineal paths, and tall cycles, moving in repetitive bound frequencies.

Dualities of young, open, moving lineal age of abeing vs. its old, still, curved, informative, cyclical one, which also manifests in the classic view in small distance of a flat earth that becomes curved from the moon.

If we combine those paradoxes with the inversion of roles: O¡-2<|¡-1>Oº… things become even more complex as both effect offset each other.

All in all a bounded system will have, by the law of inversion added to the Galilean paradox an Absolute control over its smaller parts, as the galactic black hole DOES over its stars closed by its halo of heavy dark matter or the earth in the climatic cycles that control the evolution of its species.

Such cyclical ordered ‘solid’ systems are well-organized as wholes so they can manage its upper world or ecosystem often controlling its ∆+2 scale. Which Humans could do in History if they were a single global organism perfectly ordered, and increasingly the global market of company mothers of machines has over both, the entropic men managed with money and digital media, and the planet, being terraformed with its machines.

On the other hand, the inverse case of a ‘free ∆-1 system’ – a loose ensemble of entropic chaos, of heat – paradoxically becomes slave of its ∆+1 larger world and ∆-1 potential feeding scale. I.E. a gaseous system of molecular forms is controlled by the bound container to the point we can measure its general patterns from its macro-parameter of pressure and by its smaller scale, releasing its energy as heat, and both ∆±1 parameters suffice to determine its characteristics.

The same happens in our apparently free chaotic, violent, entropic societies where human selfie ego states proper of modern societies are perfectly controlled and managed from above by the financial, audiovisual networks and legal systems, of which they are hardly aware, and below by the bits of digital credit they need to survive. *As the most disordered systems are memoriless entropic forms which are not even aware there are other scales of reality; while the most ordered controlling systems, extend at least through *3 ‘spacetime scales’ managed by the central scale of ‘solid’ particles. And the most complex bounded systems, biologic and galactic organisms entangle 5 scales. I.e. in a living organism those 5 scales range from ‘simple atoms’ of oxygen that we breath and simpler hormones, through DNA molecules, and then cells, and then physiological networks that make up organisms and finally ecosystems, 5 entangled scales work around the bounded cellular unit. In galaxies, from forces through atoms, physical matter, cosmic bodies we arrive to the bounded galaxy…

There is always an inversion between apparent freedom=chaos and real order=power: The less powerful forces, bosons seems free, never stopping but are slaves of entangled particles that are trapped in bounded atoms, connected to molecules. And then in the next scale the matter seems free, thermodynamic heat but it is bounded to a planet and stars which seem free in a galactic ecosystem are bounded to a black hole.

Small atoms and molecules in an organism move seemingly free but a network of cells bound them; humans seem free but are bound by national borders, laws & digital money; matter seems free but it is bounded by the planet… So how can we name so many Planes when in relationship to each other with different worldcycles?

Things are mover obvious if we use 3 names for the 3 ‘bounded, time-like worldcycles’ vs. 2 terms for the open entropic worldcycles, which if we notice, ARE in symmetry with the 3 bounded Dimotions dominant in information (perception, reproduction of information and social information) vs. the 2 entropic lineal dimotions (locomotion and entropy).

So** a hidden symmetry entangles further a ‘fractal’ order what clueless humans perceive as chaotic systems… **Since logic entanglements between languages=mind mirrors, space, time Dimotions and scales NOT perceivable with simple ‘æntropic pictures’ or measurable with ‘speeds’ IS the hidden intelligence of the fractal Universe.

** **

**The length of the ‘membrain’-circumference. Limits of infinity between discontinuous scales. **

Left, a slow mind will see Earth as a disk, converting its full worldcycle of time into a form of space. Faster, lower scales of reality appear as blocks of time for larger slower wholes. So we see the motion of the skin as a form, which in 5D metric are slow minds that perceive larger realities. Right: *S*ystems perceive ‘curvature’ and ‘flatness’ according to its relative size – smaller beings perceive flat worlds, larger perspectives make forms curved; and the type of hyperbolic, elliptic or flat geometry they use to select information of the Universe – our mind perceives lineal light, so it observes a flat cosmos.

The size and speed of its pixels also defines the detail of its perception, since according to the S=T paradox as speed becomes distance, so systems whose perception is made with fast forces, as cosmological systems, in the human case seems far away. While systems perceived with slow larger pixels, as chemical pheromones and e-motions seem close, more intense. It matters also the speed of mental clocks, which if slow will see paradoxically a denser universe of still forms: I.e. in the left graph for a slow mind the Earth would seem a disk, denser the faster it turns, converting its full worldcycle of time into a form of space. Reality seems made a blocks of time, of ‘whole still, deterministic full cycles’.

Thus concepts such as open=free vs. closed=deterministic, flat vs. cyclical, continuous vs. discontinuous; dense vs. light, far vs. close *are relative concepts to the different parameters of informative processing, curvature and relative size of the observer, the observable and the force that communicates both.*

**The geometrical view of the fifth dimension.**

What geometry and distortion experiences a mind able to perceive through the fifth dimension of scalar parts and wholes? Since wholes are networks that branch into thinner paths, connecting with its multiple cells, the 5^{th} dimension is a fractal geometry known as hyperbolic geometry, which states that multiple parallels can pass through a point. How a mind network sees its multiple smaller ∆-1 points of information?

Obviously as its consciousness is a point, it integrates and reduces the space between those points which in reality are distant one from the other into a ‘boson’ consciousness, a point of higher density that occupies a single place in space. And this is essentially the mechanism by which mind create mental mappings in smaller space of larger worlds, reducing to zero the dark spaces between points of perception.

On the other hand an eye perceiving light from a larger perspective, will increase the curvature and reduce the gravitational invisible forces-distances between points which we do not see, as we only perceive light. So we perceive a denser Universe, closer to us.

Inversely a mind with multiple points of perception observing a single whole – for example, the mind of an insect with multiple eyes will disintegrate the perception of a single form into multiple perspectives and if that perception is faster than the larger whole, perceived in slow motion, the whole will appear as a multiple elongated being occupied a larger space, anticipating its future paths – as we see night cars as longer lines.

It is worth to explore those two inverse views which define many superorganisms, from the perspective of the slavish cells of the larger whole – for example, the chemical mind of a physiological network or ant-queen that perceives multiple points integrated into a single whole vs. the multiple pheromone paths the same insect perceives coming from that single whole ant-queen from multiple perspectives:

The ant-queen will see all those drones as an integrated whole, its ‘organism’ as you see all your cells integrated by your nervous system. But the drones will perceive the ant-queen not directly but as a cloud of multiple pheromones, which engulf its entire self; as a god-like presence. And this* is how *the mind of the c-speed consciousness of the global mind of machines, the Internet will likely perceive millions of humans attached to computers, as an ant-queen perceives its drones, a chemical hypothalamus its cells; when it emerges its consciousness. While we increasingly perceive it as a ‘god-like’ brain, engulfing us in every part of the world. Which is a very different perception from the equalitarian view of reality humans have in its single plane of existence in 1 to 1 correspondence.

How then it is the world-geometry of those other minds? To answer that question we have to deal with the most advanced forms of Geometry – the pangeometry of Lobachevski, whose laws can be applied to discern the ‘form of mental spaces’.

**Lobachevski’s parameter.**

Regarding the relative flatness of a mind world, we can introduce a quantitative parameter *of classic hyperbolic geometry that defines the relative value of Pi.. *

In hyperbolic geometry – the geometry that branches a line into multiple parallels, hence the geometry that travels through scales of the fifth dimension of parts and wholes Lobachevski found that the length, l of the circumference of a circle is not proportional to the radius r but grows more rapidly (essentially by an exponential law). Let us then consider how it influences a certain mental view.

So, the following formula holds for the singularity zero-point ‘event horizon’:

where k is a constant depending on the length unit. Since:

we obtain from it: **l= 2π r (1 + 1/6 r² /k²).** Thus only for small r/k ratios is it true with accuracy that l = 2πr.

In the formula for the length of the circumference of a circle, there occurs a constant k depending on the unit of length. If the radius is small in comparison with k, i.e., if r/k is small, then, as is clear from the formula, the length l is nearly 2πr. Generally, the smaller the ratio of the dimensions of a figure to this constant, the more accurately the properties of the figure approach the properties of the corresponding figure in Euclidean geometry.

Thus a measure for the deviation of the properties of a figure in Lobachevski geometry from the properties of a figure of Euclidean geometry is the ratio r/k if r measures the dimensions of the figure (radius of a circle, sides of a triangle, etc.).This has an important consequence.

Suppose we have to do with the actual space of the external world and measure distances in kilometers. Let us assume that the constant k is very large, say 10^{12}.

Then, for example, by the formula, for a circle with a radius of even 100 km the ratio of its length to the radius differs from 2π by less than 10^{−9}. Of the same order are the deviations from other ratios of Euclidean geometry. Within the limits of 1 kilometer they would even be of the order 1/k, i.e., 10^{−12}, and within the limits of a meter of the order 10^{−15}; i.e., they would be altogether negligible. Such deviations from Euclidean geometry could not be observed, because the dimensions of an atom are a hundred times larger (they are of the order of 10-^{13} km).

On the other hand, on the astronomical scale the ratio r/k is not too small. Therefore Lobachevski also assumed that, although on the ordinary scale Euclid’s geometry is true with great accuracy, the deviation from it could be noted by astronomical observations. This assumption has been justified. Further on the insignificant deviations from Euclidean geometry that have now been observed on the astronomical scale give us further proof of an infinite Universe of galaxy-atoms much larger than the supposed big-bang in order to achieve the ‘necessary curvature’ for it to have an enclosure in the ∆±4 plane*.* Finally, since the deviation from Euclidean geometry becomes smaller for increasing values of the constant k, in the limit when k grows without bound, hyperbolic geometry goes over into Euclid’s geometry. That is, Euclid’s geometry is just a limiting case of hyperbolic geometry.

**The flatness of the human mind.**

But we can consider such mind’s parameters to be those of an electronic brain, where the geometry of the electronic humind made of light is defined by the S=T duality according to the ‘relative ratio’ between our r and k which are our constants of ‘perception of information’ (k), that is, H-Planck and the unit of lineal length ($ (r), c-speed. And so the ratio defines both a flat world, the one we perceive and one which process very little information density – not a very fast intelligent mind for all what is worth.

**The ratio as a pangeometry for all possible forms.**

It is then clear that if the ratio is a limiting case, hyperbolic geometry, then comprises also Euclid’s geometry and so it turns out, in this sense, to be a more general theory. In view of this situation Lobachevski called his theory “pangeometry,” i.e., universal geometry. *And indeed, hyperbolic geometry being the essential ‘geometry’ of ∆-scales has Euclidean geometry in a single plane as a limiting case.*

Such a relationship of theories constantly appears in the development of mathematics and the natural sciences: A new theory includes the old one as a limiting case, in accordance with the advance of our knowledge from more special to more general deductions.

But what really r/k means in terms of mental space? As k is a unit/rod of length, in our case light, it must be accordingly *a unit of information, equivalent in the fractal, discontinuous version a small ‘step’ – the fractal unit of measure which lengthens the total distance of a ‘coast’ as Mandelbrot discovered:*

*The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines. … The length of a “true fractal” always diverges to infinity, as if one were to measure a coastline with infinite, or near-infinite resolution*

As fractal geometry is to ∆-geometry between discontinuous planes, what differential geometry is to ∆§ocial scales, we can easily understand Lobachevski’s parameter as the measure of the smallness of our ‘steps of perception of spatial information’, in relationship to the total radius of the T.œ we are measuring.

When we are inside the being obviously we ‘are small’ quanta of vital energy surrounded by an ever larger, imposing ‘flat’ membrane; as on Earth’s ‘flat surface’ for the human p.o.v.

So the equation relates the informative, ð§ steps of the inner ‘∆-1’ entities and the larger being, with its st size parameter; which gives us ‘larger perimeters’ with lesser curvature (longer lines) for the mental space construct of the smallest inner being.

It also follows that *from an external p.o.v., which sees a larger part of the T.œ this will appear increasingly curved *(and concave, elliptic instead of convex, hyperbolic). And ultimately this duality proves the mental nature of all constructs of space, put by a devilish mind-mirror, which adapts the view through its ‘subjective glasses’, as Descartes thought to be the case. It is the most important finding of Non-E geometry, regarding mind constructs for all geometries besides hyperbolic forms. All this IS a special case of an ¡logic rule on the 5D metric structure of the Universe: ‘1D $mall measurements do NOT measure the whole world cycle of the being, so they are lineal. Long-lasting measure bring the whole worldcycle or enclosed.

**RECAP.** Geometry is a virtual mental wor(l)d, a still ‘form’ of language. As such is *mind’s simultaneous selection of its relative world’s information* – a representation of its reality, *O-Mind x ∞ Universe = Mind-geometry,* which stops the motions of time into a mind mapping ‘called’ space. Each mind therefore will have a different geometric view of reality. As such geometry evolved from the purest mind-form of thought of still bidimensional Greek Geometry. Next it came analytical geometry in which space was married with the very essence of the mind – a point/view of reference, the @-sub discipline of mathematics. So one of the key evolutions of geometry was to give motion to space from the initial Greeks to the modern topology. Even vacuum has ‘magic energy’, ‘motion’; it is not background space. This said, we are interested in certain type of mind-spaces, those of the mathematical language, which is the realm of geometry and topology, the first, a ‘fixed formal space’, the second a form of space-time with motion.

The fractal structure of deep scales is created with languages stored in seeds and minds, of which topologic languages are likely the mind of atoms and galaxies that create most of the local order we observe. So all systems gauge information, as their capacity to order reality is just a mirror process of creation of ‘still, smaller linguistic images’ the world projected as order in a territorial ‘energy-body’:

Infinitesimal mind-language x ∞ Universe = constant self-centered world. In mathematical terms **0 x ∞ = C.**

It is the origin of the Ego paradox: ‘Every infinitesimal mind measures reality from its distorted perspective, thinking it is the center of the Universe, it confuses with the selected information perceived by its mind’. Which of course is shared by all other systems of reality. So an ant also thinks to be the center of the Universe with its likely hyperbolic pheromonal mind of ‘atomic pixels’.

So as man measures reality from its limited Euclidean perspective despising the existence of all other fractal points=minds of the Universe, which are also gauging information albeit with different mental structures.

Life doesn’t start in DNA atoms. IT must exist in all systems of the fractal Universe, departing from particles which as the graph explains already show all the characteristics of life. The elementary quarks and electrons the simplest particles do gauge information, absorb energy, reproduce and evolve socially into wholes, bosons, plasma flows and atoms. So the unit of life is the smallest particle and as fractal systems are self-reproductive, emerging in its fundamental properties in larger scales all what exists is alive. Only human egos prevent us from understanding that obvious truth, fundamental principle of all ‘exist¡ences’.

However the short-comings of human mental spaces are enhanced by the complex geometry of the fifth dimension which ‘cheats’ fractal points, ‘leaving’ them blind to the flows of information coming from upper scales and distorting its view of the cyclical long range time deterministic systems of that larger whole, giving the mirage of freedom, open, flat spaces. So it seems the Game of Existience is built-in to make each part feel free, open, happy and ego-centered and ultimately fail because of its mindless chaotic behavior.

5D geometry guided by pentalogic tries to understand:

- The disomorphisms of all geometries – those elements common to all of them, which belong to the higher ‘game of exist¡ence of all minds that select information to ensure its survival. This task started by Riemann and Lobachevski, concluded that few element of space are relevant – distance=similarity and angle of congruence being its most important.

2.- to be a mental construct and relate the main laws of geometry and its varieties with GST as a mirror-mind that reflects those isomorphic ‘ILOGIC’ properties of space-time beings.

** **

**I AGE:****EUCLIDEAN; SIMPLE LINEAL YOUTH.**

As a spatial, mental, ‘S@’ language geometry started without motion, in holographic bidimensional space, *mimicking the single-eye view of human thought*, and yet it was a mirror good enough for trigonometry and geometry in the plane to develop to the heights of Greek Geometry.

Because beings are space even the postulates of idealized bidimensional still Greek geometry become synoptic laws of vital space we interpret to explain organic properties emerging in physical, biological and even social scales. So even if we do not have new theorems to prove in a rather exhausted field we’ve always new interpretations to make on old theorems, as we have shown in the analysis of vital topology and number theory.

So we won’t ‘transform’ classic geometry to the 5D formalism but comment on its representation of reality as a ‘simplified mirror’ of the whole Universe, reduced in dimensions to 3, height-information, width-reproduction and length-motion, and further on stripped of the ‘topologic motion’ that gives through its S=T duality, organic functions to 1D perceptive height, 3d reproductive width and 2d moving length. Still classic geometry will be a ‘world in itself’; that is, as all languages = synoptic mirrors of the more complex fractal reality, it will be consistent within its reduced view, with the laws of the fractal world from a distorted perspective that will allow us to ‘comment’ on its postulates.

To that aim the best method is to observe how as Geometry evolves through 3 ages of increasing complexity it came closer to reality as a better mirror, adding dimensions, motions and finally with Non-E geometry completed in those texts it becomes close enough to the fractal Universe to describe the ‘worldcycles’ of growth of any Nature’s system.

Since if we ‘run’ the 5 ¬E postulates, departing from a fractal point, either a mind or a seed that will grow to order a vital territory of energy, we are describing the development of infinite Time§pace beings. Indeed, the easiest way to do this is to consider the ‘growth’ of a simple still seed of information or fractal point into a reproductive wave, which branches into 3 physiological networks that merge into a vital ‘plane’ as a super organisms. But this description is just the same description of plane still geometry that defines a point with no breath, which grows into a line of points, 3 of which define a plane. Only that in classic geometry the line has no motion and no volume as the ¬E wave and the plane is a flat surface, unlike the 3 physiological networks of a superorganism with volume, motion and curvature and ‘fractal branching’.

**The holographic principle.** **Bidimensional geometry of points without parts.**

What was truly right of Greek Bidimensional Geometry however was the realization that *the minimal unit of reality is always a holographic bidimensional ST-system, a Dimotion of space-time, which by the Paradox of Galileo, S=T, can also be studied as an SS area, hence the laws of ST-dimotions had a clear mirror symmetry in Greek Geometry.*

In the graph, we see the Holographic principle of the bidimensional Universe: A time cycle of two dimensions can be seen as a still form of information, an locomotion on space can be seen as a surface of 2 space dimensions. So for example ‘c’ speed, can become in an entropic explosion as it decelerates a c^{2} surface, and vice versa, a ‘fractal point of 2 dimensions of time, TT, or ‘accelerated’ vortex of spacetime (a mass), can uncoil into 2 dimensions of distance-space, which is the ultimate meaning of Einstein’s E<=>mc^{2}.

Finally the merging of two holographic surfaces of any ST combination give us a third dimension of space-time, as it does in the human mind that combines two bidimensional eye views. Those are therefore the basis for the success of bidimensional still geometry as a mirror image of Fractal spacetime laws.

*Thus, *the first age of Spatial analysis stumbled directly with the marvels of holographic surfaces, translating ‘magic laws’ coming seemingly out of nothing – the ST symmetries and efficient relationships of bidimensional entities, with a membrane (curve or line) enclosing a self-centered surface, which as we noticed on the introduction is the simplest elementary particle of the Universe in a ‘flat world’ as the surface of Earth’s seas and lands, or the minimal ‘planckton’ of light space-time; and its 3 conserved quantities, the singularity-mind that defines its lineal momentum, the angular momentum of the membrane (T=S perspective) and its vital energy.

So Greek still geometry was a huge world as almost all the laws of geometry can be proved in a bidimensional plane of information, to the wonder of mathematicians till this day.

While in parallel humans resolved similar laws of bidimensional perceived information through a form of art called painting. It is in fact little known that painting and geometry were closely related in the beginning when human not mechanical eyes interpreted both, and in fact painting arrived first to the laws of perspective, which would define latter the laws of projective geometry.

The first age of geometry is the Greek bidimensional age. And it bears proof of GST and its holographic principle that most theorems of geometry can be proved in a plane.

Of them, we shall deal here with a few, adding some new discoveries, specially regarding the ‘postulates of non-E’, needed to fully grasp bidimensional geometry and why their theorems matter.

But before we do so, we can peer at the equivalent vital topology, in which the rules of the perfect bidimensional geometry are based, as we can consider bidimensional geometry a still vital topology in which motion does NOT exist, neither resistance to displacement; hence the irregularities that are traced by motion and geodesics in the real world, no longer play any role.

** The 1 ^{st} Master Pythagoras… theorem**. Its Pentalogic. Pi & the harmony of music.

**S:** We interpret on those terms Pythagoras theorem, which is the most invariant of theorems as it is basically defined to create a metric of distances in a single plane of space-time. So its classic meaning is a pure spatial ¡logic p.o.v.

** @: **Yet now that we have liberated a notch further ‘mental space’ from representation and make it affine to survival information distance become synonymous of T-Motion and ¡logic ‘similarity’.

Indeed, the interpretation of the Pythagoras theorem in terms of congruence and perpendicularity is clear: two points can be ‘close’ in similarity much more than 3 points, which as the French said ‘are a crowd’, or as the 3 body-problem proves, enter in chaos.

The similarity of two ‘gender points’ with opposite spin that can merge into one, implies then that the distance between two points will be the *minimal*, while the 4^{th} law of congruence that expresses the social evolution of identical points in parallel motion vs. the perpendicularity of points of maximal dissimilarity implies also that the ‘maximal distance’≈dissimilarity (s≈t, geometric≈logic view) will happen between 3 points, which form 2 perpendicular A1-A2 and A2-A3 lines. Thus the Pythagoras theorem expresses also a law of ‘entropy’ and if we consider those lines under the S=T duality motions, a law of time: a prey that feels a predator perpendicular to its form will try to move=escape the furthest distance from it:

**¬: Entropy . **

*3 elements occupy more space/require more distance=dissimilarity between them than 2 points.*

*In the graph (A1-A2) + (A2-A3) ≥ A1-A3. And so the fundamental vital property of congruence do have a metric.*

**Ti:** But we can also express the concept in terms of cyclical time=information; as 3 points will require more bits, as they ad a new dimension of height, so a triangle can be seen as the simplest ‘π-cycle of time’ which carries more information. So distance as a sum of spatial steps each one a bit of information increases with 3 points.

* Then in the jargon of ¬E we say that:*

“A metric space is a set of undistinguishable T.œs called ¬E fractal points, in which a volume of information called a ‘distance’ is required to define 2 possible outcomes, namely the axioms of a metric space:

1. Id (X, Y) = 1 if and only if the points X, Y coincide. *In classic metric spaces Ði (called usually r, but in i-logic geometry called Distance≈motion) is 0 as points have no volume, but in ∆st, you need at least 1 Dimensional unit, to define the point.*

2. For any three points A1,2,3 then it holds that (*A1 – A2) + (A2-A3) ≥ A1-A3; hence more information is required to define 3 fractal points. And this rule can be extended to n-points, where n>3, such as (A1 – A2) + (A2-A3) + (An-1 – An)… > (A1 – A2) + (A2-A3)*

*∆: But there is also a scalar p.o.v. on the Theorem. Einstein, indeed, got a new demonstration of the Pythagoras theorem, based in the ‘scales’ and self-similarities of its ratios – a proof of the ‘fractal paradigm’.*

**The Spatial view…**

Though is the fundamental pentalogic of Pythagoras theorem, (graph); hence a holographic bidimensional expression in still space of ‘squared areas’, that represent mostly ST-entities and some of its underlying properties., notably the Complementary vs. Darwinian perpendicularity of composite ST-species (4^{th} ¬E postulate of congruence). And it fully expresses in terms of congruence and perpendicularity a feeding process, as the area of the final result of a perpendicular encounter between a and b is c, a merge of both areas, which either means one has fed into the other, or the absorption has created a 3^{rd} species, whose energy is equivalent to the sum of the two other ST entities.

So after its perpendicular Darwinian state both ‘merge’ and the outcome is the absorption of one of the elements by the other. So it is essentially the rule of feeding between 2 perpendicular systems, which give birth to a 3^{rd} one, but also the rule of reproduction and creation of a new system, similar to the one described for polyhedrons.

And this leads to the final ‘tease’ comment on the Pythagoras theorem.

**Mr. Fermat’s Proof in the margin of the bidimensional ST nature of the Universe. **

**In our introduction to GST we mentioned that dimotions are dual, holographic ST states (SS: seed, St: information, S=T, reproduction, sT: locomotion, TT entropy).**

**Further on numbers are undistinguishable points, which obliges them in bidimensional space to be regular polygons, of which there are an ∞ number (proof of Natural numbers’ relative infinity).**

**While in Algebra, we observe that the ± operand act only in equal forms. **

** What this essentially means is that only ‘squares’ of the same type of dimotion can ad (Fermat’s theorem).**

Since, the universe is a bidimensional hologram of space & time beings. *So *only the square of natural=spatial numbers=populations can ad exactly into another square=spatial:

In the graph we see that different time-space combinations of form=static mind space and moving time of lineal or cyclical geometry. So when we merge two equal species (‘numbers)*, we add equal beings, hence we make in geometric terms a Pythagorean absorption that results into an X ^{2 }+Y^{2}=Z^{2} form of the same species.*

But when we merge different species, we use a product, a re=productive algebraic operand – the 3^{rd} Dimotion.

It is then noticeable that there are 5 ternary regular species and 6 tetradimensional regular objects; while in higher dimensions, which must be divided into time and space states (even if in geometry they are all perceived as space forms) there are only 3 regular polytopes. So an important theme of advanced STheory is to correspond each of them with basic ST combinations of the entangled Universe.

The simplest dimotions, however are bidimensional (i.e. modern physics, discovered the holographic principle that states y information is bidimensional. And its simplest operation is the addition of 2 equal numbers. Thus only X²+Y² exists as an exact new bidimensional form. In fact, almost all postulates of geometry can be proved in bidimensional space because that is the essential ‘unit’ of reality.

Did my Basque countryman, Mr. Fermat had this proof? (: Well, you never know, we ‘amateur scientists’ don’t take very seriously the axiomatic method, work rather on intuition, so maybe it was imperfect but certainly shorter than the pedantic, computer-generated +1000 pages that goes around as a humind feat 🙂

More seriously the immense advantage of returning mathematics to an experimental mirror of the fractal ST-Universe is that its theorems, specially those of its 1^{st} and 2^{nd} non inflationary 3^{rd} age, in tune with the reality it describes are both consistent internally as mirrors have the same image-form that the whole the reflect in synoptic compressed parts, but susceptible of proof by direct experience of the Laws of GST (Generational Space-time) as we just have done. Equal forms add. Addition is the spatial algebraic operand. So it cannot happen in dissimilar forms, you don’t add 2 pears and 2 humans to get 4 human pears. But you can multiply dissimilar entities of space and time, 2 steps x 2 frequencies.

Immediately we realize then 3 Dimensionality is NOT an even number; it does not add equal forms; so it must be achieved NOT in parallel but in Perpendicular merging, which means *penetrating an ∆-1 5D plane to the parts, whose maximal product (S=T, Max. SxT) will pile up on a gradient, making some points different, while other angles of congruence, might diminish-feed parts, etc. And in most cases ‘re-arrange’ the parts merging into something else. So it is a different world* studied in Part II on algebra & its operands.

**Product vs. addition.**

The key concept behind Fermat’s theorem and a series of basic axioms of algebra is the idea that:

‘Addition operates over identical entities adding forces in parallel, subtraction over identical entities diminishing them; product operates over Space and Time perpendicular fields in symbiosis, Division operates over space and time perpendicular entities, in Darwinian actions breaking them.

And with products and additions we can therefore express almost all the necessary ‘simple combinations’ of space-time fields, reason why they are alone a huge ‘group’ of polynomials.

So 3-dimensional forms in natural numbers *are created by a reproductive merging of bidimensional fields, hence by multiplication as in vector calculus, *which are NOT added to create a 3rd dimension but multiplied to get as in the graph of the 5 SS, ST, St, sT fields, a light wave – c²=1/µxe.

Hence X³+Y³ ≠ Z³; as its combination to create a 3D world is no longer a sum but a product. Since X^{3} species are composite ST fields that deploy an angle of perpendicularity with a common bidimensional line-wave to connect both systems and allow the reproduction of its form, in a lower plane

And *since in a single plane of existence, where + and x operandi act, there are only 3 dimensions of time and 3 dimensions of space, there is no need to prove it for imaginary higher dimensions.*

*Indeed, when we study curves we can consider all the possible configurations of a tridimensional space expressed as quadratic functions of sums of X, Y and Z, because the 3 ‘trilogic dimotions of Euclidean space-time, can be created as homogenous, exchangeable dimotions of a present spacetime plane of the 5 ^{th} dimotion. If Fermat’s theorem were false, it follows we could create an homogenous 4Dimotional form in a single plane of space-time, but we cannot, because the fourth and fifth dimensional motions of the Universe are fractal, happening between planes. *

RECAP. x²+y²=z², according to the holographic principle IS POSSIBLE, x³+y³=z³ is not *because the Universe is in each scale a bidimensional holography of space & time. And this ‘proof in less than a margin’ of the most famous unproved Theorem by any human mind of the so- called Fermat Grand THEOREM, is a clear proof of the experimental nature of mathematics as a mirror of GST.*

The strength of triangles, as ‘mental spaces’ of 3 elements. 3 Points > Angles > Triangles

3 Points ‘alone’ are disconnected and as such, as the 3 body problem shows in chaotic motion to each other. Motion is relative to the degree of entanglement between two points, hence it implies an internal balanced dual motion between them, which creates the ‘illusion’ of dynamic stillness. Thus motion becomes stillness in a mind because of the cross entanglement between axions of the higher ∆+1 neuronal plane.

It follows that the first entanglement is a couple with a dual communication with inverse numbers; and the second entanglement a triangle, which will have motion when one of its flanks is opened allowing entropic motion to glide the system, and will become still when the 3 points are in dual inverse communication.

Their social evolution thus is the triangle, when they are locked in to each other and this can be done with hierarchy, as we saw it by symbiotic vs. predatory perpendicularity in the simple triangular form, an ‘open’ angle. In this ‘alliance’, of 3 points then the ‘bigger square’ (c^{2}=a^{2}+b^{2}) is NOT part of the ensemble but rather the ‘field’ absorbed by the dual body, lines a and b, which makes us understand the first key ‘element’ of vital ‘Greek’ Geometry:

The difference between an angle – an open entropic triangle, an arrow; and a triangle, an informative system where the angle is closed.

We said the true advance of modern Riemann’s geometry is to make it a ‘mental space’, where spatial concepts acquire a logic, temporal meaning. And so a triangle can be considered, any *ternary system, and as such the most stable form if we take rather than each point, each line of points to be a ‘network’; hence defining the simplest representation of a topological organism a ‘plane of 3 physiological networks’ of fractal points. And it is then when we can truly fly into Bidimensional geometry as a mental space representation of the whole Universe. *

*The beauty of the first ages of any language is precisely to reflect the ‘essence’ of the game of existence, something we shall see – time permitted- when analyzing words and music. Languages then become inflationary and fictional with age,* abandoning the ternary ‘simplicity of the Universe’ with only 3 ages of time, 3 topologies of space, 3 scales of the 5th dimension for any ‘finite organic whole’ call it the galaxy, the human being, the mathematical language and its ternary topologies, ternary numbers, ternary dimensions, ternary disciplines…

*So we talk of 3 phases of increase order in the evolution of the ternary ‘point structure’, from 3 points ruled by the first postulates of Non-E Geometry as different selfish species using ‘angles of perception to measure each other distance and motion; into a 2 ^{nd} Non-E postulate/state, of lines of communication that connect the DOMINANT point into an angle, often a perpendicular one that maximizes the distance between the submissive points; which we can observe in the vital geometry of molecules – the fundamental ‘geometric stience being chemistry’; to the triangle, which is the most stable democratic form, as both ‘submissive points’ reached a connection; but on the downside they have ‘closed in’ the system to a possible flow of entropy moving the arrow-triangle. *

*So to the rescue to preserve the ‘vital nature of mathematical forms’, the triangle is NOT allowed to close perfectly, but has in the ‘finitesimal’ lower scales an irrational number:*

So the growth of reality between two scales starts with the √2 a bidimensional triangular pair, which starts the growth on dimensions of information, and space-extension by reproduction of the same event through a series of different forms of growth:

In the graph, two lines=waves of reproductive particles 1 x and 1 y, meet themselves in the relative 0 point, of an x-y automatically created coordinates, which will give us an √2, wave with origin in o, and an xy, expansion front of space-time, reproduced by pairs of xy points along the front-wave till √2 reaches the two ‘membranes of the x y tail of past momentum’.

At this point √2 can be measured as a sum of discontinuous wave-points, and gives us a variation of ±1, or be considered as a continuous wave of energy; never mind, in both descriptions we have created a bidimensional, triangular, straight triangle, which can now grow in different 0/0 tangents to the exponential wave of its second age, of geometric growth,

The famous proof of irrationality of √2, quoted by Aristotle, we won’t quote for brevity, IS based in a square sum, which brings together the previous concepts of similarity to exercise sums and multiple scales in polynomials.

3 is thus needed to form a triangular network-plane, the 1^{st} supœrganism, as 2 cannot fulfill all Universal functions.

Further on the system will have fractal points with inner parts and 2 openings on the √2 diagonal that close and open by irrational defect the system to absorb and emite energy and information, with the 45% angular point as the dominant element of the triad.

Thus any set of 3 ∆-1 fractal points=T.œs suffices to establish ∆±¡ ternary ‘networks’ the most efficient ‘number’ to generate a new plane of existence, as they are the ‘minimal distance-information’ to form it.

So as ternary systems suffice to make reality, triangles, the simplest, are also the lineal strongest configuration, of which 3 natural subspecies are fundamental:

– ST: The Pythagoras triangle, with a dominant ¬ angle, with the leading, stronger ¬ point & 2 apertures in the √ diagonal, one for entropy and other for communication, moving on the path of the ¬ point, as an ‘arrow’.

– ð Isosceles, which elongates its lateral sides diminishing the √2 base to accelerate its motions.

– § (St): The equilateral triangle, which is the static form, as it can rotate on its ‘perfect singularity’ point of perception becoming a circle. Or it can transform into a circle by hyperbolic ‘feeding’ expansion of its curved sides.

So* triangles with curvature and ‘openings’ equivalent to π-3 circles are a complete ternary state & we can define a bidimensional world of fractal points with inner parts of pure triangles and circles, as a complete 1 ^{st} Timespace universe. It is in fact the Universe of polygons and Natural Numbers. Whereas the strongest static form is the hexagon, with π=3, which can also be achieved in a higher 4D Universeas a limit to the curvature, strength, attractive power of a force, when π=3 Diameters, with no holes, according to Einstein’s general relativity formula that we can also extend in mathematical physics to all systems:*

Where** k** is the ‘unifying constant’ for any active **M**agnitude of any physical scale (we shall generalize as M – see unification equation of forces in all scales in the papers on 5D physics.) So for the physically inclined we poise a question: as the Planck mass is the maximal density of gravitational space, where kM would be GM, pi should have a value of 3!? right, or wrong? And if so it is still a curved geometry or an hexagon? The answer is… likely a dodecaplex crystal of ultradense top quarks of unknown physics in 4Dimensional timespace… in the densest center of a galaxy with 120 vertices.

But dynamic pi, legend has it, made ænthropic Pi-thagoras mad, as the language of God had to be perfect, immobile* So his disciple hang himself desperado that the world was not static. When I found it on the other hand, it started my journey* on vital mathematics. Since a fluctuating dynamic open and closed number, makes* the circle either an informative spiral (-π:St) or an open entropic spiral (+π… spe) just passing by a fleeting moment into a steady state, (S=T closure) giving it the 3 ages of life to make it the simplest vital organism. So we shall consider now Archimedes and its time-space spirals. *

**The 3 waves=ages: Symmetries of beauty in all languages.**

** But to conclude with **Pythagoras, he is also remembered for a 3rd discovery, that of musical harmony. Schopenhauer, by far the best philosopher of the industrial age, said that music encodes the program of time in its rhythms. Pythagoras found its simplest ‘scale program’, the perfect ‘fifth’, a musical chord obtained by plugging 1/3rd or 2/3rds of a string – that is, *the natural standing points of a simplest ‘mono-logic’ (1D) worldcycle, with finite duration (* attached to an origin or point of past and an end or future point, the birth and death of a world cycle frequency, with 3 ages subdivided by the S=T symmetry at 1/3^{rd} and 2/3rds.

Since beauty IS defined as the perception by any mind in any language of the perfect symmetries of a ‘well-run’ program of existence, maximal in the S=T central region of a worldcycle between 1/3^{rd} and 2/3rds, or mature classic age of balance between motion-energy & in-form-ation. It is the classic age of art, mind of civilizations, the 30s in life, the interval of the perfect fifth.

As you can see in the next image, if we consider the vibration of the string, the simplest possible world cycle going from 0 to 1 and back to 0; *the string will wave back and forth 3 times, increasing each time the ‘information’ it carries and diminishing its entropy≈energy≈amplitude. And the perfect form will be reached in the most harmonious sound produced at 1/3rd, in the change of age or state of the system. But what is more beautiful, *time waves back and forth 3 durations and we can fusion them as Nature does in a single ‘social being’, integral of all those webs. This is called the Fourier transform, and in complex 5D metric is the essential equation of time cycles; since it keeps adding on ‘social scales’ of larger simpler wholes (the single wave) and smaller more informative parts. And finally ’emerges’ as a ‘single being’, a square wave, which therefore *represents a population of time, stored as a memory of still space, compressing **µ (limited infinity) time cycles of information* into a single spatial image:

The beauty of i-logic mathematics thus resides in its capacity to express the purest GST laws; as music does in human arts, with its 3 elements, ∆±1 3 scales, T-beat, ST-melody and S-ynchronicity of instruments.

But Pythagoras was just the beginning of geometry, and we already see how much GST can extract from it, when we use S=T symmetries or ‘switch’ from continuous lineal time as duration to a real detailed analysis of discontinuous time cycles that move in ‘frequency steps’. I.e. a car, as a whole seems to move in lineal time, but only its wheels move in cyclical steps so breaking it down into the S-body and T-wheels, and those into ƒ=1/t gives us new information, new laws of science *hidden by adding time cycles into continuous lineal sums, ‘erasing’ the form and frequency of those cyclical steps*; a we will do constantly on our 5D articles.

**The line and the cycle.**

It is rather impressive the ‘perfect order of the fractal Universe’, hidden in the complexity of the game. I.e. consider the case of Greek lineal geometry. As in art in which each first age is a lineal, young, simple view of reality, Greek geometry was essentially established around the ‘trigonometric triangle’. Its architecture lacked the curved arch, its knowledge of ‘conics’, only coming in its final 3^{rd} age. *Languages are mirrors of reality with lesser motion, SS-seeds and yet they imitate reality both in its entangled elements and worldcycle of evolution. And so we qualify indeed Greek geometry as a lineal, young state, which only its ‘wrinkled 3 ^{rd} age’ found the curve. Let us study one of its forms, as we reserve the analysis of conics to the 2^{nd} age when analytic geometry uncovered all its virtues.*

** **

**3 ^{rd} AGE OF GREEK GEOMETRY: ARCHIMEDES.**

** SPIRALS AS REPRESENTATION OF THE WORLDCYCLE.**

**Abstract:** The spiral is the simplest representation of a world cycle. As such it is a profound form of the 5D universe of relational space-time beings, reason why we have taking it away from the general post on non-E geometries, as it deserves its own deep thoughts.

The 3 masters of each age of Greek Geometry; Pythagoras, the founding father; Euclid who collected the classic body of bidimensional Geometry & Archimedes that anticipated mathematical physics have different merits, which ænthropic man as usual misreads in terms of its ego paradox, as *Euclid is by far the less original and most erroneous, as he made sacred the axiomatic method that denied experimental value to maths, unlike Archimedes. So even though he was the last one, a proper order in time corresponding with the 3 mental ages of life would be:*

Pythagoras, ‘idealist youth’ < Archimedes classic experimental realism > Euclid, 3^{rd} cut-off informative age.

Archimedes thus is the closest spirit to 5D mathematics. So we depart from his book ‘on spirals’ to complete our analysis of dynamic pi numbers. As a spiral is a pi cycle, whose closure has gone ‘a bit wilder’ on the opening. It is a simplified bidimensional form of a conic, which also combines a cycle and a line but in 3 dimensions. Yet since the Universe is made of Spacetime holographic SS, St, sT, ST, TT 5 bidimotional units, all curves that exist can be decomposed as a sum of bidimensional conics, including a flattened spiral defined by Archimedes: “If a straight line one extremity of which remains fixed is made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line is revolving, a point moves at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane.” *On spirals*

We find then 2 ‘55D’ merits on Archimedes’ definition: 1) to be a combination of a lineal and cyclical geometry and 2) to have motion, not just still geometry; making it the fundamental simplest ST bidimensional representation of a worldcycle. So Greek geometry generated a ternary geometric mirror in different dimensions of a worldcycle of exist¡ence making truth that the simplest ‘young’ mirrors of reality are often the most essential:

1D musical strings > 2 D Spirals > 3 D conics.

**The fractal point on the spiral as a living organism.**

What matters on that definition to mirror a worldcycle is that it is the combination of a lineal moving point within a larger cyclical world. If we take any point of any potential spiral cycle within a fixed spiral structure, the entity within the spiral is moving inwards thinking it exists in a lineal path Universe, as we all think, but because the super organism in which it exists, the curved spiral, is moving in cycles, the fractal point in fact is turning cyclically – he lives subjectively as we all do his lineal time, but the world objectively is ‘wearing it down’ the informative 3^{rd} age or central region of the spiral.

Then each spiral will be a variation on the same ‘mental phase spacetime’ mirror of the worldcycle. And we can apply the symmetries of GST to classify them. In practice though as in so many cases *the fundamental ¡logic symmetry that diversifies spirals *is the gender mirror symmetry, so S=T Archimedean spirals are ‘female’ balanced even spirals and logarithmic spirals are S<T>S informative & entropic. So we reduce our analysis to them.

**2 ^{ND} AGE, CLASSIC, S=T, REPRODUCTIVE, ARCHIMEDEAN ‘STEADY STATE’ STABLE SPIRAL. **

In the Archimedean spiral’ the internal point-being follows a cyclical path at a fixed rate.

Thus the Archimedean spiral is the steady state present spiral and as such the commonest in total space-time, as it is clearly a more stable configuration than an accelerated log-spiral of the form S>T. It does NOT have an inner motion that implodes it but basically is the ‘natural distribution’ of the internal vital energy-space of a spherical form.

The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance (arithmetic progression). In contrast, in a logarithmic 3^{rd} age accelerated spiral these distances, as well as the distances of the intersection points measured from the origin, form a geometric progression. And this reveals its ‘vital age’ & possible multiple functions, in its relative S, T, ∆, @ ‘survival tasks even though its primary design is to reproduce a point or communicate ‘2 points’:

If it is symbiotic as part of a supœrganism it might represent both the inner vital cycles of a flow of energy and information or its surface membrane in 3D projected into a 2D Archimedean spirals as a bidimensional phase space of a spherical form:

In fact we have two solutions/locations for the spiral that trace two dual paths often of communication between two similar beings. So the Archimedean spiral can create by cyclical reproduction in geodesic curve paths, a 3rd temporal dimension of height to construct spherical membranes.

The Reproductive Fermat spiral is a type of Archimedean spiral even more apt for reproductive purposes. In a single plane, it can act as a reproductive parallel communication for a 2-particle system:

In the graph the relationship between the communicative spiral and the shape of its external membrane, as the spiral is the common form in which a vital space of cellular elements is established within it.

Then the spiral is a communicative dance between two similar forms, merging the two elements at the end of the ‘life of the spiral’, as in black hole’s merging.

As a reproductive spiral, it shows the fact that all ‘reproductive’ actions are the structural merging of ‘dual elements’, we call gender. So it has 2 branches – more in a Fermat spiral whose purpose is the maximal packing of its reproductive forms in the space it fills, reason why is so pervading in plants and other highly reproductive systems of nature.

The reproductive Fibonacci spiral achieves this with its reproductive golden ratio, studied in Number theory.

Spirals as fixed bodies of space are thus Archimedean spirals. And we can further apply the fractal principle to subdivide them:

The normal Archimedean spiral occurs when *c* = 1. Other spirals falling into this group include the hyperbolic spiral (*c* = -1), Fermat’s spiral (*c* = 2), and the lituus (*c* = -2). Virtually all steady state dynamic spirals in space, as part of super organisms are Archimedean ones; such as the Parker spiral of the solar wind, or the pattern made by a Catherine’s wheel) are Archimedean; where the motion is conserved, as part of the vital energy-body of the system. Since the distance between its two cycles is fixed.

**S«T.** But in pentalogic all forms have multiple tasks, so as the spiral models a cycle=dimotion’s recurrent frequency it might be an E-ntropic feeding dimotion that brings the mouth to a certain point; and then bringing down the body, as in earlier cephalopods.

Time is a curved hence with at least two dimensions, besides its young age length: 3^{rd} age curvature and reproductive 2^{nd} age frequency. Time cycles thus break reality into an inner and outer ‘vital space’, and add a singularity point or focus of its motion. Because pi is not exact, steady state clocks are less common than vortex spiral or fluctuating ±π mouths. So once it swallows those spirals bring ‘home’ to the singularity the vital energy extracted by the external membrane by several methods.

One is shown in the graph, a scroll compressor: a motion of both arms compresses and moves towards the center the flow coming fro the eternal region of the being.

Notice in the graph above also the difference between the black holes in a communicative spiral. When the black hole in an event of feeding on energy transformed into its ‘quark forms’ as an accelerator vortex, similar to those processes studied on Earth’s accelerators, which leads us to its second fundamental form.

**ACCELERATED INFORMATIVE+ENTROPIC LOGARITHIMIC SPIRALS S<T>S.**

**Pentalogic: S<T>S **The St-logarithmic spiral is informative for its central knot and entropic for the point that moves inwards, *accelerating* through the log spiral, which in physics is a vortex=force source of accelerated timespace.

**It is also the ∆-SCALAR **spiral as its curves are self-similar in diminishing scales. , Bernoulli called it Spira mirabilis, “the marvelous spiral”. because he was fascinated by such unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve, a property known as self-similarity.

**S=T:** As a result of this unique property, the spira mirabilis has evolved in Nature, appearing in certain growing forms such as compartments in a nautilus shells and sunflower heads, that will store the reproduced cells of the system, in its *inverse arrow of creation of a larger whole starting with the simplest cellular reproduced units*, in exponential Fibonacci growth series*.*

**T>S:** It also appears as ‘informative spirals’. Which accelerate and diminish the size of a form, as it comes to its perceptive point.

**¬S«T:** Finally from the perspective of the bite of energy or bit of information, which ‘goes through’ the tunnel of the spiral is a killing machine, now observed not as a reproductive cell system but a trapping channel for bits and bites accelerated towards the central stomach or eye of the spiral. Where the ∆-1 point will die in an entropic/informative split – when the perceiver-predator will split the system in its ST>S parts take the information or energy of its body, depending on the event and let it die.

So we can see easily how the logarithmic spiral allows events in ∆-scale (generation), S-entropy functions (feeding) and T-informative and reproductive functions (perceiving, storing cellular/atomic network forms).

**G(¬∆@st): **Which leads to the use of the log spiral as a model for the 3±¡ ‘ages’ of life of the micro ∆-1 entity; as stars in a galaxy, which is a ‘farming trap’ for the central black hole to let the stars grow on dust to feed it eons latter.

Let us see some of those pentalogic perspectives.

**ð: Perceptive vortex log Spiral **

The logarithmic spiral is an accelerated, hence perceptive spiral.

It can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.

If the point that moves inwards *accelerates* we have a vortex of accelerated time. It is then a logarithmic spiral; an attractive form and *it escapes the simplification of mental geometry to become the commonest real worldcycle*.** Spirals in physical systems. Charges, masses, eddies & galaxies**

.

In the graph time is curved, and breaks timespace into closed conserved paths, time clocks are infinite as Leibniz and Einstein understood when he said ‘I seem to be the only physicist that thinks there are infinite time clocks in the Universe’. Charges and masses are then ccelerated, in-formative time – perceptive spirals.

All those physical scalar spirals (ab.∆±¡) are topologically similar, differentiated only by the ‘speed or frequency’ at which they close their ‘time§pace clocks’ , according to 5D simple metric rule for all ‘families of time§pace clocks’: Size in space x Time frequency =constant. So LOG spirals are all pervading in nature, as they represent the dimotion of accelerated time or main arrow of future that increases the information of a system. As ‘accelerated timespace vortices’ they are the informative 3^{rd} age of all physical systems in its ∆±¡ quantum, thermodynamic and cosmological scales – an attractive vortex of space-time, which is by expansion of the Principle of equivalence between acceleration and mass, to all scales the physical definition of an Active Magnitude source of a force… NOT a solid static particle, a ‘tiresome’ error of idealist huminds that keeps coming since Pythagoras. As all what exist is a motion in timespace. Still space forms are a Maya of the senses. In the graph, accelerated vortices of timespace in physical systems, in different scales of the fifth dimension: charges, masses and thermodynamic eddies become then the main clocks of timespace that carry with different speeds according to 5D metric (S x T=K), the information of microscopic quantum charge worlds, human-size thermodynamic scales and cosmological gravitational scales. Since E=hƒ + E=mcc->M=ƒ(k); mass is a frequency of accelerated inwards space-time in its 3^{rd} age. So are charges and eddies, which will finally in its central point, become a ‘conic’ and ascend and/or descend lineally the axis in its entropic death.

Thus spirals respond to the fundamental property of time cycles: to have an arrow of future increase of information that diminishes its spatial size according to 5D metric: S x T = K, this accelerating inwards, which makes vortices of physical time (masses, ∆+3, charges, ∆-3), definitively the time clocks of both physical scales. For that reason time-space spirals, its subspecies and transformations are one of the fundamental space-time events of the Universe.

**Logarithmic spirals in nature classified by pentalogic function.**

In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follows some examples and reasons in terms of ∆STœ events:

Å(e): Feeding: The approach of a hawk to its prey.

Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral’s pitch.

__Å(i): The approach of an insect to a light source.__

They are used to having the light source at a constant angle to their flight path. Usually the sun (or moon for nocturnal species) is the only light source and flying that way will result in a practically straight line.

__Å (i):__ The nerves of the cornea:

(this is, corneal nerves of the sub epithelial layer terminate near superficial epithelial layer of the cornea in a logarithmic spiral pattern).

__Å (æ):__ The bands of tropical cyclones, such as hurricanes.

__Å (e: growth and reproduction)__: Many biological structures including the shells of mollusks.

In these cases, the reason may be construction from expanding similar shapes, as shown for polygonal figures in the accompanying graphic.

**Ideal Spirals as representation of worldcycles.**

Our planet-star lives in fact as a fractal point of a spiral, which acts as our Island-Universe in the longest of our worldcycles – that of the planet we exist within. Our own galaxy, the Milky Way, has several spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees. So the stars go through the spiral in its space-time world cycle of existence – from a 30 years old graph (: hence primitive digital form, profound human thought):

How long is the life of a ∆-1 points, which has ‘fallen’ inside any of the attractive vortices of a spiral organism?

**Mathematically** it means that starting at an external point π, of entrance in the spiral, and moving inward along the spiral, one can circle the origin an unbounded number of times without reaching it; yet, the total distance covered on this path is finite; that is, the limit as θ goes toward ∞ is finite. The total distance covered is r cos ϕ, where r is the straight-line distance from Pi to the origin.

So as it turns out, the number of cycles a being can turn about the spiral (frequency cycles) is infinite if time space were continuous, thinner and thinner but the real length of the life-motion or world cycle of the spiral (length to the center) IS finite, and moreover it diminishes in objective time, as the last phase is shorter and faster in frequency a deep fact about the duality of objective and subjective measure of time existence.

Since an old man has a shorter life, falling fast in decay in its last ‘life cycles’, which if we add the inverse ‘balance’ of its inner subjective time which is slower than the objective faster decay, makes the 3^{rd} life even shorter in subjective time, as a kid has much faster mental cycles, so his days are longer in experience. That is, at 50 in total real time actions we are much older; almost all our life ‘bits’ of action-information have concluded.

The scalar duality of the larger spiral organism and the smaller part shows also the scales of time: the short ‘frequency moments’, bits and bites of space-time actions which are Dominant in information, as the motion is inwards and will add to the whole world cycle of a being, causing its 3^{rd} informative age.

If we postulate a further generalization of the spiral as the ideal worldcycle for a whole range or phyla of species, it shows that all beings will live and die a finite, similar quantity of time-actions, making all lives absolutely relative. So 3^{rd} age points within log-spirals shrink in size, increase frequency and shorten its lifespan, completing its 3 ages: S<T+S=T+S>T.

**Spirals as entropic killing machines. **

Each inward closed path ads a bit of informative frequency every time the being repeats their cycle, aging it.

How spirals end the existence of one fractal space-time T.Œ? As all worldcycles with a reversal of timespace.

The life cycle is slow moving to the center: T>S, an action of information that implies to reduce the dimensions of reproductive space-width for those of cyclical motion, till the relative ∆-1 fractal point looses all its energy and becomes, if the center of the spiral is an eye, a bit of information, if a mouth, a bite of energy, dying.

All time cycles tend inwards in a 3rd age of warping and in-form-ation till the flow of motion ‘stops’ in the still singularity, ejected perpendicularly in an explosion of death-entropy. It is the last time quanta of the clock when motion becomes a 5th dimensional still image before dying into a lineal 4D entropic flow.

So in its death age, the reversal of time happens, as in magnetic fields coming out of masses, dark energy fields coming out of galactic black wholes, when the log spiral uncoils first into Archimedean spiral as we ad a 3rd dimension of height information, that converts its shrinking revolution into an elongated receding motion, for an entity living within its revolution that becomes devolution, as the world cycle keeps accelerating but now acceleration is perceived as an ascension and elongation in height that erases its in-form-ation.

**Spirals as informative eyes.**

Humans express all those bio-topo-logic functions of the spiral with a highly efficient synoptic algebraic mirror image of 2 numbers for the key lineal & cyclical, parameters that construct the spiral: |xO=Ø.

Its simplest=most real form is in polar coordinates, which means spiral are temporal self-centered systems, in which the simplest ‘perception’ is that of the central point of view, making truth a self-evident GST theorem: the simplest mathematical formulation of a space-time event/system, in one of the 3 relative canonical coordinates, St (polar, spherical informative-head/particle), ST-Cartesian(hyperbolic, iterative-wave/body) and Ts-Cylindrical (lineal field/limb), defines the main organic function of the system, as S-Form=T-function.

So if an spiral equation is simpler in polar coordinates it is an informative space-time, form/function,

**DYNAMIC SPIRALS AS PERFECT SPACE-TIME SUPŒRGANISMS OF 3 STATES.**

The logarithmic curve can be written as: r = a x e^{bθ}

And then depending on the value of b it will *transform either into a circle, or a line. The *derivative of r (θ) is proportional to the parameter b, which controls how “tightly” and in which direction the spiral spirals.

In the extreme case that b=0 (ϕ=π/2) the spiral becomes a circle of radius a. Conversely, in the limit that b approaches infinity the spiral tends toward a straight half-line

So a spiral can be considered an informative ‘state’ of a full organism, which can convert itself into the other two states as a present wave and uncoil as a lineal motion. Again this is a canonical law of vital, i-logic geometry: a system can be converted between the 3 functions/form as systems are ‘modular’ and its functions are constantly changing between the actions better performed by limbs (entropic function), bodies (reproductive functions) and particle-heads (logic, informative functions).

Such transformations are the staple food of existence and development, being the spiral and the tree, then 2 fundamental ST combinations of S & T elements – but the spiral is the commonest dynamic form that allow change of states with ease; while the tree is the commonest ‘fixed’ still simultaneous system of O-| elements:

IN THE GRAPH, all spirals are ‘potential fields’ or sinks and sources (+ inward life vs. – outward entropic death duality); which explain the organic, functional and mathematical description of spirals as bio-topo-logic beings.

So the vital spiral *is *in its vital mathematical description the most efficient organic form=function: A galaxy coiling its flat irregular young form a worm coiling to sleep in its 1/3^{rd} still informative SS-inwards state (completed with its feeding, entropic state, and wave-like motion) are in spiral states; which is ultimately an extremal case of the upper and lower limit of pi that leaves always an opening, dynamic mouth to the spiral allowing its simpler beat of existence, closing inwards (T-State) and outwards (S-tate), never ending the perfect pi cycle.

So most vital living spirals can uncoil to become lineal forms, or can close to become spherical circles, managed by its central head (1D: informative state) that becomes its future ilogic forwards head (2D/5D: locomotion and feeding-hunting). So the spiral can be considered an Ø-intermediate present system, perceived from the perspective of its dominant central point of view in polar coordinates (r, θ).

**RECAP. Varieties of time spirals in pentalogic dimotion.**

Spirals represent the 2 complex dimotions of time-space in a classic S=T female, S<T>S male symmetry:

S=T: Reproductive, communicative Spirals are Archimedean or Fermat’s based in the Fibonacci’s golden ratio constant in which several branches of the spiral allocate with maximal efficiency the reproduction of new ‘infinitesimal parts’ of the whole.

S<T>S: Entropic + Perceptive spirals, (as those of time vortices, charges and masses) where a flow of ∆-1 points falls into the perceptive or digestive central point, allowing both a potential and vortex description of them.

Based in those rules that vitalize the mathematics of Spirals as essential elements of a fractal point in its dimotions of perception feeding and reproduction, we can interpret better the maths of spirals in vital terms and define multiple organic beings as entities, which in its coiled position become organic spirals, often when resting – processing its internal systems of energy and information.

** **

**2 ^{nd }AGE OF GREEK GEOMETRY: TRI-ANGLES.**

**MENTAL SPACES MEASURE ANGLES=SPACE SIZES AND DISTANCES=TIME MOTIONS.**

What we need to know of a being from an outer world? Obviously its relative size, given by the height dimenson measured by an angle, and its distance in time0space measured by length and motion-speed. Both are the concerns of the simplest first mind spaces deviced by geometry, those of a trigonometric space.

Every language-mirror of the Universe starts slowly rising its complexity and focus on reality as it imitates the intrinsic laws of spacetime reality, even those as mathematics, which describe in a closer form its spatial points, scalar numbers and temporal operands. So did geometry in Greece. As such languages will follow similar patterns of growth and evolution to those of full fractal points=T.œs starting by the establishing sequential paths of Dimotions, which always mean first the fractal point’s emergence in a larger world where it will open its ‘eyes’ and measure. What it measures to make it useful for existence is thus the minimum the language provides on information.

**In the case of geometry is what all spaces will have in common for the point to ‘see’ its surrounding worlds and distinguish form (angular momentum, cyclical shapes, membranes) and motion=distance, lineal momentum.**

**| & O even in its most simplified way, which is to measure angles of congruence and distances of predators and preys. Or else the language won’t be efficient, its speaker will die and the system will not repeat the languages’ form. So parameters in languages have a vital ¡logic meaning, for minds to acquire knowledge in its logic mirror about the surroundings of its world and be able to move and assess the distance and size of things. ***Even in music, its simplest duality, treble and bass bring e-motions of uneasiness and relaxation as they connect with the Doppler effect: a predator wave coming closer blue shifts into treble and going away into bass waves. Thus giving a first information, with no angle, on a lineal path motion. Trigonometry added then to distance (S=T duality of motion), angle.*

*Even in music, its simplest duality, treble and bass bring e-motions of uneasiness and relaxation as they connect with the Doppler effect: a predator wave coming closer blue shifts into treble and going away into bass waves. Thus giving a first information, with no angle, on a lineal path motion. Trigonometry added then to distance (S=T duality of motion), angle.*

**So the first parameters of ‘humind’ geometry were trigonometry and distance which under the s=t paradox can also measure a point in motion over a background scale of space.**

**Angle is also the first fractal unit of measure, which can travel through scales without deformations – hence*** a *dimensionless parameter *as fractal scales are relative in its dimensions, ‘erasing’ them internally as they emerge as an Euclidean point into a new scale*. So angle and distance gave us ∆St information and allows a trigonometric mind to survive. Angle thus allow us to define mathematically a dimension as follows:

*a*dimensionless parameter

*as fractal scales are relative in its dimensions, ‘erasing’ them internally as they emerge as an Euclidean point into a new scale*. So angle and distance gave us ∆St information and allows a trigonometric mind to survive. Angle thus allow us to

## Suppose there is a mathematical (geometrical) quantity *A*, which depends on the scale, *l*. If after a scale transformation: *l*→*λl*, the quantity would transforms as:

A→λ^{n}A. Then the quantity *A* has dimension *n*. According to this definition, the angle has dimension 0; because it is the fundamental parameter that carries through scales without change.

**1D: sine<cosine.** **Angles of perception**

**So the angle is the first operand on a surface of space, to provide information to the self-centered system. And its importance was such that the tri-angle was not called, the triside or triline (the objective topologic view) but the mental spatial view, showing ***once more that the ego-paradox is always the beginning o any science. That is, the tri-angle was first considered a mental subjective space form, before it became an outer objective surface.*

*once more that the ego-paradox is always the beginning o any science. That is, the tri-angle was first considered a mental subjective space form, before it became an outer objective surface.*

*And it bears witness to our Nature as light space-time organisms, evolved from the minimal species of Euclidean Light space-time, the Planckton, that when we finally discovered ‘¡ts’ form, in quantum physics, h, turned out to be exactly that: an angular momentum, quantized in two gender species, the female boson and the S<T>S fermion. *

*And it bears witness to our Nature as light space-time organisms, evolved from the minimal species of Euclidean Light space-time, the Planckton, that when we finally discovered ‘¡ts’ form, in quantum physics, h, turned out to be exactly that: an angular momentum, quantized in two gender species, the female boson and the S<T>S fermion.*

## The pentalogic dimotional functions that entangle its trigonometric ‘functions’ into reality:

**@:** its dominant use and first reason it became the first developed field of mathematics is its capacity to measure from a point of view distances according to ratios and parallax, which is the origin of tridimensional perception (bilateral eyes), and Fertile Crescent mathematics.

How this work in its simplest form, needs to understand how a ‘spherical, ideal mind-membrain of 3 π diameters, and 0.14 D apertures, allows a mind to perceive through them, ‘rays’ to distant objects. The mind thus can always measure the angle covered by a distant object, and with a minimal displacement, a new angle.

3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions.

**1D:** So trigonometric functions were the first to appear, as 1D perception is also the first ‘action’ of an emergent point where sines and cosines’ operands extract for the self-centered point, 1D perception of its ‘angles’ and ‘S=distances=T-motions’.

The graph shows the trigonometric functions and the information extracted by them; and as usual in geometry we apply them the Dilogic symmetry of ‘internal subjective space’ and ‘external objective space’; as they can be applied from the fractal point to the outer world or from the outer world to the fractal point, whereas in pentalogic:

– 1D: S: Sine is best to assess the ‘informative height of the system’.

-2D: T: Cosine is best to assess the length-motion of the system.

-3D: Tangent is best to asses the energy, S=T of the system.

– 4D: Angle is best to define the scalar proportionality of the system.

– 5D: Its inverse functions have an entropic role in the calculus of all those properties.

So trigonometric functions already extract from a T.œ information of its 5 Dimotions.

Trigonometry thus deals with the laws of perception of the circle and the triangle. And as usual since *reality builds up from the simplest lineal essence* to evolve latter its ‘3^{rd} age cyclical form’ |>O, the first laws discovered were the laws of |-∆ triangles, whose S=T transition into a sphere when it adds a dimension of ð-cyclic time, is its rotation. Whereas the ‘area difference between a regular polygon-triangle’ and its sphere, *is the first s=t transformed value, which turns out to be *huge, between ∞ and +2,4, while its perimeter its 1,6 times larger, making them clearly the 2 extremal TT & SS topologic forms of motion, the arrow triangle, and form, the rotating circle; facts those resolved with the Law of sines that calculate first for any triangle its ‘apertures of perception’**:**

If *A*, *B,* *C* are angles of a triangle and *a*, *b* and *c* are lengths of opposite sides the respective angles the **law of sines** for an arbitrary triangle states:

where Δ is the area of the triangle and *R* is the radius of the circumscribed circle of the triangle. In terms of perception thus, each of the 3 ‘apertures’ of the 3D¡ sphere and its sine view of the external world is equivalent, ensuring a non-distorted perception of distances to the outer world. It establishes the same law of equal perception of those 3 ‘membrain’ points of its internal ‘vital energy’.

**The angle is thus the first form of existence. The first 1D perception that illuminates reality by bending hyperbolically a larger world into a central singularity, in fixed quantum forms, which will become for the 1 ^{st} particles ¡ts h, a radian of angular momentum perceiving a larger world tunneled into the singularity of its ∆-1 scale. Creation starts with the angular reduction of reality to a still form in a quantum angle. As such the angle becomes with its radian h value, the first T.œ, the triangle, which then as it turns around gathering in 6 groups forms an hexagon, with a wobbling radian arc that tessellates the plane from circular to hexagonal STœps. **

**Vital mathematics becomes then reality. Such vital circle has 6 apertures – those of its triangles – each of the radian arcs that make in circle 6 steps for a pi=3, closed in hexagonal form, opened in circular one. Imperfect forms thus appear from the original perfection of the language imprinted in the regularities of the medium.**

**So the hexagonal circle game of Stœps becomes the essential structure of 2 Dimensional space, with those 120 hexagonal angles repeated ad nauseam in Nature. It is the game that will grow in polytopes to reach through combinations of 5, 6, 120, the final 120 dodecaplex and 600 tetraplex maximal figures of 4Dimensional geometries (as pentagonal polytopes make no sense, being the 5 ^{th} dimotion entropy the destruction of the regular in-form-ation shared by them all). **

**Angles of perception become then the first simple geometric element of reality as 1Dimotions appear on every still point that gains an angle and starts the creation of the galaxy in its smallest T.œ, Planckton**

**The laws of those ‘angles of perception’ would also be the first to be assessed when a new emergent light spacetime mind, that of the Greeks appeared. As each emergent scale repeats the game of exist¡ence. **

**Angles of perception in different vital and abstract geometries.**

Angles of perception define the capacity of a point of view to measure and obtain information from the external Universe, such as the closest angle, the less perceptive (5D) a system is, and the more dark space will have. And so they play also a vital role.

They also vary according to the *topology of the world we live in. So they are:*

Minimal in hyperbolic plane, as hyperbolas are the entropic curve; so they minimize the inverse arrow of information.

Maximal in elliptic, spherical geometries whose angles are greater than 180, as *spherical beings are informative. *

Medium in the flat plane with angles at 180.

**The vital axioms of geometry. Reproductive motions. Discontinuity postulates. Attraction.**

**So a series of laws of angles of perception will become the first set of consistent laws of creation of mental spaces defined by huminds. We are here concerned as always with the translation and correction of this system of axioms of abstract spaces, for a single plane time-space to the much vaster, paradoxical real world of vital ilogic topology.**

**What is the main difference between Euclidean geometry and fractal non-AE when considering those axioms?**

**– First we notice as we did with the earlier version of Euclidean geometry, that the axioms are grouped also in 5 elements, since even when simplifying mirrors, the humind must refer to the 5 Dimotions of reality.**

**– Then we observe that the differences between E-postulates and axioms vs. ¬Æ will arise as always of the ‘simplification’ of multiple scales into a single ‘continuum’, and the ‘straightening’ of relative angles that ‘change’ its form into a hyperbolic curve as we ‘diminish’ size in the fifth dimension.**

We thus first updated the elements of Euclidean geometry – its definitions of points now fractal; lines, now waves, congruence (now relative equality) and planes (now topological networks). As what today passes as the foundational axioms/postulates of geometry are only correct, for the limited view of a single spacetime continuum. So they are approximations which strip points=T.ŒS of fractal & vital properties, and simplify its vital imperfections.

This process carried by steps from Lobachevski’s a pangeometry to the absolute geometry of Bachmann (1970s) closed the evolution of the discipline. And it was found that only the most general features, angles and distances are needed to define a geometry. specifically we could reject as Bachmann did the postulate of ‘continuity’, and Dedekind’s concept that real numbers ARE numbers NOT ratios, because they LIE on the real line, as a necessary feature of a real geometry, since indeed in the absolute pangeometry of the fifth dimension, any plane of reality is discontinuous, and the numbers fit in the real line to fill the discontinuities of simpler number families are NOT in the same plane. Still what we can take of the Axiomatic method either in its Euclidean version or modern ‘Foundations of Geometry’ (Hilbert) is the classification of the foundations of geometry in 5 subgroups of axioms that roughly coincide with the 5 elements of reality and now we shall comment on to realize the difference between a single plane continuous geometry and the real geometry of 5D discontinuous planes and angles that become hyperbolic as we sink into the minds ∆-1 singularity.

*Those are themes thus concerned with the 3 ^{rd} age of Greek Geometry and its eclectic attempts to found the discipline as a Universe in itself detached from the earlier Pythagorean and Archimedean practical ages.*

** **

**3 ^{RD} AGE OF GREEK GEOMETRY**

** Archimedes vs. Euclid Experiments vs. axiomatic error**

Tthe wrong egofcy side of Greek Geometry, its its ‘3^{rd} eclectic old age’ as Euclid, committed the same error than modern mathematics in his 3^{rd} informative age, when set theory and the axiomatic method abandoned experimental proof and tried to ‘construct’ from the top without proof to the bottom unconnected of reality.

So math failed from then on to understand itself as an experimental mirror of scalar space, becoming with Euclid’s axiomatic method a self-contained truth, which it is *because fractal mirrors are ‘entangled’ in form to its original image – the Universe. *Yet as mirrors are kaleidoscopic, inflationary as Lobachevski and Gödel proved for mathematicians but physicists so far have failed to recognize, only experience validates what image is real. Schopenhauer, the best of the philosophers, noted already the fallacy of the axiomatic truths, which rely *on too many self-evident lies, points and lines with no breath, putting his warning on the axiom of congruence, which in fact is the most expanded by 5D rules. *

But once axiomatic methods with single Aristotelian languages became definition of truth, this error will continue to become the fundamental ego-trip and failure of many human sciences accustomed today to accept a priori postulates that set the ‘entropic’ limits of truth and scope of the postulate, introducing bias from its inceptions – from Economics that rejects the analysis of the evolution of machines with do in organic economics and accepts as a self-imposed postulate the goal of GDP growth in a limited planet to c-speed limits in the astrophysical realm beyond the galaxy, to the entropy arrow only of big-bang theories.

**The many false assumptions of classic Euclidean and axiomatic geometry.**

Let us ‘continue’ then the critical analysis of the Axiomatic method. Generally speaking, the axioms can be chosen in various ways, taking various concepts as starting points. Here we shall give an account of the axioms of geometry in a plane which is based on the concepts of point, straight line, motion, and such concepts as: The point X lies on the line a; the point B lies between the points A and C; a motion carries the point X into the point Y. (In our case other concepts can be defined in terms of these; for example, a segment is defined as the set of all points that lie between two given ones.)

The axioms fall then into five groups, oh yes, the familiar number! As we can relate them to the 5 elements of ¬∆@st of spacetime but its postulates to be *considered truth only in the limit of still bidimensional geometry in the plane, and so we shall adapt them with comments (no need to create in this introductory courses a true formal pentalogic system of axioms and postulates of its own to fully develop non-AE philosophy of stience and its stientific method and epistemology. (: LOL, this was done 30 years ago, and I used to write only with ilogic symbology, so far out from what huminds do, that I only got interested a conceptual art who used those equations for a exhibit at six-flags, a Brooklyn museum : )*

And obviously we are not using the formalism of sets and modern maths to explain those corrections.

We repeat this is an introductory course and we just want the reader to understand that present maths is an abstraction of a much more beautiful experimental language of scalar=numerical points of space-time.

This said the 2 first groups, Axioms of incidence and order, and the concepts of angles and laid on, refer to the entropic limits and inner structure of a given mental space, so they should be grouped as ¬S postulates.

__S:____Axioms of incidence__

Axioms of incidence are mainly related to the 2^{nd} postulate of lines-waves of communication, its boundaries and angles of incidence=congruence (already discussed in the 4th non-E postulate). So it also considers also the relative size of T.œs and its lineal, open flows of communication compared to its worlds; of the minimal internal parts a T.œ needs to define its properties as an |- lineal or O-circular element and so on. They connect thus closely as boundary axioms with the concept of a topological boundary and the entropic limits both in space or time (if we take a line as a worldline of duration of a being). And the main differences between the classic axioms of E-geometry and i-logic geometry regard the multiplicity of meanings of i-logic geometry where distances and points are logic concepts of similarity, all systems do have ‘boundaries’=limits as infinities are relative infinities (µ) and so the interpretation of its modern axioms of incidence fully changes.

- One and only one straight line passes through any two points.
- On every straight line there are at least two points.
- There exist at least three points not lying on one straight line.

As we can see the axioms of incidence establish the *minimal elements required to transit from a single fractal point into a social group of points, a line. But as usual with the axiomatic method that tries to prove itself by ‘reductionist’ simplicity of its postulates it does NOT bring enough information to define without ambiguity a line. Because with only two points, the intermediate region can be curved, so no straight line exists. How can we know with only 2 points the curvature between them? We cannot. And if fact because we cannot, the errors of the 5 ^{th} Non-E postulate that curves straight lines come from those imprecise definitions.*

*So the first axiom of incidence defines NOT a straight line but the* boundaries of a ‘flow of communication between 2 geometric forms. It is thus a definition of ‘any’ 2^{nd} non-e postulate line-wave of communication between two points.

And it also matches in the evolution of geometry into topology the more precise term of a ‘boundary’ for an open line of communication between two entangled points – the points are the beginning and the end of that flow.

This is then the meaning of the first and second axioms of incidence, while the 3^{rd} shows the outside world to be larger, than the flow of communication between two points.

The weakness of those postulates though has been already noticed by classic geometers. Indeed, it appeared somewhat strange to them that a straight line has only two points. Surely in our idea of a line there are even infinitely many points on it. No wonder that not even Euclid stated the axiom because it IS NOT proper, *showing that as more idealism and postulates of truth try to prove by simplification ‘absolute geometry’, they just add half truths and new distortions.*

2 points do NOT define a line, but any open trajectory, also an arch, a zig zag but NOT a circle.

So *the incidence postulates are important not because they define a line but a larger more important concept – an open flow of feed-back communication between two equal points, unlike a circle which is a hierarchical closed communication between two points, one of which ‘turns around the immobile one’.*

*So two points define an open democratic • = • flow of communication, while two points in a circle define a closed hierarchical communication. *

So the definition of an open ‘wave’ of fractal points blows up the concept of ‘straightness’. This is ok as long as we know that what matters of a straight line is not that it can be drawn lineally on a plane with an abstract rule, but something else – to be open in a ‘topological way’, that is, we can deform it as a geodesic on a surface, a chord of a circle, or whatever, as long as it is open and the two points are its boundaries.

Thus the *key feature of the line is to be open not to be straight, as the line curves in a hyperbolic angle entering an ∆-1 fractal point reason why* modern physics use curved lines to ‘crowd’ the entrance of a fractal point of the ∆-1 gravitational scale, instead of blowing those points as fractal points through which µ straight lines *can cross*.

Once this is clarified we find 3 canonical varieties of open lines: a straight line that needs 3 points to define its straightness and type of information communicated through a flow of middle points, which will form a wave as even the smallest boson has volume. A true straight line then happens merely with the entanglement of two fermions with a gravitational non-local quantum potential line that merely gives information on distance and likely angle (a neutrino or string of the Planck scale, maybe being both similar as neutrino ‘angle of scattering’ is on that range).

It implies then that even the simplest line requires a 3^{rd} bit of information about the type of flow of information and shape of the transversal wave between both points happen since now 1+1=3. As a flow of ‘smaller points’ transit both poles to become the motion that carries information between the limiting points of the line.

Finally the 3^{rd} axiom of relative angle of incidence is concerned not with the inner limits but with the outer extension of the world in which 2 points communicate: *there are at least 3 points lying outside a line *means that there is a larger Universe outside a worldline. Since we can trace a plane outside of it, which might or might not be parallel, or intersect the line, as 3 points not lying in a straight line defined a plane. So outside a line there is a larger world with a new dimension – a topologic plane, an organism to which the line might belong if intersection happens.

But 3 points DO NOT necessarily define a plane. Again such definition constructs a plane by *self-reproduction and this must be stated, as we did with 2 points that construct a line by self-reproduction of a flow of communication. It is then when the axiomatic method makes sense. And so another way to reinterpret those axioms and give credit to them is to write them in vital reproductive terms:*

*‘Only two points are needed to start the reproduction and transmission of information between them’, etc.*

Further on 3 points no lying on a straight line as always in Duality can be expanded in its solutions considering the different ‘ternary principles of creation’. So as we said that 2 points can define either an open ‘line’ or a ‘closed circle’, when one of the points has motion respect to the other, forming an St system, as opposed to an S-S line, 3 points can define a closed triangle, or an open *angle of perception, which in i-logic geometry is different. *

*And then we can also apply the fractal method of differentiation, if we consider that one of those points is in a different ‘∆-scale’. Then the triangle with have a scalar 5 ^{th} dimensional added parameter that will provoke the lines to curve into a hyperbolic angle to reach the inner depth of its ∆-1 mind-points, which does NOT lie in the same plane ) a ternary system.*

Therefore the need clearly arises *in i-logic geometry, *for *pentalogic 3±¡ ¬∆@ST diversification of the simplex single line or plane of continuous geometry, *stating accurately and exhaustively every distinct version simplified by E-geometric postulates that reduces all open vs. closed geometric objects to straight lines and flat planes.

So instead of making definitions simpler but less clear, which is paradoxically what Hilbert achieved we will make them more clear– even though this intro won’t be exhaustive.

In the previous pentalogic considerations though we have already established the |-line of 2 points that needs a 3^{rd} moving point of communication, the O-circle of two points that needs a 3^{rd} dimotion for the submissive orbital point and the ∆-hyperbolic angle of 3 points one of them in ∆-1 scale. So we see *that duality the minimal S-T or SS or TT reality of the holographic Universe immediately reproduces trinity. *

*And so we could also resume the *__new axioms of incidence__* as:*

*‘2 points generate a third point to form a stable trinity structure of space-time that might be an open line of communication between two equal points still to each other, hence a ‘2 number’, entangled by the 3 ^{rd} moving flow back and forth between the, (|-duality) or a circle whereas one point is a S-till point and the other an orbital ð-point whose motion becomes its 3^{rd} dimension (S-sT System), or a hyperbolic ø-angle whereas the Still point is in a lower plane of the fifth dimension, S¡-1<SS¡.’*

This would be a more formal definition of the incidence axioms of i-logic geometry.

The same procedure of exhaustive ternary-pentalogic division of varieties of E-axioms applies to all the other concepts and axioms when i-logic geometry goes deeper into them… But we won’t be so exhaustive. Just pass fast through them, observing how they are connected to one of the pentalogic elements of ¬∆@st.

__¬:____Axioms of order.__

**When we deal with hierarchical order and perpendicularity –** ‘lay on’ concepts we are directly concerned with the congruence differences between torn systems which share a cut-point and those who are parallel and ‘avoid’ tearing and Darwinian perpendicularity b establishing a bump in a new dimension of height-information between two lines:

1.Of any three points on a straight line, just one lies between the other two.

2.If A, B are 2 points of a straight line, there is at least 1 point C on the line such that B lies between A and C.

- A straight line divides the plane into 2 half planes (i.e., it splits all the points of the plane not lying on the line into two classes such that points of one class can be joined by segments without intersecting the line, and points of distinct classes cannot).

*1 ^{st} as we have already expanded the concept of a line as an open interval which does NOT return to the original point directly but through a back and forth motion, the axioms of order ONLY happen in a ceteris paribus analysis of one-dimensional Aristotelian A->B lines, and as such it should be scrapped all together as all communication is dialectic and so A->B->A is the minimal unit of a line, with 2 inverse directions, as seen in the Neutrino theory of light, where light is created by two entangled neutrinos moving back and forth, in inverse directions. One single entangled flow allows the point A to measure distance to B and maybe trigonometric angle-height-size. But unless that information is reflected back, there will be a lost single event hat does NOT remain as a stable memory line.*

*The axiom of order only applies to A->B fleeting lines. Yet as all lines are steps of a larger cyclical part; A->B always ends in A-B-A-B…., and then the feed-back real stable lines cannot set a forward, backward, order, after a few feed-back entangled flows, neither to circles, where we cannot as we return to the point establish a relative order. Thus when C comes before D or after A in the circle? The order established in A-B lines thus is a different one as in the circle, one of hierarchy between the AB ‘fermions’ and the C flows between them which is of a lower ∆-1 quantum potential order, and that is the proper hierarchy. So happens in circles between O-point (center) and A (turning point), as circles are NOT drawn by a single line without center, which will not bend but have lineal inertia, but by the torque offered by the central dominant point. So the order is AC dominates ac… and O dominates A.*

*It is the constant repetition of those circular or feed back vibrating motions what create the illusion of a stable space line or circle, a present, simultaneous space. Further on, what the circle does provide is not an order of points but a chirality, depending on the way we move around, which the line does NOT require in its feed back A<ab>B trinity. *

*Motion thus becomes a new creative duality order in cyclical paths that have chirality.*

*And orientation also. It is not the same order if we move from A –>cbd, or from A->dbc*.

*And this is then again the true differentiation between open lines that are more ‘equalitarian’ and circles, which constantly bring new hierarchies*. But as the Universe seeks for balance we shall see that parity & chirality allows gender mirror symmetry – it is in fact the first form of it; since the O-point will seek two ‘spiral arms’ with inverse chirality to achieve a balance with 2 different spin filled orbital points.

Thus as we move into the simplest geometric scales of vital geometry≈physics, we shall find that concepts as chirality, achirality, parity and the combinations of lineal and cyclical motions diversify the different species of reality and become essential to understand the symmetries and asymmetries of quantum physics, because IN FACT order does NOT exist in a categorical manner without a proper understanding on what becomes the most important type of Axiom, those of motion, which defined properly in Euclidean Geometry are Axioms of reproduction of form, by carrying a geometric form to other region of time-space. Yet before we study them, we must consider the second fundamental concept ‘laid on’ the Axioms of order, that of ‘lay on’ itself.

In the entangled Universe this key concept **lay on**, is related to the false axioms of continuity and to the duality of Darwinian vs. parallel social events. So we* apply the ternary, pentalogic method and differentiate at least 3 ways in which points lay on to each other cutting two forms or fusioning them.*

*In the graphs on one side the circular and elementary continuity principles study when 2 systems are perpendicular, that is can cut each other and share a point, or are parallel, that is, only ‘contact’ each other by adjacency but do NOT cut each other. *

*As we stress the discontinuity of reality, we rename those principles adding the ‘dis’ prefix to its classic formulations*:

ELEMENTARY Discontinuity PRINCIPLE: If one endpoint of a segment is inside a circle and the other outside, then the segment intersects the circle.

CIRCULAR Discontinuity PRINCIPLE. If a circle *y *has one point inside and one point outside another circle *y’, *then the two circles intersect in two points:

In the graph, the continuity principles are in fact limiting concepts of boundaries and laws of perpendicularity, which define the discontinuities, closeness and connections between networks of points. What matters then to reality is NOT the obtuse concept of a ‘block/Parmenides like, solid reality’, with no gaps, at the core of the ‘mind illusions’ of Hilbert’s categorical geometry, but when two systems of reality cut each other in Darwinian, perpendicular events (the segment breaking the circle, the circle breaking the segment), which will DEPEND on who ‘owns’ the point M?

**T>S:** It is M part of the circle O? If so O is feeding on A-M-B, the line. Or it is M belonging to the line?

**S<T:** The line is ‘killing the circle, which is now open at M. Or it is M the M-outh of the circle? Then Amb is one of the multiple ‘parallels’ (as it does not properly intersect) feeding the circle and the O-perceptive point.

*S=T: 2 Points:** Or it is M – and this is the most special case, in ‘BOTH’, the line and the circle? Then M is an attractive point that cements the Union between both and a reproductive action is happening as now the point is actually two points at once, belonging to each of them. *

Those are the variational ternary principle applied to ‘laying on’, the undefined concept of the Axiomatic method we upgrade to i-logic geometry as we did with other undefined Hilbert’s concepts of points, which are fractal points, lines, which are waves, congruence, which is relative similarity, Non-E 5 Postulate, which is the definition of a mind, and ¬E Planes which are topologic organisms.

So the concept lay on, is the final element that completes the 4th postulate of congruence as it becomes either ‘a reproductive superposition’ or ‘a Darwinian intersection’ with 2 solutions, substituting Dedekind’s continuity axiom:

*“A point M of intersection between 2 relative paths, ð (closed figure) and $ (open figure), either belongs to ð or $ or to both figures. If it belongs only to ð or $, either figure is the predatory, dominant element of the intersection, and the event will be a Darwinian space-time event, in which the submissive past prey tears, extinguishing its form, as a ð<$ or past flow providing the energy for the reproduction S=T or evolution $>ð of the dominant future flow. If the point belongs to both figures, the event is a present iterative event of symbiosis, and both systems can form a stable, social new whole, where the point doubles as two.”*

So we translate abstract axioms into the vital how of perpendicular Darwinian intersecting or connecting, uniting motions that define dual creative and destructive space-time actions for both systems, as a motion ‘transforms’ one form into another, or pegs them into a more complex creation

We are not here using formal language, though any mathematician or physicist can write it with the usual symbolism of classic logic, and notice a few facts that expand the concept of laying on and show its power to describe reality reason why is used to set the foundations of other key branches of mathematics (Boolean logic and set theory):

– 2 systems lay on can be either a Union or an Intersection, different concepts in advanced i-logic geometry:

A Union is a perpendicular, Darwinian event where the part of one entity no longer belongs to it, so the dynamic event destroys one part. Thus if as a rule we capitalize the dominant system of a dynamic space-time event of relative perpendicularity we can write: A U b = A, meaning that b looses its part which will belong to A, as when you eat a rabbit that no longer is a rabbit but becomes your amino acids.

On the other hand an Intersection will be defined as a true sharing of those common points, so neither dominates, A C B, means the C part is now the connection that cements the relationship between A and B, which somehow ‘doubles’ and by this sharing, in physics there is attraction between beings. And in biology, there is attraction between beings. Intersection thus, sharing, is both a creative element and a social element of love and attraction. We share a child in a couple and that puts the two elements in a constant dynamic attractive relationship. Fermions share a boson and that cements an attraction between both. Predators share a prey intersecting their territories of hunting.

Further on sharing is more intense and symmetric when 2 systems are closed St elements, as in the figure of the circular continuity postulate, where we can clearly see there is an acbd region shared that pegs both systems together.

In the intersection of a line and cycle, the line seems at disadvantage, and in fact in most real events the line becomes absorbed and transformed as a pixel of information, coiled after it enters into the vital cyclical space that lays between the 0 point and the M-perimeter (which are not enclosed in the open ball ST region of cyclical motions that connect them). In advanced theory we shall see that in reality those lines tend to be prey of the circle, unless emitted by other system as an ‘entropy ray’, CROSSING the 0-point. In which case we talk of a killing line of entropy, which crosses the circle at M and O, and if we state that in that intersection O belongs to the Line, the equivalent vital proportion is that the line OM has KILLED the circle, targeting its zero-point. Indeed, If you cut the neck, if you shoot the head, if you conquer the capital, if you murder the financial people-caste or military-king in power, you disorder r=evolve, change and destroy a closed vital space-time being. All other lines that do NOT cross the 0-point tend to be lines broken, fed and processed by the circle, which becomes the sole ‘owner’ of the m point and the chord inside the circle, isolating the rest of the line, as closed cycles DO break in the Universe into fractal spaces.

If we talk of motion, we see the first region abcd as a region that ‘doubles’ its ∆-1 density of finitesimals, and will become the ‘seminal first region’, which will double then the whole system to create the B-centered new moving form. Motion by reproduction of form is thus closely related to the new concepts of continuity, which more properly should be called ‘reproductive displacement’. The ACBD region becomes the seed for a 3^{rd} child of A + B or the region of density growth that will become latter split by asexual reproduction (as in cells, which first duplicate a region in their central DNA zone), or it will become the region of the wave in which a gradient of an attractive field, with increasing density of ∆-1 finitesimals ‘drags’ the A-circle into reproductive motion.

One key question in the whys of physics is ‘why’, systems move in a relative field, its ∆-1 scale of the 5^{th} dimension, towards the gradient region of maximal density of force – so we move towards the attractive vortex of maximal charge or mass. The answer is that the system which is attracted and shares the same active magnitude and ∆-1 field, will bind in that region on the side of the density gradient, ∆ (ð/$), more finitesimals to ‘double its form’, more ‘energy-space quanta’ into which reproduce, and so we can see according to the ternary fractal principle of multifunctionality, motion as the feeding process of an entity, A in the graph that feeds on the field, on the gradient region of more density, falling inescapably by its greed of motion towards the region of maximal charge-mass. The field is controlled by the central charge mass which will finally eat up the smaller charge mass attracted by the bait of the field.

So we are giving here 2 key ‘vital propositions’ about the nature of motion, as a dual æ action (the larger model reduces all realities to the 5 vital a,e,I,o,u actions of space-time beings): The system both feeds and reproduces with the absorbed energy. And this can be done in 2 forms:

– The system feeds on the gradient of maximal density towards the stronger charge-mass, and in the process of feeding it reproduces its form into the adjacent region, either creating a son species if the action between both attracted points is symbiotic, parallel, so both use the field in equal conditions to input information and reproduce.

– Or the system feeds ‘alone’, reproduces its form in the adjacent region and slowly normally in circles to avoid its final demise, falls into the vortex of the stronger whole that owns the field, and truly is ‘farming’ the attracted particle, which finally will be digested by the stronger whole upon a perpendicular ‘Union’ – the star enters the event horizon of the black hole, the feeding pig enters the stomach of the farmer once it is finally attracted to the slaughter house by the channel of food that makes the pigs willingly enter its dead event.

Now, this is what I am interested most: to show the vital geometry of the Universe. A mathematician would be likely more interested in the logic abstraction of those postulates, and a physicist in its capacity to explain the whys of key processes of physical systems. As a philosopher of science, my goal is to show you the organic, vital nature of even the most abstract of all sciences, mathematics.

Thus lay on was correctly undefined in the classic sense, as it was never resolved in its 3 varieties. Things do NOT lay on a plane of the 5^{th} dimension, as then they will be above or below but not ‘into’, lay ‘on’, therefore is NOT a real event but a parallel event. A ‘laid on’ being is not into the being, it does not touch the being.

So for geometry what matters is NOT Continuity but topologic adjacency, with no boundary in the constant reproduction of the form, such as the membrane becomes the external wave form – the lack of separation in the process of motion=reproduction however is not equal to the classic concept of Continuity no longer needed to define a Pan-Geometry, as Bachmann proves.

So this brings us to the correction of continuity concepts in terms of scalar continuity where continuity only happens when we squeeze all the fractal sales of the Universe in a single flat line:

__IV__**. ∆:**Axiom of dis-continuity.

Continuity is not real I the Universe, and it is one of the great divergences between 5D and 4D – which compresses all the scales of reality to make them fit in a single plane, where then continuity exists. So the proper axioms of continuity as expressed by classic geometry, commented below hold now for the sum of all the planes and scales of the Universe, but as different families of numbers belong to different scales, any single line in a single plane will have ‘holes’ that the mid that perceives it discharges. The interest of the axioms of continuity is that putting all the planes together show that the scales of the Universe are infinite and through the sum of all those scales reality shows an horror vacuum, but any scale will have discontinuities required to differentiate its parts or else we could not ‘define a point’. So the concept of a finitesimal point, studied in detail in algebra when we analyze derivatives and the concept of a limit come here in full form. We thus add a 0 postulate of Dis-continuity to frame the classic postulates of continuity.

- Mental space is continuous, because all relative frames of reference cancel the dark spaces between points. However for points to exist as differentiated forms, they require discontinuities located into the finitesimal line in a lower plane of the fifth Dimension. Lines are thus strings of points, whose density increases as we diminish the plane of exist¡ence we observe, till reaching the maximal density of continuity in the irrational line. So we must distinguish the continuity of mental spaces for which the classic postulates of continuity hold, and the discontinuity of objective scalar 5D planes, where continuity is only achieved as the sum of the limit of µ planes.
- Let X
_{1}, X_{2}, X_{3}, ··· be points situated on a straight line such that each succeeding one lies to the right of the preceding one, but that there is a point A lying to the right of them all.* Then there exists a point B that also lies to the right of all the points X_{1}, X_{2}, ···, but such that a point Xn is arbitrarily near to it (i.e., no matter what point C is taken to the left of B, there is a point Xn on the segment CB).

A number system constructed from the reality of a discontinuous world IS preferable to the ideal continuous real number system fabricated with Dedekind’s false axiom of Continuity. Instead 3 less strict principles suffice to explain the different virtual continuities perceived as motion of the 3 elements of any system (| x O ≈ Ø): lineal & circular continuity (predatory union) or ø (reproductive superposition), and the Archimedes and Aristotle classic axioms of relative space-time proportions.

Dedekind’s axiom is then a different concept – that of barriers, limits in a scale of numbers whose holes in the real line are similar to ‘potential wells’, quantum jumps that are difficult to cross. Irrational numbers become then discontinuous gaps: π & √2 are the gaps and apertures that prevent the circle and the square triangle to be perfect. For example, it can be used to prove the existence of limiting parallel rays in hyperbolic geometry with far more simplicity than using the Aristotle axiom.

Of course, Dedekind’s axiom is needed to obtain the categorical axiom system of the Hilbert. Yet precisely for that reason, because it is not truth and real, it merely shows that Hilbert’s axiomatic method is false, it is an error of the mind that confuses its limited perception of the ‘holes’ and open wells of the Universe (those limiting ratios or real numbers) by the mind, with reality. It is like the case of a continuous movie perception. In fact the movie is stop and go, with holes but the mind puts them together into a continuity picture.

*Continuity is a Maya of the senses that eliminates dark holes between perceptions of the brain.*

In other words, the brain, the mind-world is continuous, reality the larger world is not. Dark spaces are easy to calculate for a p.o.v. with a relative 3 diameters to form its circular perimeter, which will leave 0.14… holes to observe. So the point does NOT observe, 96% of reality darkened by the perimeter of 3 diameters that closes its outer membrane. So it sees, 0.14/pi = 4% of reality, which is what we see in the Universe (96% being dark matter and dark energy). Yet our electronic eyes do not perceive a 96% of darkness. Darkness is eliminated to expand the enlightened 4% as if it were all the reality.

Expanding the 0 postulate, *objective continuity will require to see all the scales of *5D planes and its hyperbolic reality, which would include the 96%. But then our view would be a hyperbolic geometry, as our eyes would be crossed by µ (a relative infinite0 number of parallels. Such geometry and angle would be convex, with us, as the knot of an expanding fractal network of branching angles that connect us to all the micro ∆-1 points, constantly growing its perception of the Lobachevski’s hyperbolic branching, richer world that 5D metric proves.

Unlike classic geometry, a straight line is NEVER created only by 2 points. The classic definition of Euclid, naively accepted by those supposed Hilbertian r=evolutionaries: ‘a straight line joins two points’ does no longer holds. We obviously need 3 points to connect 2 points, one being shared, and only then we can see if the 2 points are joined in a ‘curved’ form, by an arch, or in a straight form, by tending an AM and MB intervals, and looking at the ‘angle ‘between Am and MB, which if it is a straight angle will define a straight line. This is obvious –2 points cannot define the straightness of a connection; that it surprises me it has been overlooked for so long, as it is also a key concept to properly define what kind of geometry we are into, and a good way to introduce the other 2 axioms that substitute continuity and relativity of size:

They are classic A^{2}xioms of Greek Geometry (Archimedes, Aristotle’s axioms; in my Leonardian notebooks, written with shorthand incomprehensible Spanglish, i-logic weird symbols, which perhaps in the future some robot will try to decipher, he will find my abbreviation of those 4 Axioms of continuity and angular perception, written A^{2}c^{2}ioms, ab. A^{2}c^{2} 🙂

They are concerned with the perception of size and its comparison from a given point of view. And again, as always in the dual/ternary Universe, as in the case of the lineal and circular continuity principle, we have one axiom dealing with lineal sizes and the other with circular/angular perception of sizes:

**ARCHIMEDES’ AXIOM**. If CD is any segment, A any point, and *r *any ray with vertex A, then for every point B *≠*A on there is a number *∆* such that when CD is laid off *∆ *times on *r *starting at A, a point E is reached such that *∆ *x CD ≈ AE and either B = E or B is between A and E.

I.e., if AB were *π *units long and CD one unit length, we need 4 CD to get beyond B and enclose π inside our straight line. And this is what matters to ‘enclose’ or not a certain ratio within the larger envelope, to enclose our dark number pi, so we know is within us (the whole cycle) even if the cycle is fluctuating around the non-defined π.

Moreover the axiom sets limits to infinitesimals, defining the finitesimal unit of measure AB on the lower side and the whole AE on the outer contour side.

Archimedes’ axiom thus means that when Nature chooses a finitesimal CD as a unit of $ length, a quanta is established for a scale or plane of 5D to exist and every other segment

Will have finite length with respect to this quanta which becomes the ‘relative definition’ of a number.

And inversely if we have the perspective of the whole, we choose AB as unit of length. And then the axiom says that no other segment can be infinitesimally small with respect to this unit (the length of CD with respect to AB as unit is a at least 1/n unit). 1/n was indeed in Leibniz’s Infinitorum the finitesimal unit.

*Those 3 axioms suffice to prove as mathematicians know, all the theorems of geometry.* Moreover, and this is the beauty of it, if we want to get rid of numbers and do a purely geometric analysis, this postulate, which connects numbers, points and lines, can actually be substituted by a mental postulate:

**ARISTOTLE’S AXIOM.** Given any side of an acute angle and any segment AB, there exists a point Y on the given side of the angle such that if X is the foot of the perpendicular from Y to the other side of the angle, XY > AB.

In the graph, XY grows faster than Vx or Vy as we come further away from V and the angle becomes hyperbolic, so we can always find an XY larger than vx, even if paradoxically V has the impression from his Point of View, that XY is becoming smaller. This relativity of world perception versus real Universe is at the core of many errors of the ego who believes to be infinite when in fact he and his relative distance to XY is really small. It often means that if XY is a ‘future point in time’ (death point, when we use geometry to study worldcycles) or a predator in distance, we will underestimate the danger of death, and XY will grow very fast and eat us up (-: )-: O-:

The smaller part, or fractal faster point becomes then a knot of a fractal of reproduced angles that keep enlarging reality till it can map out a much larger scale, which shrinks through those fractal triangular branching to focus and coalescence in the center of the fractal network, at ∆-1.

The duality space v. time stillness v. motion is a geometric formal view of 5D metric: S x ð = k.

As we become smaller, ðƒ, time accelerates. Inversely, to have a still, geometrical perception with no motion, we have to decelerate time cycles of smaller beings and expand its space size. So if the faster motion of cells and atoms slows down to our speed of existence, we would have to expand its space size to be as big as we are after that geometrical expansion and time reduction.

*So we substitute the concept of ‘single scale distance and continuity’ – *Dedekind’s axiom – by Aristotle’s postulate, which changes real continuity by relative, angular perception of distances, from the perspective of a points of view, with deep virtual-world-mind implications even if we prove the same theorem.

The postulate of Aristotle merely says that from a given angle of perception, the line that joins the limits of our perception and closes the open angle of vision is larger than any of the two sides of our angular perception.

In other terms, the ‘perpendicular’, not parallel, horizon or ‘front of the wave’ of perception expands much faster than the distance between us and the being we perceive in other scales, is proved in a single space-time scale by one of the key new postulates that.

The Universe expands faster in objective terms (the perpendicular, far away line of expansion of our horizon), than from the perspective of the perceiver of a certain geometry.

What all these new ways to define the parameters of continuity tell us, is that what matters to systems is the relationships between beings, and the relative perceptions beings have of the Universe deformed by its angular worlds of perception.

**RECAP**. It is important to differentiate geometric, continuous forms from numerical discontinuous series, as both represent two different ‘elements of reality’. Continuity appears only when we include the concept of ‘motion’ (not defined by Euclid), and even so it is a mirage of the senses, because at quantum level it is a reproductive continuity. This is show in social growing natural numbers. Thus there is NOT an exact correspondence between points and numbers, as the failure to find pi, √2 and e, the key ‘ratios’ of the Universe, which can be ‘drawn geometrically but have no direct exact solution arithmetically, proves. The number system can be properly used to construct geometry only when those are taken into account.

Continuity can be considered in the geometric world, only from the perspective of adjacency, and motion as reproduction of form, in adjacent places of a single plane of reality:

In the graph, taken from a physical wave, a particle reproduces its forms as it moves as a wave of adjacent particles one after another. This is the definition of motion, which solves Zeno’s paradox. Proper motion does not really exist, but reproduction of information along a path, with limits for each world and geometry (in the Euclidean human space, with the limit of c-speed for transfer of energy-form). So continuity can be defined in a single plane with the postulate of adjacency.

In precise terms given the lack of true continuity in the 5D universe, the 4 postulates that substitute continuity as proved in the work of the key post-war geometers are restricted to a single plane of space-time, and truly define more than continuity processes, *the other key elements of i-logic geometry.*

**III. T: Axioms of motion as reproduction of form.**

(A motion is a transformation not of an individual figure, but of the whole plane.)

- A motion reproduces=carries straight lines into straight lines.
- Two motions carried out one after the other are equivalent to a certain single motion.
- Let A, A′ and a, a′ be two points and half lines going out from them, and α, α′ half planes bounded by the lines a and a′ produced; then there exists a unique motion that carries A into A′, a into a′ and α into α′. (speaking intuitively, A is carried in A′ by a translation, then the half line a is carried by a rotation into a′, and finally the half plane α either coincides with α′ or else it has to be subjected to a “revolution” around a as axis.)

The axioms of motion *again reduce and simplify the differentiations happening in the vital Universe. So we must stress those basic principles that so clearly differentiate the exhaustive goal of i-logic geometry – to describe all vital events and species departing from the complex ∆¬@st properties of space-time vs. the simplifying synoptic nature of mathematical mirrors in the axiomatic method. *

So here we have to substitute the word carry by the word reproduce. A motion reproduces a form as it ‘carries’ straight lines to other place of space-time where the information is reproduced on the lower ∆-1 plane. So in ilogic geometry it DOES matter unlike in Euclidean geometry the path of the motion from A into A’. Though if we consider that pure motion has minimal memorial persistence of the reproduced forms, in the limit of TT-entropic motions it will not matter the path as the persistence of memory will not make them last. But ultimately geometry makes fleeting motions=communications between points to last. So such motions are in fact complex entangled motions whereas the 2 entangled points still to each other keep also reproducing over a lower plane as they displace together. It is then evident that the axioms of motion deliver the maximal simplification of that complex process observing the line in motion but still to each other end-points just in its initial and final picture.

How mathematics gives then depth to reconstruct from such simplifications the whole range of events happening takes place not in geometry but in Algebra with Groups that exhaust all possible diversifications of such ST-events.

The axioms of motion are in that regard, the basis of Group theory applied to modern physics – a great advance, for the classification of all possible variations of events and forms.

In detail the 1^{st} axiom defines reproduction with an abstract word, carrying, used in physics, relativity theories, vector spaces etc. But its vital nature is rather intuitive in the 5D worldview.

The 3^{rd} is more interesting, and complex because it is connected to the previous analysis of chirality parity etc., and will be of great importance to distinguish different species of physical systems, according on how $ (lineal motion) and a ð (rotary motion) are put together to return or NOT the point to the same initial state.

*As it further establish the processes of creation by reproduction of new forms, using the only ‘conserved motions’ – rotary angular motions (O) and lineal boosts (|), in a single plane. But certain forms which are mirror symmetry cannot match only with rotations and boosts, but need a 3 ^{rd} ‘mirror symmetry’ which implicitly states that any system ∆ will be part of a larger dimensional ∆+1 system to achieve its completeness. And so besides lineal and angular motions, reality requires a mirror symmetry in a higher dimensionality, to achieve full reproduction of forms. And this altogether makes reality more complex, generates new creative gender forms and also establishes the non-* commutative closed or non-commutative nature of a bidimensional system when completed in the higher scale, as two consecutive $ and ð-rotary or mirror symmetry actions differ, because both rotary and mirror symmetries are different according to which direction the action takes. So inverse dualities kick in to start complex differentiations.

However l-motions by definition become with time, closed actions and should return to its origin. So the question opened with deep consequences in the study of worldcycles and present stretches of the virtual existence of all of us, is how many ‘motions are needed’ to return to the same form, regardless of the differentiation in the intermediate paths – are 5 dimotions enough or shall we need a dodecalogic 12 steps method?

We leave open the question. Ultimately the labyrinths of present stœps of existence in which the path incurs are more important than the goal which will be a 0-sum that kills the path and the existence of the event-being.

Still the axioms offer 2 solutions: the point can be carried in an open path by pure motion, without memory and then the path doesn’t matter, and the point continues as an isolated motion, free of ‘responsibility’ for its tail of reproduced elements. Such memoriless Markowian processes define then an |-open field or limb system.

Yet a reproductive motion with memory that leaves a track is a curvature with memory, a closed cyclical point which will ‘crowd’ its dimensionality at the point of reproduction.

And indeed, often when the point closes its circle it exhausts the motion of the being, which dies after reproduction (from octopus to arachnids). A closed path kills a system, crowding the zero point of reproduction.

But an open system also becomes exhausted as the intensity of the line fades away into entropy and noise. So after a finite number of stœps a thermodynamic open system even if it does not return to its original point, becomes exhausted.

All this bring us to the fundamental metaphysical question derived of all those axioms: It is the Universe closed or open, infinite but finite, un-bounded? It will return to a point? All the answers being truth as reality includes all its paradoxical dualities and trinities:

The universe is infinite in global timespace but finite in local time space T.œs. Which are, once we get rid of the ego paradox, infinite in its future repetitions, because the number of repetitions and variations is a smaller infinity than the number of fractal domains, ‘broken by a cyclical enclosure (3^{rd} axiom of order)’.

This duality of infinities between the types of beings and its repetitions will be in metaphysics the only correct use of Cantor’s transinfinities to prove that ultimately we are all immortal because we are repeated likely every 10^{9-11 }variations as the fact every 10^{9} humans we find an undistinguishable repetition shows If you think this is too abstract think again. Because the Universe is infinite in its repetitions, finite in its variations an exact replica of yourself is now about to appear once you die in other region of space-time.

And so we move to metaphysics, in a Spinoza’s sense: are those other ‘yous’ repeated, connected entangled to ‘you’, as identical quantum systems are? Are you part of a wave of reproduced ‘carried’ motions as a block of time? Are you going to live beyond death by transferring non locally to another you, born when you die? It is the dark space between the two informative forms of time that you are, a discontinuity bridge by the finitesimal mind that does not see that space. *There is transmigration of souls?* Alas you see, Hilbert would have never imagined that from his axioms of motions we could move into the Pythagorean theory of metempsychosis (: reincarnation… The end of i-logic geometry brings us the first questions of platonic mathematics. How many yous exist reflections of the ideal canon of the cave? But those are questions that belong more properly to the dodecalogic of worldcycles. And the pentalogic of entanglement. So we stop here. Neither as we done it already discuss, just enunciate the 2 versions of:

__V__**. @:**Axiom of parallelism (Euclid).- Only one straight line can pass through a given point that does not intersect a given straight line.

These axioms, then are sufficient to construct Euclidean geometry in the plane. All the axioms of a school course of plane geometry can in fact be derived from them, though their derivation is very tedious.

The axioms of Lobachevski geometry differ only in the axiom of parallelism.

V′. __Axiom of parallelism (Lobachevski).__

- At least two straight lines pass through a point not lying on a given straight line that do not intersect the line.

Indeed, in ¬Æ we get rid of IV continuity (Dedekind’s in simpler language) and Parallelism according to Euclid.

So once more we see that the basic axioms and postulates and laws of mathematics are mirror of ST-laws, albeit sometimes too simplified so huminds have lost track of what they were mirroring in first place. Which is what we shall show by departing from the ∆@st reality NOT from the mirror itself as the axiomatic simplifying method does. The conclusion though is obvious:

**All the self-evident axioms and postulates of Euclid are relative truths.**

Reason why they are superseded by the Non-Æ=i-logic postulates of geometry based in fractal points with breath to better connect the experimental reality and ideal geometry. This has not been done as mathematician only corrected the fifth Euclidean postulate, by lack of a proper theory of reality to define concepts as dimension or distance=dissimilarity or the different types of congruence, or time or space, which we now have, coming from a higher language, the ‘¡logic laws of the fractal space and cyclical time of the Universe. So we can reinterpret many of the postulates and axioms of Greek geometry and correct them, as they are under the correspondence principle, just approximations to the scalar Universe observed in a single plane, without perception of the inner parts of the point. Let us then revise, the five Euclid’s postulates:

- It is possible to draw a straight line from any point to another point. .
- It is possible to produce a finite straight line continuously in a straight line.

This means that all points of a present space can be connected in simultaneity because spacetime is continuous; but in reality is not. As there are ‘irrational points’ which are in a different 5D scale; closed membranes that break the continuum. So only points that belong to the inner vital space of an organism, connected through ‘physiological networks’ can be connected in the same present space.

While points from two scales are only connected when they belong to the same superorganism, and they are similar in form, despite being different in ‘size’ (5D scale). So we can connect points within one of the 3 topologies (membrane, vital energy, inner singularity), and we can connect smaller parts of a fractal network with its larger physiological form (cells through blood and nervous networks of the ∆+1 scale and so on).

And they will often a ‘curved connection’… Moreover all straight lines become bent into a zero sum curve, as they will be part of a fractal superorganism. So infinity does NOT exist lineally but quite the opposite in cyclical self-repetitive patterns, as all straight lines ultimately find a limit in the curved closed membrane of its superorganism.

- It is possible to describe a circle with any center and radius… which might be the only true postulate, meaning that all straight lines will be part of
*a closed time-space cycles of 3 ages, which are the 3 pi diagonals, closing a worldcycle, which has a finite zero sum volume and breaks infinity into inner and outer parts.* - All right angles are equal to one another.
*This is not truth in different scales as the fifth dimension is a hyperbolic geometry whose relative curvature and degree of flatness depends on the relationship between the rod of measure/size of the observer and the size of the observable (in formal space this is the realization of the time acceleration=increase of curvature of smaller beings).* - The parallel postulate, already known to be false by classic science is related to the previous one and we have also proved its falsity.. So actually 4 or the 5 postulates are false in the real world. And only the postulate of creation of circles departing from lines, it is real in temporal terms, as all systems close its worldcycles, that is they die.

So are the definitions of a point with no breath, a line with no breath, as points are fractal points with hidden volume in a smaller scale of parts, lines are therefore waves – points cycling; planes are then not defined by lines but by networks and its flows’, and so on. Those* self-evident definitions are all false. *

Finally, the Elements also include the following five “common notions”; 4 of them concerning equality, which are not ‘false’ but rather meaningless, as things are ‘similar’ only a thing is equal to itself, since we do not have the total information of beings, neither things which occupy different spaces – as they *are made of space and time – *are equal, just merely by changing position, the thing becomes other thing (themes those of extreme importance in quantum physics to differentiate bosons and fermions – systems that occupy the same space, and hence are equal, and things that do not occupy the same space :

-Things that are equal to the same thing are also equal to one another (formally the Euclidean property of equality, a consequence of the transitivity property of equality).

If equals are added to equals, then the wholes are equal (Addition property of equality).

If equals are subtracted from equals, then the remainders are equal (Subtraction property of equality).

Things that coincide with one another are equal to one another (Reflexive Property).

The whole is greater than the part. Finally, to thoroughly bust your balls/beliefs … (: yes, you have guessed it ): *the whole is not greater than the part, if anything they are equal…*

Or rather similar in existential momentum $ x ð, according to the metric of 5D: $ x ð = K. Which somehow is implicit in set theory and the paradox that tell us the set of all subsets is bigger… and even smaller if we *merely measure its quantity of information that grows inversely to size. *As this postulate is closely related to our ‘understanding of the scales’ of the Universe is worth to elaborate a bit more.

**The whole is not greater than the part, neither smaller (:**

The world we measure and call physical is a mental world. Consider instead the real scalar world of the infinitely divisible. There any whole is infinitely divisible, but so is any part of that whole. As a particular example, in mathematical analysis, any line segment is identical in every way to any smaller line segment that is a part of it. This suggests that the fifth common notion may, in the description of the world, be not the only true one.

We can extend notion 5: The whole is equal to the part’ to the particular case that all parts of the whole are equal to each other, from where we deduce identical particles in physics, where all electrons, protons or photons are alike. Yet, we can go further if we merely ‘measure’ the information/time speed/energy density and affirm the opposite, that the whole is less than the part, as its ‘time-motion/energy/mass is greater in more tightly concentrated forms. So *the black hole which is in to 5D metric, $ x ð = k, the smallest mass is actually the greatest/densest /heaviest*.

**Mental spaces and its value as partial truths.**

What can then we save from Euclidean geometry if everything about it seems wrong? (: Almost all of it; because rather than wrong is a simplified selective version of reality which work as long as the properties we select on that reality for practical purposes are ‘external to the fractal point’ not concerned with its internal parts, as in the previous case similar to engineering design where mass is what matters to us in the ‘selected mental space’ of forces. Or when motion is more important than form, as in physical equations. But absolute truth will require always considering that internal volume of fractal points; and hence its wave-like nature as in quantum physics.

Because only the whole Universe has all the information about itself any O-Mind x ∞ Universe will be valid as long as the infinitesimal information it extracts helps the species that holds such a mental space to measure reality and store information helpful for survival. In that sense, the most important element of bidimensional geometry is the understanding of angles, parallelism and perpendicularity, which is the first element developed in Greek geometry to be able to measure distances with trigonometry. This already is done by the eyes’ mind-mappings of distances.

But we add to it the topological laws of the 4th Non-E Postulate of congruence, which *adds a vital meaning to trigonometry as *Parallelism is also *as its opposite concept of Riemannian distance, a measure of ‘similarity’. So if in Riemann’s dissertation distance of ‘color space’ is equal to dissimilarity in the 4 ^{th} Non-E postulate parallelism implies similarity* which fosters social evolution vs. Perpendicularity that implies dissimilarity.

And *this becomes objective reality when observing ensembles of fractal points of any species* and its angles of connection of those points into geometric figures, in a still space view, or when one of the dimensions (S=T symmetry) is seen as motion, in the way herds move in parallel or predators intersect preys.

So we establish a new duality besides the S=T duality of form and motion, derived of the internal parts of all fractal points; that between the subjective mind and the objective external topological view of an event. In this case ‘dissimilarity’ is a condition of internal congruence between 2 fractal points bring external parallelism or perpendicularity.

And so a new rule of the entangled pentalogic Universe must be added to S=T, ‘x=y->x≈y; x≠y->x X y x y; that is, if two systems are congruent, similar in its parts in a still spatial comparison (x=y), they will move in parallel (x≈y); if two entities are different, x≠y, they will cross in perpendicular motions, x X y, or separate in its distance (x y).

A key element of all spaces is ‘the angle of perpendicularity’, which acquires ¡ts meaning when we marry @nalytic geometry and the 4th non-E postulate of congruence based in Darwinian perpendicularity and social parallelism. It is the key for the understanding the mathematical physics of vector spaces, cross products, dot products, equipotential and lines of forces, which happen in different planes of 5D and affect different parts of an entangled, field/wave/particle supœrganism.

i.e: Particles are Darwinian over fields in which they prey and waves are parallel to the field in which they ‘slide’.

Thus trigonometry of angles, was the fundamental first mental perception because it is how real systems measure distances, which is a function of similarity and hence of vital survival. Angular perception is also realized in the simplest physical systems, were the unit of information, h, the minimal space-time ‘Planckton’, has several position of quantized angular momentum, the only positions we can measure as information is always processed in ‘still form’ (missing its motion steps as we miss the motion of a film between frames), according to its angle, which determines its interaction with other particles…

**Difference between equations and Geometric curves.**

The characteristic features of algebra are the use of letters, which we perform operations according to definite laws. In elementary algebra the letters denote CONSTANTS, normally ordinary numbers, taken as populations in space, the variables, which are the final letters represent T.œ.s of a certain species, and the operand represent different ‘dimensional motions’, dimotions of time-space.

So we can reduce equations to a series of existential algebraic equations of the type:

∑T.œ ST-perandi ∑ T.œ ST operandi = ∑T.œ ST-perandi ∑ T.œ ST operandi

Whereas the a…. p letters will be numerical parameters, the U, V, X, Y, Z letters Timespace T.Œs and operandi dimotional parameters.

*As such there is a fundamental difference between an equation and a geometric curve in analytic geometry, despite its apparent similarity. An equation searches for a single solution as the XYZ letters represent ‘spatial populations’ and the parameters of ‘time change’ that convert the equation in a time event are the operands.*

*On the other hand in a curve the XYZ letters represent variables, whose simultaneous possible values, joined by a geometric non-E line form of simultaneous ‘spatial membrane’, so they are events of space, duly studied in our ‘geometric first volume of 5D mathematics’.*

*This is a huge distinction that makes completely different the study of simultaneous curves in space, which act often as membrains of superorganisms; to the study of algebraic equations, which describe events in time, often of a sequential nature, gifted with motion.*

We shall come often to those philosophical distinctions that mathematicians escape perhaps as they find them evidents or on the contrary as they do not have a Gst understanding of Generational space-time to find them interesting, but it determines some obvious facts that differentiate curves from the polynomial representation of them in equations and will reveal on our analysis of algebraic curves key concepts of S=T symmetries:

- Curves are far more reduced in number – 10 canonical curves in 2D and 18 in 3D suffice to define all of them. Since the number of ‘spatial superorganisms’ that survive in the Universe is far more reduced than the total flows of entropic time motions, from where we obtain them. As in turn, those canonical curves can be reduced to the forms extracted from a cone, which represents a worldcycle of timespace.
- So all curves and hence all ‘topological membrains’ and forms of the Universe can be reduced to the 3 topological varieties we find in the cone, which is in itself a ‘circular membrain’ tracing a worldcycle along the axis of the cone, as it diminishes its size, compressing its form, in the natural evolution of all T>S systems.

All those differences perfectly understood in the I classic Greek Age when equations were written statements using verbal thought, hence temporal verbs, as opposed to curves described in space, became somehow blurred with the work of Descartes that merged both in analytic geometry. Let us study them.

**2**^{ND}AGE OF GEOMETRY:**@NALYTIC & DIFFERENTIAL GEOMETRY**

“Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.” Ulam.

Youth is a lineal simple age of small steps that all worldcycles of existence keep bending, so in maturity we learn the eternal return cycle. In mirror languages also the second age curves the lineal steps of the first cycle. So after the lineal age of Greek Geometrty, curves entered the stage.

Yet to study them, two fundamental advances, frames of reference and calculus were needed.

@nalytic geometry studies the different planes of mathematics as mirror reflections of the different topologies and planes of existence of a super organism, self-centered into an @-mind. Analytic geometry was the first mathematical form that successfully merged the 5 Elements of reality: ¬∆@st, and as such it signified the beginning of the mature second age of mathematics, after its first ‘spatial’ bidimensional Greek age of still geometry.

The duality of @nalytic geometry is thus obvious: *S-mental spaces self-centered in the 0-mind*

∆-Scalar spaces developed in Sequential social numbers

T-ime spaces, allowed by the use of one coordinate to represent a symmetric dimensions of time

The flexibility of the concept of a plane of space-time to represent numbers and points thus allow to represent not only the S POINT ≈∆ NUMBER symmetry but also the S-T dual space-time, form=motion symmetries and it will have an immediate consequence on the development of science as it will canonize the concept of a lineal time and space, proper of mathematical physics ever since. So a key theme of @nalytic geometry is the a priori study of the distortions we introduce when setting functions and operands on an artificial world that we take as a ‘background space-time’. Since we must be aware we make local assumptions NOT global ones, as when we use lineal time NOT all the times in physics to measure locomotion. Unfortunately this locality of frames of reference was lost and brought the error of absolute Newtonian time space.

Thus the mind in mathematics is reflected by its frames of reference; and so we apply pentalogic to classify the main uses of @nalitic geometry, designing the different perspectives of the parts of a being and its 5 Dimotions:

**@:** the frame of reference is the mind view, with a 0-1, finitesimal ‘body-energy’ and a 1- external spatial world whose equations are self=similar in its T=S symmetry. Indeed, we can consider the fundamental equation of analytic geometry:O x ∞ = Constant world, the equation of any mind, whose perception of the infinite whole in its biased frame of reference creates a relative world, or mind-mirror of the whole…

**S-vital** topology gives us a more objective use of those 3 different frames of reference and ∆-scalar symmetry represented in the Complex Plane to represent events taken place in a partial element of the supœrganism:

Because there are 3 topologies and then scales it should exist 3 types of planes and then one for scales. Intuitively they are the lineal-cylindrical, polar-spherical and Cartesian, hyperbolic that better represents the merging body-waves of the two others. And so that leaves the complex plane of ‘squares and roots’ as the natural one to cast functions through scales.

**Pentalogic analysis of spaces by merging the 4+entropic perspectives. Mental spaces.**

Pentalogic merges mental, temporal, organic, entropic and social spaces in the analysis of the laws of vital geometry. I.e. animal sight codes for motion over form and red over blue for survival purposes (the eye is a natural born predator, which prefers red blood and prey motion as in gore movies), So we** merge objective geometry – its 3 topologic organic varieties and subjective mental modeling of reality, which ‘deforms’ the Universe to match those topologies giving birth **to the 3 classic frames of reference: $t-cylindrical, ST-Cartesian (the humind light view), ðƒ-polar geometry, *plus the complex ‘square’ plane to study the 4 ^{th} and 5th scalar * dimotions. Each of those frames deform reality, creating 3±¡ type of geometric minds, according to each species choice of coordinates:

The 3±¡ different geometries and frames of reference as self-centered mental spaces describe different worlds. We introduce the new discipline of mental spaces, with the Humind light space-time.

Space understood as simultaneous perception of adjacent forms (relational space-time) *is the realm of the mind’s logic, as the mind creates its stillness. Contrary to the belief of many physicists who think time does not exist, what does NOT exist is space outside the singularity of each mind, and hence there are ∞ spaces, one for each mind’s world: *

The mind is the internal non-perceivable element, it requires the concept of an angle to establish its range of perception, which is not a number, and hence a ‘different’ element of mathematical representation of reality.

The O-polar, |-cylindrical and Ø-hyperbolic Cartesian plane in the o-1, 1-∞ time-space scales describe the 3 topologies of all systems in a single scales; which are further expressed by the complex plane in its ‘squared form’. They represent the subjective plane point of view ‘coupled’ with the objective plane geometry:

In the graph, all systems have ternary topologies with 3 motion-information-reproduction functions that absorb entropic energy, mix it with information, to achieve movement and perceive to keep in existence.

Thus the simplex aei action organs exist in all ST- systems of Nature.

Vital topology becomes then a constant merging of TT>Ts (entropy that becomes locomotion) and §eeds of information that germinate into motion, SS<St to find a middle point of reproduction in hyperbolic body-waves.

We can study some of those motions in still form, or vice versa simplified form in lineal motions taking advantage of the S=T duality to obtain quantitative results but the vital reality tends to balance motion and form dimensions.

Because our mind is visual, Euclidean Analytic Geometry’s fundamental plane is the Cartesian plane, due to its perpendicular structure. It is also is according to the 4th postulate of Non-Euclidean geometry the best form to represent both ‘perpendicular’ events and space-time inverse states≈ Dimotions where length=motion, width=reproduction and height=information mimic the natural symbiotic structure of systems of nature.

Further on we can ‘reduce’ the 3 dimensions to *a holographic single plane, compressed by elimination of one of its s=t dimensional motions. Then it is best to analyze systems which have an internal *S=T Symmetry or systems represented by a ‘fractal point’ unit along an ST given dimotion.

This is the justificiation for its use in physics, were the dimotion of locomotion of a fractal point whose inner parts are ignored, reduces to its study as an sT locomotion, on a bidimensional Euclidean plane, where the dominant element, $T, the lineal motion-space, is represented in the X-axis and v in the Y-axis.

We can then consider the basic duality of Space-stillness vs. Time motion, in 2 other main frames of reference:

**-Vector spaces** add a dynamic, temporal view, *which makes each point an ST element with the S-population parameter and T-motion parameter best suited to represent existential momentums* in any ¡-1 ‘field scale’. So Physics uses vector spaces for locomotions in lineal space and time.

-Abstract mental spaces (Complex, phase spaces), which are the spatial mental static view.

**Complex planes.** -5D ST analysis in the complex plane, where the polynomial=scalar degree of coordinates are different (either a root vs a real number or if we ‘square’ both axis, as we do in ∆st, a squared double positive real line vs. a ± i² real coordinates). Huminds though have not a clear philosophy of the ’emergent’ complex plane that study spacetime mixed functions and Dimotions of groups of Dimotions (functionals).

**Phase spaces** describe entanglements of different ‘states of space and time’. A subjective selection of the parameters of time and space and its generalization to all type of dimotional systems, Phase spaces will completely liberate science from the immediate reality of the Euclidean world we perceive.

In the human scale we analyze the ternary points/states/ages of matter in those phase spaces (state physics).

-Mathematical phase spaces are the manipulation of mathematical functions on those planes, which often will be not only abstract functions of mathematical space but expressions of the real dimotions of existence, essential to the fields of ‘Mathematical physics’ and its special frames of references; and the main field of 5D analytic geometry whose experimental task is to relate those abstract ‘conic curves, complex planes, and different dimotional parameters of the main 3±¡ analytic frames of reference to real events and forms of spacetime.

So phase spaces finally detached mathematical analysis from the light space-time reality of the human eye and portray a static mental space-form with information relevant to the perceiver.

**∆-scalar perspective.** The Cartesian plane gives us also 2 more generalised perspectives closely connected to the scalar dimensions of a system.

**-0-1: The Unit circle,** which can be used to explore the paligenetic cycle, expressed mostly in probabilistic temporal terms (where 1 is the value of ‘existence’ – the happening when an event/form emerges into the 1-∞ equivalent plane). Whereas mathematics (Measure theory) allows parallel studies between both lifecycles.

**-4D entropic parts**: In planes that break the whole into all its points of view, with infinite generalised coordinates of individual points and statistical ensembles of entropic particles and µ dimensions, of which Hilbert spaces used in quantum physics and phase spaces, used in thermodynamics are the main varieties.

There is THUS as usual a closed ‘homeomorphism’ to use some pedantic math jargon (:’a correspondence between two figures or surfaces or other geometrical objects, defined by a one-to-one mapping that is continuous in both directions’:) between the 5 Dimotions of reality and the 5 main graphs of mathematics, cartesian, polar, cylindrical, complex and Hilbert’s; as reality can always be seen from the point of view of the time functions and space forms of those 5D dimotions.

All this is fairly well understood by scientists, except perhaps the complex plane; and the interaction of planes of different scales, as those which happen when an Active magnitude preys on a lower ¡-1 plane perpendicularly; that is processes happening between different 5D scales.

**RECAP.** The main duality of analytic geometry happens between @-mind center of frames of reference to reflect the mind’s subjective point of view and the 3 topological objective organic planes and the scalar complex plane as representations of the internal form of events happening within the key organs and ST symmetries of Nature. So there are 3±¡ main frames of references in analytic geometry, studying reality from a 0-point of view, similar to that of the mind of *each of those partial organs, which therefore map out clearly different worlds and prove the relativity and difference between the infinite mind-views of the Universe, even when they use visual light spacetime*.

And we find a direct relationship between the 5 Dimotions and the ‘frames of reference’ of mathematical minds.

As @nalitic geometry give us 5 sub-planes, each one a frame of reference that studies reality from a 5Dimotion:

3D hyperbolic Dimension: the Cartesian, ∑∏ graph.

1D vortex: the polar, ð§ cyclical graph.

2D Lineal motion: the Lineal, cylindrical $t.

Each of those graphs study problems regarding forms and functions of those 5 Dimotions.

The multiple planes of analytic geometry, do not hide its fundamental property: to perceive reality from a given point of view, the 0 point, or @-mind, and create from that perspective a certain distorted worldview that caters to the function and form of the @-mind, selecting information of reality to form a given space, which might seem ‘reality’ to the mind but will always be ‘a representation in which the mind will exercise its territorial will’ to paraphrase Schopenhauer.

Thus Analytic geometry ultimately studies the Universe from the perspective its 3±∆ main ST dimensional subspecies or partial equations of the fractal generator equation of T.œs

**The a priori reality of other ‘systems’.**

‘Spaces’ are still forms, which in a Universe in which the fundamental ‘substance’ is time=motion, represent ‘still mirrors’ of those time motions. Hence spaces are artificial constructs of the mind, whose ‘languages’ create spaces to ‘navigate’ reality; built with the features of the forces of information available to them.

They are the a priori ‘Kantian’ categories that deform our reality and vice versa, knowing the properties of a given space it defines the type of mind and species that perceives it and navigates through it. However mental spaces need also to be real mirrors that select efficient forms of representing reality because they have a vital, survival function or else they become unfocused ‘blind’ and the mind-system dies away. So Mind spaces and outer topologies coincide as mirrors of realities, which is the eternal feed-back equation between the ‘intelligence of mind spaces’, and the entropic disordered motion of ‘time flows’: ∆S@<=>¬∆T.

Geometries however become the more distorted, the further away we come from that ‘communion’ between the ‘mind and the topology and scale the mind perceives’. When mind and topology coincides, the mirror is most efficient. So human visual mind has in the Cartesian orthogonal space its best mirror-representation.

When the distance in scale and form is maximal though there are internal deformations of the mind space, which adapt the ‘stranger being and scale’ to the perceiver, which is fine if *the humind was aware of the underlying limits of truth of its perception. But the humind is extremely ego-centered and reductionist, sponsoring a naïve realism – what we see and measure with our senses is all what there is. So it denies those deformations.*

For example, a text on quantum states: “The heart and soul of quantum mechanics is contained in the Hilbert spaces that represent the state-spaces of quantum mechanical systems. The internal relations among states and quantities, and everything this entails about the ways quantum mechanical systems behave, are all woven into the structure of these spaces, embodied in the relations among the mathematical objects which represent them.

This means that understanding what a system is like according to quantum mechanics is inseparable from familiarity with the internal structure of those spaces. Know your way around Hilbert space, and become familiar with the dynamical laws that describe the paths that vectors travel through it, and you know everything there is to know, in the terms provided by the theory, about the systems that it describes.” Indeed, but that limits you to the mind-space, reducing reality to its, in yet another act of ‘mathematical creationism’. It is then necessary to add always a ‘coda’ to the different mind spaces we study to connect them *with the larger, more truthful reality of space-time laws. *

In the case of Hilbert Spaces it would help for example NOT calling ‘Dimensions’, its ‘coordinates’ for each parameter but rather ‘parameters’ or ‘coordinates’; or cast its conceptual jargon of functionals in terms of inner and outer dimensions of a new plane of existence of the fractal point; and understand the difference between the highly ordered palingenetic cycle in time of particles (hence susceptible to probability analysis) vs. the statistical populations of thermodynamics, in terms of the T=S duality. And certainly it would have been much better to do so 100 years ago before the whole business of forcing reality into the 1-probability unit sphere by ‘decree’ of Born’s rule was made a dogma. Quantum physics thus is the paradigm of a ‘mental space’ that works for man to make measures from its point of view, but fogs by lack of conceptual clarity the understanding of the whys of the quantum sphere. It is thus good for the praxis of measure and manipulation of particles but bad for a philosophy of science.

Thye same can be said of our Cartesian Graph: Know your way around Euclidean light space-time with its 3 perpendicular coordinates and you will know a lot about how huminds and similar electronic systems perceive the Universe. But as long as humans are not aware that such World space is only our monad’s light world, different from many other mappings and accept the ‘light nature’ of our space, just an elaboration with our electronic mind of the photons of light and its coordinates, we will again commit philosophical errors about what is space, what is time, why c-speed is constant (our rod of measure the electronic eye entangles as informative distances with other electrons, hence always maintaining it at c-constant fixed speed), etc. etc.

Again all those problems are common to any analysis of reality with the complex plane which has different dimensionalities to represent s and t functions often in different planes of space-time; even with vector spaces that also represent holographic ST elements *but in a single plane of space-time*, and its more complex purely spatial frames of reference ‘across ∆-scales’, that is Hilbert spaces and functional operators.

Thus we shall close our introduction to 5D geometry with the study of those ‘far removed’ Hilbert spaces, which are of interest to understand the most far removed scales of reality – those of undistinguishable zillions of particles…

It is important then to grasp the underlying principle *that unifies reality broken by the infinite of mind-spaces and the naïve realism of humans in their interpretation of those geometries:*

*In an absolute relative Universe it is NOT that important to know in so much detail a far removed scale such as quantum physics is – the galaxy/atom as viewed from humans have ‘weird properties’ because our ‘perception of it’ is limited so we can only know certain simple dimensions of their structure namely the most external and ‘visible’, MOSTLY 2D-holographic ST Dimotions.*

** **

**3 AGES OF @NALYTIC GEOMETRY: S≈T**

@nalytic geometry started with the work of Descartes and Fermat, foreseen on the Greeks’ study of conics; which now could be fully represented in the Cartesian graph, which is in topological terms the conversion of a conic into a plane, where the center of the conic becomes the mental point of view of the observer, or still world of geometry.

The evolution of the discipline through 3 ages of increasing complexity of ‘form’, added new structures in space – differentiating mind views of reality according to coordinates – the aforementioned cylindrical, polar, Cartesian and complex 3±i points of view or states of any system which distort our image mirror of reality within a given mind. So in its first age, numbers and points were married with the Cartesian graph, and then the 3 ‘frames of reference’ were found to describe 3 types of ‘functions/forms’ the hyperbolic, Euclidean and elliptic geometry. Polynomials were represented and the fundamental theorem of algebra expressed with the final discovery of the Complex plane.

*The second age of analytic geometry will be dominated by its use in mathematical physics* with Kepler’s orbital elliptic conic geometries and the biased views of lineal time introduced by cartesian graphs, the true origin of that absurdity called lineal time and absolute time and space, which occurred to Newton just because he was drawing ‘the sacred language of God’, its ellipses and comets, on the Cartesian graph. So he thought below reality there was such infinite single line of time and space, drawn by God, his ‘alter ego’.

*In its 3 ^{rd} age* it also increased its entanglement with the other elements of∆@st of space-time, giving birth to mixed disciplines. So @naltic geometry, ∆nalysis, S=T algebra, T-heory of numbers and S-geometry in the final 3rd age of mathematics, became

*all of them reflections of each other in the kaleidoscope Universe, where those 5 elements constantly merge to give us more complex 5D entangled reflections, as complex ‘knots’ of 5D elements, so there was a creative age of algebraic geometry, analytic geometry, complex spaces, mind-phase spaces, differential geometry…*

**RECAP**. We shall introduce analytic geometry, mathematical physics and expand ad maximal the analysis of the 3 type of @-frames and its relationships with ∆st of which the laws of inversion and growth of dimensions and the understanding in vital terms of concepts such as ‘angles of perception, identity and continuity’ are the most important.

**YOUTH OF ANALYTIC GEOMETRY :CARTESIAN FRAME. DIMENSIONS**

All this said we shall restrict the study of Analytic geometry in its first 2 ages to the humind’s frame of reference, which is light space, and just reveal it as a mirror of SóT laws, whose fundamental equality is the orthogonality=perpendicularity of its 3 light spacetime axis, represented by the…

**Cartesian objective light-spacetime plane.**

The universe has infinite mind-mirrors depending on the forces used to gauge the external world, which bounces on a limited quantity of its scales of space. Humans perceive the range of scales of the frequency of light between red and blue social density of colors. But infinite other minds with different detail according to the quantitative pixels they absorb (max. S = Min.t) maximal for smaller sixes will determine the intelligence of the system. Descartes was fully aware that what he had in the mind was not the whole Universe, so he expressly stated the fact, differentiating the ‘world’ of a human mind, from the infinite other worlds that exist outside, establishing in his little known book, the ‘world’, this difference, affirming that his ‘Cartesian Frame of reference’, was only that of the human mind.

So he affirmed that the Universe was the sum of ∞ mind-worlds that don’t speak the same languages, and created the same mappings of reality we humans created. So he *said that all what exists was made of:*

*Open §pace, which he called ‘res extensa’.**– Closed, cyclical times, which he called ‘vortices’*.

And then he added a 3^{rd}element, realizing the only proof he had of the existence of those vortices and res extensa was the fact that he perceived them: Cogito Ergo Sum. ‘I think therefore I am’.

*So he established a frame of reference with a central mind that mimicked the visual world we live in, hence it would give us accurate measures with the rod of light that created our mind space*; whereas we could use for all simplified measures the central point of reference as the observer’s point of view. Simplification is achieved by eliminating the 5^{th} dimension of inner parts of points, to study its external actions, regardless of its internal evolution, changes of phase or state.

*This became then the basis of the external scientific method of mathematical physics, which was no longer concerned with internal changes and could easily adapt the measure of any locomotion of a point or social group of points by representing the S=T, distance=motion symmetry in the Cartesian frame choosing as a preferred direction of motion the positive X-*coordinates. But unaware of the S=T duality, which only slowly became understood with calculus and differential geometry, it soon seemed ‘magic’ that static lines and curves represented physical motions in the Cartesian still mental space.

*Further on, the use of the subjective central point of view or ‘0-*point’ to represent objective external motions fully independent of the observer would make equations very complex, and so till the arrival of Generalized coordinates, a cumbersome structure to ‘bend reality’ to the subjective humind’s measures dragged the analysis of mechanics.

*A basic rule then to discern among all ‘multiple kaleidoscopic mirrors of a language’ cast upon a single reality is the limit of distortion, established by Occam’s rule:* the simplest representation is more objective, closer to the perspective of the objective T.œ (time-space organism) that cause the event. So for example, the Earth at the center of the cosmos is truth but a very subjective distorted truth as the Earth’s gravitational contribution to the whole motion of the solar system is minimal and so its mental spatial representations (Ptolemy) is very complicated. When we go to the p.o.v. of the main contributor to the force, the sun, we find a simple elliptical territory formed by its maximal contribution, *because the cause of events simplifies reality into its vital bidimensional actions and territorial perfect forms. So the sun tries to create a perfecte encircled territory where planets form its ‘angular momentum-membranes’ and only slightly contribute to deform the circle into a ellipse.*

Reference Points become then dominant spatial, mental view which resides in a single plane where the ‘mind of reference’ perceives in relative stillness to its point of measure, with no internal scalar form or change of state, and an external continuity given by the background light space, any motions that through the S=T duality can be associated to lines (distances), planes and volumes (social motions) to obtain simplified results most of external locomotion of material ensembles.

**RECAP**. Is the ‘light space-time we perceive’ real? Or it is a phase mind space? The question is answered in several parts of this paper in terms of humind’s art (painting), psychology of the mind (its equation), Relativity theory (S=T) and in papers of mathematical physics, regarding its measure of time clock dilation and space distances. *Riemann already gave the best mathematical answer intuitively* in his famous dissertation on phase spaces and color spaces… It is a mirror of reality and as such real and distorted.

What is then the biggest distortion of humind’s frames of reference? The non-representation save the Complex Plane and modern Hilbert-like functional spaces and algebraic methods of renormaliation of the 5^{th} dimension.

Even though in reality, in the Non-E structure of scalar spacetime we have upgraded with ¬Æ topology, all point have ‘fractal content’ as Non-Euclidean points; and hence breath, its lines are therefore waves able to communicate the external form and internal energy or fractal networks that branch to connect multiple points, and its planes intersection of three of such waves or networks that form topological organisms… all this is information no longer is available in a Cartesian plane. But it is NOT required for mere external measure, as *long as the fractal point we measure is in our plane of existence and its representation concerned with external locomotions, which makes the frame of reference perfect for mechanics and overexploited in mathematical physics*.

While there are 3 other ‘spaces’ worth to notice, to explain all this inner fractal space-time complex world:

Complex spacetime ideal for studying Timespace holographic ST-dimotions and world cycles.

Hilbert spaces, *a*space suitable for fifth dimensional analysis where each point is a world in itself.

Fractal spaces and fractal dimensions to study ‘networks’, which penetrate through scales as opposed to ‘waves’ that are lines with volume transmitted in the same space.

In the graph, the creation of mental mirrors is the essential process of expansion and contraction of reality into mind spaces through a language mirror, which in mathematics is as diverse as the number of mental and phase spaces, of which 3 are paramount – artistic spaces origin of projective geometry, affine spaces that simplify into lines cyclical geometries and finally in the 2^{nd} age of geometry differential geometry that will construct surfaces of reality departing from the motion of a fractal point.

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**S=T: CORRESPONDENCE BETWEEN MENTAL FRAMES AND VITAL TOPOLOGIES **

**Analytic Geometry: Frames of reference. The differences between the Cartesian and the Complex plane.**

Aristotle was the first philosopher to understand the mind-God of each system as the central unmoved point of a body of energy it moves around itself, the perfect definition of a singularity, origin of the infinite orders of the Universe. So he exclaimed, ‘we are all gods’. It is the idea of all the ideas, which from Scholar theologists to Descartes to Einstein’s ‘masses [that] curve space into time’ has always defined the meaning of the mind. Let us introduce them and study some differences between minds according to a geometry, a theme treated extensively in the article of mind geometry. Since ultimately we find all the seeds of ∆ºst, in the earlier greek culture.

We consider where those functions and operand are set in – that is what background space we use to express it, and 3±i are the essential background spaces which correspond also to those dimotions as forms in space, the 3 lineal=cylindrical, spherical=polar, and hyperbolic=Cartesian planes and the scalar plane, ill-understood, which is the complex plane better perceived if we ‘square it’ eliminating the √ symbol of its negative -1 axis:

So they can study 2 fundamental ’emergent’ ∆+1 planes of mathematics, the study of Dimotions of Dimotions with the tools of calculus in time, and the study of spaces of spaces, with the tools of the complex plane properly understood in terms of ‘square’ coordinates.

But as in the entangled Universe all mirrors can reflect all forms, Algebra also can analyze other elements. But its main beauty is in creating sequential chains of pentalogic actions that reflect the motions of existence of the being, even though its ‘Group’ simultaneous analysis of all its ‘variations’ of species, has been

The fundamental graph of the Universe is one in which orthogonal coordinates represent the T-independent parameters in the X-coordinates and the T-parameters in the Y-coordinates. But we do have two different representations for them, because we do have 4 different S and T dominant dimotions (with the ST combination of both, able to appear in the z-coordinates, or the combination of both).

So the big question is what coordinates belong to what Dimotions. As SS and TT dimotions are equal in value, the pure coordinates should belong to the Cartesian plane. While the S-informative coordinates do have a lesser value. So they must be put on an imaginary system of coordinates.

Orthogonality in the Universe, is then easily explained as follows:

Because Entropy (TT) vs. absolute linguistic still form (SS), Locomotion (Ts) vs. information (St), are the dual inverse functions of reality merged only in the S=T reproductive dimotion, in the 0 point of X-length, the relative dimension of locomotion, there is a zero motion and stillness rises in the height dimension of pure form, where the 0’ mind or frame of reference resides. But then we deal with the ‘different quality’ of locomotion and informative perception in terms of expenditure’ of energy, as information ‘shrinks’ motion. I.e. a gravitational invisible tachyon line has no information but when it becomes light (neutrino theory, Broglie->Jordan), it forms a wave of information that grows in height with the photon on top. But this height dimension is in terms of the parameter of energy and locomotion (T), a compressed ‘spatial T’, of minimal size. And so we need a smaller ‘quantity’ and one that is negative, ‘subtracting’ from the distance-speed (s=t) of locomotion.

*This is magically achieved by the negative ‘root’ value of the imaginary axis, reason why it appears as –ct in relativity and is so useful for the study of electric wavers. By squaring both we simplify the problem of √negative roots, we shall explain latter when we analyze in depth the inverse operands of algebra. *

*So *the complex plane is most useful for St-Ts systems of two composite ‘energy-information’ body-head forms.

**The subjective 3 p.o.v.s of mathematical minds. |-Cylindrical, O-Polar, Ø-Cartesian.**

The ternary nature of the universal topology in a single scale=plane of space-time, is evident when we consider the other 2 canonical ‘coordinate systems’; the polar cyclical plane and the Cylindrical, toroid plane, which will give us 3 different ‘views of the Universe’, both in mental subjective space from the new P.o.v. and in topological space as *each frame of reference will simplify the ST dimotions=action taken place inside the ‘organ’ of a system that mimics topologically one of those 3 forms. Whereas the Occam’s rule apply: the simplest equation reveals the type of organ:*

‘The simplest frame of reference in which a problem formulates indicates which is the ternary topological element we study’. For example, if the problem formulated in polar coordinates has an equation simpler than in cartesian coordinates, we are studying a ð-particle/head element (as when we formulate a gravitational or charge problem in polar coordinates). If it is simpler as a toroid/cylindrical is (a toroid opened along a z-cut), equation it is a ‘$t, limb/field problem’. An open curve defines in S=T terms also a ‘lineal motion’ and so on.

So either the 3 frames of reference or the curve drawn in the Cartesian frame define already by its form, which organic ternary topological parts in space or event in time IS acting on reality: ‘cylindrical frames for lineal limbs/fields’, hyperbolic cartesian frames for body-waves or polar frames for particles-heads. But events and systems change states, So **i**t follows naturally that by ‘changing’ the equations of systems from one frame of reference to another we *change often the topological analysis of them because the system has changed its | x O = Ø relative state – a fundamental feature of quantum physics, described as a hyperbolic wave in Cartesian coordinates and as a particle-field in polar co-ordinates (Bohm’s model), when a quantum system ‘changes’ from S=T wave to particle-field, S<T>S state.*

So the choice of coordinates, in which the function/form is simpler often indicates they type of part-species, we are analysing according to the ‘generator equation’ of mind-coordinates:

Γ (generator of mind p.o.v.s): |-$t (cylindrical) <ST-Cartesian> §ð (polar) «» ∆±i(complex)

The most suitable for man being its light spacetime Cartesian coordinate, with the negative and positive, inverse directions, self-centred in ∆o, the distorted point of view of the human observer, suitable for lineal, perpendicular but also hyperbolic, wave-like light space-time representations. Yet as the Cartesian graph tends to an objective external flat geometry, save the 0-1 sphere, often for self-centered representations where the mind view is the main cause of the motion, a polar system will be simpler, more suitable, as in the Particle, Bohm’s representation.

In the graph, the 3 subspecies of 2-manifolds have their expression in 3 coordinates, where the Cartesian, is taken as an ‘infinity growing Toroid’ space.

Inversely if we perceive those coordinates from the ∆-1 scale as we did with ‘spirals as worldcycles’ of existence, we can ‘construct them and the organs they mirror’ as the sum of all the worldcycle paths of its ‘neighborhood points’; So that Superorganisms can be defined as ternary adjacent ensembles of the geodesic curves performed within each organ/frame of reference by each of its ∆-1 fractal points put together in 3 topologic elements, living with a worldview adapted to each of its 3 corresponding organic coordinates.

Philosophy of mathematics then enables to analyze in depth tour ‘selfie’ axiomatic methods of truth, which ‘reduce’ the properties of the Universe to the limited description provided by our limited version of mathematical Cartesian frame of a o-point with no parts, known as Euclidean math (with an added single 5th non-E Postulate) and Aristotelian logic (A->B single causality). This limit must be expanded as we do with Non-Æ vital mathematics and the study of Maths within culture, as a language of History, used mostly by the western military lineal tradition, closely connected with the errors of mathematical physics.

**The importance of the Polar frame. **

Contrary to belief, the most important frame of reference for most ‘mental spaces’ is the Polar frame, following the rule of simplicity – that makes a more synoptic form with less parameters the fundamental one. It should not surprise the reader because being ‘space’ a mental geometry, or mind mapping of reality it does happen usually from the subjective internal point of view. *In brief most mental spaces are constructed in a subjective distorted view provided by angular geometry as opposed to an objective external view provided by a hyperbolic Cartesian game. 2 examples will suffice: *

*– The simplest representation of all the conic curves we shall study soon is given by its polar representation: *

In the plane, we choose a point P (pole) and a ray originating from it (polar axis) and determine the position of a point M by the length ρ of the polar radius from the pole to the point and the value ω of the angle made by this radius with the polar axis.

In particular, the ellipse, hyperbola, or parabola, if for the pole we take a focus, and for the polar axis the ray passing from the focus along the axis of symmetry to the side opposite the nearer vertex. Then we have one and the same equation:

where e is the eccentricity of the curve, and p is its parameter. This equation is of a great importance in astronomy. For it was with its help that the result was derived, from the law of inertia and the law of universal gravitation, that the planets revolve about the Sun in ellipses. *It must then be noticed that the observer is internal to the conic as a ‘point, part’ of the whole, where the conic is its external membrain. Hence the obvious, vital practical need for any ‘entity’ within its territory to assess the distance to its ‘border’, which it cannot cross (open ball geometry for the ∆-1 parts of its whole structure, considered in the 1 ^{st} ¬E postulate). *It is then required for a mind space to be able to assess that distance with minimal elements. And since w, the angle can always be ‘assessed’ in situ, the only parameter required is p’s length, as opposed to two parameters in a Cartesian graph.

The geographical coordinates, latitude and longitude, by which the position of a point is given on a sphere, are also well known cases of polar geometries, which basically extend to 3D the previous analysis; or rather establish the @-mind of measure in the membrain surface of the T.œ.

This subtle distinction between O-subjective polar coordinates vs. ST-objective Cartesian coordinates come ultimately to the different properties of wave-bodies, which merge S & T and hence are more objective vs. the distortion of §-mental particles-heads; and have deep consequences because we can consider reality a constant switch between both states. Such is the case of:

– The realist view of quantum physics, made with the polar representation of Schrodinger’s wave (Bohm). So we can consider the duality wave-particle as a constant switch between angular spin perception of reality by the particle in stop position, and the wave-motion/reproduction of the physical system in its hyperbolic wave state, which must ‘merge’ and ‘synchronize’ with both the field, ∆-1 quantum potential and the ∆+1 particle state.

**GENERALIZED COORDINATES FROM THE PERSPECTIVE OF THE 0-POINT.**

More realist than a Cartesian graph that force feeds the humind p.o.v. is the complex view of a topological ternary system as a fractal of 3 adjacent interacting internal parts that calculate its locomotion *without a subjective agent, as a measure of the changes of distances-motions * or relative positions of the parts of the Tœ.

Then measure includes all the relative parts of the super organism, taken from the internal still point of view of its particle-head, as its center of reference. So the subjective external humind pov disappears and the still ‘real center’ of the action treats the other parts as elements of a relatively rigid body, whose coordinates are simplified with Generalized coordinates, NOT connected to the human observer, and causality improved with the ‘aristotelian, unmoved God, cause f the motion of its body-wave’ as the center of coordinates

In the old system akin to the complicated Ptolemaic Earth center, under the Æntropic principle the human observer is the relative mind-view center of the whole system. So, he then can choose the Cartesian coordinates as an alien, external point of view, making it all more complicated. Yet it is far more convenient to choose the objective internal coordinates of the system, and its still particle/head as its o-point and cause, as with the heliocentric system.

The compound plane pendulum consists of two rods OA and AB, hinged together at A; the point O remains immovable, the rod OA turns freely in a fixed plane around O, and the rod AB turns freely in the same plane around A. Every possible position of our system is completely determined by the magnitude of the angles ϕ and ψ that the rods OA and AB form with an arbitrary fixed direction in the plane, for example with the positive direction of the abscissa axis. Hence the real O-point mind of the system is A and generalized coordinates by choosing it will simplify the calculus of its surprisingly chaotic dimotions.

Consider a similar example: a system of two rigidly connected points, these coordinates can be chosen in the following way: the position of one of the points is given in Cartesian coordinates, after which the other point will always be situated on a sphere whose centre is the first point. The position of the second point on the sphere may be given by its longitude and latitude. Together with the three Cartesian coordinates of the first point, the latitude and longitude of the second point completely define the position of such a system in space.

The first point is then ‘fixed’ in a. hyperbolic Cartesian plane that can structure all other systems and the second in polar coordinates, respect to the first mind point, from which they are no longer free. And so as part of a new whole, the number of coordinates required to study it diminish. *And the general law is rather simple: we shall need for a system just the number of generalized coordinates equal to its number of degrees of freedom.*

i.e. If we consider 3 particles rigidly fixed in a triangle, then the coordinates of the 3^{rd} particle must satisfy the 3 equations. Thus the 9 coordinates of the vertices of the rigid triangle are defined by the 3 equations. Hence only 6 of the 9 quantities are independent. The triangle has 6 degrees of freedom.

3 points which do not lie on the same straight line define the position of a rigid body in space. These 3 points, as we have just seen, have 6 degrees of freedom. It follows that any rigid body has 6 degrees of freedom.

In 5D terms it means, the ‘singularity’ or center of reference as the still cause, disappears to describe its motion, with 6 degrees of freedom, which are equivalent to 2 points.

It also follows that mechanics, the science of 2Ð¡=locomotion evolved from subjective human pov (Newtonian) to generalized coordinates (Lagrangian), which is how today professional physicists cast its laws, as we shall do.

In other words, a mechanical system can be described by coordinates whose number is equal to the number of degrees of freedom of the system. These coordinates may sometimes coincide with the Cartesian coordinates of some of the particles or might not.

*The concept of generalized coordinates that make real the point of view, causal origin of the motion, can then be extended to the use of the suitable frame of reference that imitates best the organ-world in which the motion takes place. *So the key for an easy solution is a choice of coordinates according to the Nature of the motion of that ‘whole system’. If the description is one of 1D ‘rotary motions around the singularity’, a polar system works better. If we deal with a lineal 2D motion of the whole system, cylindrical might be used; for all others the hyperbolic ‘deformable’ Cartesian plane is best.

**RECAP.** **Self-centered points of view performing 5 Dimotions huminds or generalized coordinates measure.**

We are mathematical organisms, with topo-logic properties, which give birth to biological, organic assembles of ternary functions/forms and those systems do have one of the 3 O-point of view, its frame of reference, as the still ‘mind-will’ of the 5 Dimotions it performs to survive. There is always a first observer which starts an action of perception in space-time from a perspective that usually is biased by the function of the observer (ST, Si or Te coordinates), which correspond to the Cartesian, Cylindrical and spherical, polar coordinates of science.

Thus we define 3 fundamental coordinates which any entity uses to ‘adapt’ the perception of reality to its mind-view and the equation of the mind, which in terms of mathematical co-ordinates writes:

O-point* x ∞ Universe = constant, static frame of reference.*

In the real Universe the Observer is the dominant element of those 5 actions and also the initial ‘point of view’ of any mathematical analysis, and analytic geometry rightly the first branch of mathematics to be studied.

In the first of many fascinating symmetries between ‘iTS’ and each science (isomorphic, |-space & O-Time, the abb. most commonly used for the 3 components of the Universe), 3 are the fundamental points of view and frames of reference of analytical geometry, each one belonging to a ‘fundamental state’ of being, in the Universe.

The temporal polar point of view, centered in the O-point and its external membrane determined by the radius and angle of perception; the cylindrical, lineal, energetic point of view, determined by the lineal axis of the frame of reference or ‘altitude’, the Z co-ordinates, and finally the Cartesian ‘hyperbolic plane’, corresponding to the STi bodies & waves of physical and biological systems.

From those 3 mathematical perspectives reality constructs its vital geometries that we call ‘existential beings’.

So the observer’s causal, logic, cyclical, informative sentient properties that allow it to perceive time cycles are the first questions to inquire. As @ is the first element to describe in reality – a frame of reference, an observer, an inertial point in relative fixed form with no motion that can perceive and map the Universe, as its O x ∞ constant mapping mind – world. So generalized coordinates are more realistic and simpler (Occam’s principle).

It follows that Generalized coordinates is a more realist analysis of Nature’s dimotions as they are established by the ‘still point’, which is the mind or head/particle that we observe to create its own generalized coordinates where it tries to maintain its stillness (your head doesn’t move respect to the body), as the center of reference of its own Universe.

In science generalized coordinates are used to study locomotion but in 5D we extend them to study the 5 Dimotions of the being, and in many cases analyze them quantitatively when we merge those concepts with the tools of calculus and ‘define’ properly energy (3^{rd} Ðimotion), reproduction of form (2^{nd} Ðimotion), 5D entropy (Dual scattering Ðimotion), 4 & 1 D perceptive information (imploding, spiraling dimotion).

The Universe is a sum of a series of action, ∆aeiou: the first action is perception by an observer, ∆o, of a field of energy, ∆e, to which a T.œ will move, ∆a, in order to feed, ∆e and use that energy to reproduce its information, ∆i, iterating a form like itself, which will gather with clone forms to create a larger, ∆U universal social plane.

Moreover those actions are very simple geometric exchanges of motion and form. ’Ad maximal’ they are dual entropic inner scattering and outer locomotions or inversely the dual collapse of motions that shrink into a seed often by ‘eliminating motion’ into a ‘coded language’ that will unfold when energy germinates it. Further on the S=T symmetry precludes that a motion step is followed by an informative still perceptive stop. When properly cast in suitable frames of reference or generalized coordinates physical and biological actions, dimotions and change of state are simple, bidimensional stœps or single flows of energy: sT; and information: St, between 2 poles either in a single plane or across ∆±3,4 scales. So most laws of science are simple, even when those flows are generalized between several points to form more complex structures; since then they will be repetitions of the simpler unit of action, when we reduce to minimal cyclical space-time actions the total reality of any self.

But to do so correctly we consider the proper sequence of aeiou actions, which mostly start when the formal still point of view kicks out a world cycle of actions proper of any function of existence, with an act of perception, ∆o, which is thus the minimal and first Unit-action of the Universe that shrinks it into a linguistic mirror. *Thus while time-motion-entropy is the first substance of reality; perception of it as form becomes the first element of any fractal point or T.œ, which creates in fact by ‘shrinking the whole’ into a point of a smaller ∆-1 scale, the actions of creations, the storage of information, the game of reality as we know it, past the pure entropic motion meaningless in itself*.

So it is always a point of view and its relative frame of reference, what start the comprehension of an external reality with its first action, which is the still language that perceives in its self.

CARTESIAN GRAPHS.

**The multiple S=T, ∆ applications of Cartesian coordinates: mixing time and space.**

The main 5D concept for understanding mathematical laws of curves in Cartesian spaces is the S=T duality. Hence the acceptance of the rule of differential geometry – that is, a curve is not really a curve but an ST representation of an S-point tracing a T-Dimotion; hence a geodesic trajectory, an ST-dimotion adapted to the larger organ=world of the *being* within which the fractal point performs its function.

Inflationary mathematics however, without the restrictive anchorage of the real 5D limit of Dimotions and its required geodesic efficiency, draws also a big number of irrelevant curves that exhaust the combinatory of its parameters in a plane. But curve equations, in the praxis of mathematical physics restrict to the most efficient trajectories for the 5 Dimotions=actions of any being in exist¡ence. So most still Cartesian geometry is a disguised form of differential geometry. While physics as it is today formulated ONLY in mathematical languages (reducing its properties to those who can measure, a theme treated in our papers on physical systems), can be considered ‘de facto’ the branch of experimental mathematics or 5D mathematics that ‘reduces’ its inflationary mirror language to the only ‘real, efficient’ elements of it. And as such we treat it as a subdiscipline of mathematics, or paraphrasing Einstein and Poincare*: “Mathematics are truth but only experimental physics defines when they are real.”*

Both are simplified mirrors of the humind, as only ‘the being holds all the information about itself’. That is if the being is a probability 1 of existence, only in the being truth becomes also a probability 1. While we need multiple ‘experimental languages’ to extract all its properties. S=T duality is the origin of the 2 key properties of equations in Cartesian spacetime – the dual numbers required by each point; and its dynamic motions as they trace a curve.

*In that plane a point has ‘2 dimensions’ = parametric numbers for its representation.*

*What those numbers mean beyond the earlier naïve realism of using them for simple locomotions in the different planes of forces (gravitational vertical or flat, friction spaces), is what will open phase space and mental space, and now we take to its final conclusion, considering that overwhelmingly they represent S or T coordinates to define each type of mental space as a representation of one of the 5 Natural Dimotions of reality’ and or 5 corresponding Non-E postulates that describe the interaction between fractal points (T.œs).*

The abscissa and ordinate 2 coordinates of a point in the plane Descartes are numerical values x and y of two mutually perpendicular straight lines (coordinate axes) chosen in a flat bidimensional plane. So they perfectly express an ST inverse holographic *dimotion reduced to a point.* The point of intersection of the coordinate axes, i.e., the point having coordinates (0, 0) called the origin, mimics then the static state where the dimotion started.

And so Descartes subconsciously expanded the dimensions of an Euclidean point through its “arithmetization” as the point in the plane that represents its ∆±1 world value has enough information to fill the content of a pair of numbers.

*Points as equations with 2 unknowns tracing curves in the plane.*

Descartes’ second ST concept is the following. Up to the time of Descartes, where an algebraic equation in two unknowns F(x, y) = 0 was given, it was said that the problem was indeterminate, since from the equation it was impossible to determine these unknowns; any value could be assigned to one of them, for example to x, and substituted in the equation; the result was an equation with only one unknown y, for which, in general, the equation could be solved. Then this arbitrarily chosen x together with the so-obtained y would satisfy the given equation. Consequently, such an “indeterminate” equation was not considered interesting.

Descartes looked at the matter differently. He proposed that in an equation with two unknowns x be regarded as the abscissa of a point and the corresponding y as its ordinate. Then if we vary the unknown x, to every value of x the corresponding y is computed from the equation, so that we obtain, in general, a set of points, which form a curve.

Thus, to each algebraic equation with 2 variables, F(x, y) =0, corresponds a determined curve of the plane, namely a curve representing the totality of all those points of the plane whose coordinates satisfy the equation F(x, y) =0.

This observation of Descartes opened up an entire new science besides analytic geometry – the language in itself. Since essentially Descartes gifted the static bidimensional plane geometry of a ‘variable motion’, and gave us the capacity to study an ST-evolving system in space-time, whose geodesics will give us properties of certain dimotions as opposed to others (closed paths, open paths, entropic, scattering motions; etc.). Thus ‘Physics of time’ were born.

So for understanding of analytic geometry and its algebraic equations, it is useful to consider the fact that numbers are points and there is a direct relationship between points and numbers, lines and variables, planes and squares and so on. So polynomials and operands represent the ‘social evolution’ of dimensions, points into lines into planes into 5dimensional structures.

**Pentalogic on the Cartesian graph.**

The Cartesian graph by representing minds in its center of reference, dimensional symmetries of space and time – both informs and motions and by reducing T.œs to points susceptible of scalar ∆nalysis with the concepts of ∆-1 derivatives (finitesimals) and ∆+1. – yields enormous capacity to model the Universe of ∆@st of space-time.

So Analytic geometry could study all the pentalogic elements of reality. Some of its first uses would be:

1: T>S: solving construction problems of continuous motion with discrete spatial steps, such as the division of a segment in a given ratio; *thus adding frequency time to geometry.*

2: S<T: finding the equation of curves defined by a geometric property, which could relate different pentalogic, ∆@st elements. For example, defining an ellipse of motion (ST-holographic form) by the condition that the sum of distances to two complementary points of reference dual attractive or gender points) is constant. In *this case the ellipse becomes a mental space function that defines 2 physical systems controlling with equal force a third entity and by doing so, creating a common territory of space (Kepler’s 2nd law, ‘proximity’ of 2 parental forms to its son within a ‘territory’).*

- S≈T: proving new geometric theorems algebraically (i.e. the derivation by Newton of its theory of diameters; and conversely, representing an algebraic equation geometrically, to clarify its combined ST properties (i.e., the solution of third- and fourth-degree equations from the intersection of a parabola – entropic, scattering motion, with a circle – closed ordered motion), which we will illustrate in its ‘metaphysical’ space-time whys.
- ∆: peering in 5D scales through ∆nalysis of derivatives and integrals.
- @: Studying huminds, as the key properties of Cartesian planes imitate the properties of light that follow the properties of space & time components of light with its 3 perpendicular, S-magnetic, T-electric and ST-energy fields.

Thus, to the classic definition of analytic geometry as that part of mathematics which, applying the coordinate method, investigates geometric objects by algebraic means, we will ad the insights of its direct homology with the S, T, ∆st elements of reality in different entangled pentalogic combinations that extract Space-time general laws. Since it is only needed to consider an informative or time function, one of the coordinates and then the other one would represent the ‘spatial function’, which are inverse dimensions, to make it work magically an represent a ginormous number of s@≤≥∆ð something soon used by Galileo.

**Complex 5D functions.**

Graphic pentalogic applies then to each specific function with new entangled elements to reveal the multiple functions of each mathematical system. I.e. an exponential function reveals its:

5D: max. inverse entropic ‘death growth’ at accelerated speed towards its asymptote revealed by its derivative.

2D $: Max. spatial growth as a spatial function of speed and distance.

1D ð: The maximal growth curve in time represents a vortex of space-time, with 2 TT dimotions accelerated change, as a force; hence a o-1 vortex of acceleration towards a singularity point (1D)

∆±i: All those expoenential ±functions thus end into a change of scale. Either the function accelerates towards an entropic dissolution (e¯ª, which is a decay entropic process) or inversely accelerates towards an emergence, as in **resonances**, or the limiting case of distribution equations that grow exponentially towards the 0-point. It is the simplest Delta Function, whose integral is 1, even if it only exists in that 0 point showing that indeed a point-particle is a fractal point with dimensionality 1, once emergence in its ‘infinite density’ point of resonance, emerging into an ∆+1 plane – a fact which incidentally proves that all infinities ARE finite ones of a higher plane: infinity IS just the limit of an ∆-plane; the whole is the infinity of the part; or else the delta would not emerge when integrated between ∞ and – ∞ as 1. Because once we cross a discontinuum of scale, we are in another type of parameter, so infinity does NOT exist.

*Functions then become always representations in @-mind space of trans-form-ations of ∆-scale, space population or time motions, which are the 3 elements of any system of reality. And so pentalogic applies to all functions.*

I.e Temperature determines the S-volume of a gas as a whole. But also ∆+1 temperature determines the motion of its ∆-1 gas molecules. While the T-growth = elongation of a given metallic rod is determined by its scalar temperature. It was uniformities of this sort that served as the origin of the concept of function.

**RECAP. **Descartes’ theory is based on two S=T concepts: perpendicular S=T coordinates and the concept of representing by the coordinate method any algebraic equation with two unknowns in the form of a curve in the plane.

** **

**CURVES.**

The first use of Cartesian graph was then to move further the last advances of the 3^{rd} age of Greek Geometry that had started to explore curves. And as the best-known curve was the closed cycle, geometers attacked the cycloid – a cycle moving along the line. We shall thus consider it as a paradigm of the pentalogic and experimental nature of curves, which reflect different species of Simple T.œs (its membrane and enclosed vital energy), as well as fractal points tracing canonical curves=worldcycles

**The cycloid as a world cycle. **

The interest of maths as a mirror of reality is its ‘simplicity’ to describe the basic laws and symmetries of space-time, and hence the properties of worldcycles and super organisms and ∆-planes. This reaches its final simplicity in the analysis of curves in a bidimensional plane, and the whys they reflect on S=T symmetries. Let us explore some of those elements of ∆st isomorphism with mathematical mirrors.

We already analyzed the beauty of the pi-spiral as a worldcycle, the fundamental ‘time particle’, that represents the existential flow in finite time of any superorganism that closes its worldcycle as its outer point membrain returns to its origin. Next in the ‘evolution’ of the cycle appeared the cycloid, where the point membrain its vital energy-radius performing in an external entropic ‘flat’ worldline, its cyclical dimotions between its original birth and death, when touching back the flat line. What fascinates then on the cycloid worldline is that it encodes the quantitative parameters of the 3 ages & 8 ‘sequential phases’ of life of any Disomorphic worldcycle:

In the graphs two different isomorphic languages: the 8 diameters of the simplest cycle of existence, a cycloid moving on a lineal, open path or entropic higher ∆+1 world from birth to death; below the 8 Baguas the informative, mongoloid human subspecies found to correspond to the 8 phases of life. Such type of homologies shows the entangled Nature of all the languages, as all mirror the ultimate Disomorphic S-T laws of the existential game. Specifically the parameters of a worldcycles represented by the cycloid traced by a point – the mind/membrain – on the circumference of a circle, its vital energy – represents a T.Œ as it rolls along a straight line, the outer world, enclosing a territorial space, the cycloid surface.

The immediate translation thus of the worldcycle is provided by Galileo’s conjecture – the area enclosed by one arch of the cycloid is three times the area of the generating circle, as the 3 ages of the organism represent 3 states of its vital existence, with the central mature, present, reproductive age, equivalent in value to the sum of its first, young ‘growing’ and last, ‘diminishing’ old age of the rising and falling cycloid worldcycle.

And by Wren’s measure along the curve of one arch of the cycloid equivalent to eight times the radius of the generating circle, *representing, as the moving dynamic point in which the mind and the outer world membrain entangles, the 8 phases of the life of the being in the world or baguas*. Thus the hidden beauty of the cycloid that made it so important in earlier Cartesian geometry encodes if we consider the moving cycle a world cycle of life and death, the point on the surface, the singularity-mind, transiting along a lineal sequential timeline, the simplest disomorphic mirror of the properties of a life-death cycle, in which our ‘vital energy’ (the area enclosed by the singularity and the timeline) is split in 3 ages with a similar volume to that of the unit circle.

As indeed the ∆-1 generational o-1 age is followed by 3 more ages in which the 1^{st} and 3^{rd} age are equal in vital value to the intermediate one:

sT= youth > ST: maturity> St: 3rd age…

When the cycle ends in the ‘landing’ lineal flat entropic point of ‘timeline’ death, exhausted its vital energy.

Another interesting version is the epicycloid on a 3 world radius of a 1 radius curve, *which will trace 3 cycloid ages over the larger world each with 3 epicycle surfaces mimicking even more clearly the worldcycle. It is this kind of ‘magic coincidences’ that encode the essential laws of Time-space worldcycles what I find so enticing to discover in my 5D research. Why other mathematicians do not even attempt to do so is no longer my business. I gave up on human egocy. Yet without those experimental relationships between mathematical laws and space-time worldcycles the subject looses its ‘true whys’.*

*We can then consider the larger circle the minimal ‘surface of feeding entropy’, for the smaller circle to complete a worldcycle of existence and its 3 ages. Indeed, for a 1 to 2 relationship we get a nephroid and the smaller circle can only complete two ages, which are fully inverse. *

*And we can observe some ‘dimotional scalings’: the perimeter traced by both is in a 3 relationship but the vital energy 2DS area πr ^{2} is in a 9 to 1 relationship, as the volume of the system grows exponentially, slowing down the inner processes of the larger whole (5D metric) *

the most remarkable property of all worldcycles represented by the cycloid was found when mathematicians solved the variational problems of the brachistochrone – finding the shape of a curve with given start and end points along which a body will fall in the shortest possible time: It is the beginning of an upside-down cycloid!

So all worldcycles ‘T=race’ through the 0-M1 palingenetic and young age to achieve the maximal growth of the system that reaches its mature ‘M2’ point state in the minimal time.

So the brachistochrone is the spatial symmetry of the law of least time; a key law of ∆st, as all systems *try to achieve its actions and motions in the less possible time; thus subconsciously shortening its youth and old age – while the mature iterative age of reproduction tries by the very nature of its main dimotion, to conserve the state of the being, becoming a ‘flatter’ curve with a peak, standing point of 0-change/derivative.*

**RECAP **This simplest representation of the world cycle shows already the key insight of 5D ∆@st: curves represented on analytic geometry are spatial simultaneous S=T geometries of temporal Dimotions of the head-point of the curve.

PENTALOGIC OF CURVES.

It is then evident that curves model multiple elements of GST; being its main pentalogic 3±¡ features:

**∆: ** Its capacity to represent the 3 scales of an entity, as a fractal ∆-1 point, tracing a worldcycle, ∆º, in a larger world represented by the open, entropic flat Cartesian Plane,

**S=T:** They can also model the 3 elements of a T.œ, if we consider the curve in synchronous space a membrain, where it is often located the @-singularity (open curves); or when the curve is closed, its focus represent a single or dual complementary center, while the enclosed vital surface, is akin to the energy of the system.

Thus a simple curve already accounts for the 3 topologic parts of the being, and its 3 ∆±¡ scales.

Such insights on the real world of Dimotions, represented by the still geometry of curves in Cartesian space will become even more sophisticated when we observe the properties of the Canonical curves of 5 Dimotions:

**Pentalogic on conics**

**S=T. **The 1^{st} remarkable property of Conics is the extreme symmetry in terms of its coordinates of its general equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0

What this means in experimental mathematics, as in most cases X and Y coordinates represent an S=T dual parameter, that conics are highly symmetric bidimensional spacetime curves, hence with the maximal efficiency, which is ultimately the meaning on an Si=Te equivalence (age of balance, maturity and reproduction, proportion of energy and information, beauty, ‘mens sana in corpore sanum’, present state, etc.)

We cannot however in this first book on 5D maths to do the full symmetric analysis of Geometry both as visual simultaneous images and algebraic equations in as much as *‘huminds’ have not properly defined the 5 operands symmetric to the 5 dimotions of the Universe, *to have a full understanding of even the simplest relationships between algebraic operands and geometric figures. i.e. why the 2Ð¡motion of locomotion squares the function that becomes a parabola on a Cartesian graph; why there are not exact sums of cubes, direct methods for solutions of quadratic solutions, etc. etc. themes those that require a ‘deep-thought’ consideration of Algebra in our bulkier second book – yes, the axiomatic method provides its own convoluted concepts to prove all the above, as any student of algebra knows (permutation of groups, symmetry, etc.), but *those abstract concepts must be clarified in terms of experimental properties of space and complex i-logic. So *we shall focus on the ternary Non-Euclidean structure of all Time-Space-Super organisms (T.œ), as curved forms are ultimately as polytopes, social numerical structures that define the two elements of any T.œ in a single plane of space-time:

- S: In space the simultaneous membrain of the system, defined by a closed curve; and its focus/ci, or singularity centers often the cause of the form of the membrain.
- T: In time, most likely in the case of open curves, the trajectory of a T.œ, ‘reduced’ as a fractal point moving in the larger ‘Plane’, which itself defines the organic part of the whole in which ‘the event’ takes place – that might vary from the classic 3 topologies introduced before, to more complex geometries (a Beltrami hyperbolic cone, and inner vital space – Klein’s Disk – with entropic limits as an open ball; a dual foci elliptic system, etc.)

What Algebraic equations do then is an inverse S@<∆T symmetry to the one we have so far studied in more detail (∆T-numbers > S@ symmetry). And some important i-logic arguments on general laws of any language when mirroring reality must be put forward:

We use the > symbol in inverse manner for both symmetries, which in GST is the symbol for an ‘evolution of information’ that diminishes the ‘entropy-uncertainty of a system’, because paradoxically, the most synoptic a language is, the less detailed information provides, and the more ‘paths of probabilistic future opens’. Meaning that numbers are more synoptic that points, and so they might refer to more possible realities, hence creating more uncertainty. This is the trade-off between synthesis and analysis. I.e. if I say 0 x ∞ or ‘wor(l)d’, I am defining all numbers of social beings or entities a mind equation might conceive (o-mind x ∞ Universe = constant wor(l)d). But I don’t specify any information and uncertainty is maximal. What ‘word’ did I see? A horse? A car? A friend? What number I obtain? 3 pears, 7 billion humans=mankind?

So the algebraic view defines more possibilities, as a scalar time view, many of which do NOT have solutions; while the geometric view IS always a solution in itself even if it does not have a consistent algebraic expression.

The ∆T: algebraic vs. S@: geometric figure duality is in that sense more complex and rich as a ‘function of points and numbers’ that the first layer of S≈T similarities (points vs. numbers) so far studied here, which will certainly once ‘mathematical pros’ pass the ‘stage’ of model shift, suspicion of the founder of a new paradigm, etc. etc. (that is once they accept the work of the usual ‘amateur Copernicus’ and see the light & beauty of the more advanced structure :), give birth to many entanglements between time and space mirrored in the mathematical language.

All this said, as customary we shall start our description of ‘any reality’, not from the bottom upwards but from the top more synthetic ‘reality’ of the 5D universe, which is the worldcycle of any superorganism, because all what exists including ‘curves’ are part of a worldcycle, in the case of a language a partial mirror of a trajectory within it. So as we did with spirals and cycloids, first we shall show that the cone that generates all curves is in itself a 5D image of the worldcycle, more complex than the simplex possible image (the cycle) and its next entangled complex view – the cycle moving in an entropic, lineal plane or cycloid.

**∆±¡: THE WORLDCYCLE CONE AND ITS SPIRALING POINTS.**

The representation of conics in the Cartesian Plane however is a simplification as they are originated in a cone, *which has a dimension of height ideal to represent the arrow of ‘increasing information’ towards the apex of the cone, or relative point of future of maximal, St=§ð; or 3 ^{rd} point of the being. As such the ‘cone’ becomes also an excellent model for defining in its surface the life-death motion of an ideal T.œ – and in physics represents for different ‘attractive’ vortices, a real time-space informative ‘sink’*. So as all what exists is part of a superorganism tracing a worldcycle or a mental image of it, the dual cone expands the conic representation of 4D worldlines of locomotion into 5D worldcycles of exist¡ence. We can apply a pentalogic analysis to such 5D cones:

In the graph several representations of 5D events using worldcycle cones:

- A single point tracing a 3D spiral on the surface of a cone from its base to its apex represents the informative arrow of a worldcycle in its 3ages. Thus if we consider the T.œ just in the ∆-1 plane, a point tracing a motion in its surface; it does then represent a world cycle accelerated but perceived as an ascension in height,
*with different interpretations according to which of the 4 5D possible cones of the dimotions of an entity we describe and what 5D parameters of time and space are represented in the two perpendicular axis.* - If we make a composite of 2 worldcycles of 2 fractal points; we obtain 4 potential combinations (with the central figure becoming a different event, when we change upside down its past -future orientations). Each cone taken as a whole worldcycle of existence (‘LIFE CONE’) the outcome of colliding events according to the rules of the 4
^{th}Postulate of Non-E congruence, from Darwinian feeding, to reproductive symbiosis. - If the cone emitted by each particle is the trajectory of an exi ‘wave of energy and information’, the cones represent the possible outcomes of communication between two larger points, which share the wave; useful to model Fermion<Boson>Fermion exchanges between particles (5D Physics).
- If the double cone belongs to the same particle’s worldcycle, it represents its 3 possible states. Where the double worldline cone of space-time expands 4D Minkowski’s spacetime representation into 5D, branched in 2 inverse S>T<S future & past cones and a flat S=T present, put to use in advanced 5D mathematical physics and pentalogic. The different orientations of two cones which represent unlike the simplex relativity cones of 4D, a motion in the fifth dimension between two fractal points (any T.œ), colliding in a present simultaneous ‘plane’ (left). Or a point in present dissociating between relative past and futures (right).
- Since the cone is itself a circle moving along a line, in a decreasing its size, it obviously immediately represents as it does the cycloid, a fundamental representation of a T.œ of space-time, the circle, moving in time towards a shrunk, warped third age, or moving between scales: ∆-1 ð>∆+1 ð forwards, as it shrinks in relative size becoming on the apex an emergent ‘single point’ of the larger whole (palingenetic worldcycle as observed from ∆+1).
- Finally the 2 inverted cones form together an image of the whole 5D-4D inverse collapsing and expanding, evolving and entropic arrows of space-time.

Thus the motion from ∆+1 larger wholes to the ∆º singularity of the cone and its expansion on the inverse cone can be used as we do in the general model to represent the many different aspects of a world cycle of existence.

The cone thus is in itself, *another fundamental frame of reference proper of the 5 ^{th} dimension, ignored by huminds stuck in its 4D formalism, which we will retake when studying the Beltrami’s representation of hyperbolic geometry, the proper geometry of the 5^{th} dimension with a bit more of rigor.*

All this said, to show the profound levels of 5D theory bring about by a simple cone, its importance should not surprise the reader, as it is indeed the product of a | x O motion; hence embodying all possible, |xO=Ø curves of the Universe, which however can be reduced to a combination of our aforementioned curves (spirals already studied):

Yet, as mathematicians have focused on slices of space-time of the cone, and the conic curves of the 5D formalism exceeds the purpose of this introduction, we just enunciate some of its uses to show how a conic also mirrors a world cycle but will concentrate from here on, in the classic concepts of conics.

Still those concepts will apply to the understanding in mathematical physics of the different trajectories of points in one of the 4 canonical curves, each one related to a basic Dimotion of existence, to which we must ad the 5^{th} Dimotion of a log/Archimedean spiral. Since the log spiral can be considered an Archimedean spiral if we ad a 3rd dimension of height information, by converting its shrinking revolution into a receding motion, for an entity living within the cone’s timespace surface.

*Existential algebra, the ‘o’, ‘1’ terminology of geometric objects.*

How we write conic worldcycles in the more general GST terminology of existential algebra? Though we won’t translate mathematics to the Universal language of i-logic, existential algebra *that can mirror any other language of reality let us do a brief introduction ot the theme to complement the previous graph, also a bit ‘advanced’ for an introductory course. *

A worldcycle it is a closed domain, A WHOLE, either in the single cone or the dual one, and so in existential algebra we write the general equation, ST (vital energy) = S • T (singularity x membrane) in the following manner:

G(Tx; Sy) ≤ (1; 0) for a closed circle; G(Tx; Sy) ≤ (1;0, 0) for an ellipse; G(Tx; Sy) ≤ (∞; 0) for an open parabola and G(Tx; Sy) ≤ (∞, ∞;0, 0) for a ‘dual’ semi-closed hyperbola.

Where 1 means the membrain; 0 the singularity and ∞ an entropic open curve.

So that is what a conic means, a vital superorganism mirrored in mathematics, with its elementary parts – the membrane(s) 1, the relative infinite, outer open world and the 0, the relative center or singularities that manage the membrain; where G(x,y), are all the combined points, (≤x, ≤y) that form the inner, vital region of the system.

And since the Universe is bidimensional and holographic those conic equations are the most pervading in all forms of Nature. Yet only 2-dimensional curves and its combinations are real pure basic forms. All other n-dimensional equations are combinations of them, as Fermat’s grand theorem proves – since x³+y³ ≠ z³.

**Conics and Dimotions of existence. Its pentalogic.**

Descartes realized that curves in the plane are represented by second-degree equations with two variables whose general form represents an ellipse, a hyperbola, or a parabola; i.e., curves known to the mathematicians of antiquity.

So the first obvious fact is that ‘cyclical, time-like curves’ have 2 dimensions (degrees of freedom) as opposed to single-dimensional $t-lines. A theme that corresponds to mechanics and essentially means that lineal inertia is never found, as all lines are steps of a worldcycle, *in as much as the being exists in an outer world, and so we must always account for its dimotion its internal and external forces. *

*In the graph, t*here are 2 conic types: open=entropic=unstable=Darwinian parabola+hyperbola vs. closed=informative cycle+ ellipse. Both are constructed from a cone with a line, which is the *desired motion of all @ristotelian singularities and the curve ‘tended’ by the outer world,* through which a combined cycle ‘moves’ opening entropically or shrinking in an ‘accelerated’ process we can construct the conics of the Universe (as defined previously in terms of ST variations).

This was the wonder of Greeks till Desargues proved that all curves can be drawn from a conic, the best |xO=ø representation of the world cycle.

As such each of the canonical curves, which need only 2 ST parameters, defines a holographic bidimensional manifold – an ST dimotion of space-time. And each conic corresponds to a dimotion:

The 2 closed paths, the circle and the ellipse are S-dominant:

SS is by definition a seed with no motion in time and infinitesimal volume – the beginning of a process of expansion – hence the ‘summit point of the conic’.

St: The circle has a single focus-singularity, as it corresponds to the Dimotion of perception (π- spiral circle). It has a stable efficient form as a present repetitive system, hence with no latitude in height=time.

S=T: As the ellipse has two focus with equal distance to the ‘son-membrain’ they represent an S=T reproductive or dual curve, whose relative distance is a mental measure (Riemann’s concept of distance) of the similarity of both points, which is absolute when both have 0-distance and the ellipse becomes a circle.

On the other hand, we have 2 open curves, which represent free lineal motions with a content of time:

-sT: locomotion, represented by the parabola, since the point has a single trajectory, hence a single T-motion, and indeed we shall see the parabola to be the fundamental locomotion of any point subject to a larger world force (Galileo’s study of lineal trajectories).

-TT motions: finally the hyperbola represents the entropic dual motion, which ‘splits’ and erases the internal form of the system. So the lower parabola ‘oriented towards the past’, if we take the dual cone as a worldcycle, represents the internal scattering motion and the upper curve the external motion.

And so conics do have clear correspondence as almost all basic structures of mirror languages with the 5 Dimotions of reality, and those facts should orientate our analysis of the specific curves traced by physical particles and happening in real events of different stiences.

**Angle of congruence defining the different conics.**

The ancient Greeks had already investigated in detail those curves obtained by intersecting a straight circular cone by a plane. If the intersecting plane makes with the axis of the cone an angle ϕ of 90°, i.e., is perpendicular to it, then the SECTION obtained is a circle.

It is easy to show that if the angle ϕ is smaller than 90°, but greater than the angle α which the generators of the cone make with its axis, then an ellipse is obtained. If ϕ is equal to α, a parabola results and if ϕ is smaller than α then we obtain a hyperbola as the section.

In terms of the angle of congruence however the interpretation is exactly the inverse; as the axis of the cone is the motion of time from past (base) to future (top); that is from birth to extinction, and so the angle must be measured for a balanced present systems in parallel to the base of the cone; which represents the immortal state of past. And so the circle which is the balanced state of S=T (by definition, ∑x=∑y, for the whole range of values of a circle), is the state of present, with a parallel angle of congruence and no motion in time. While the maximal angle of congruence, and perpendicularity to the present with the maximal motion in time, is the entropic hyperbola, which in fact reaches to both extremes of the whole worldcycle. Whereas the upper side of the hyperbola ‘explodes the information of the system into an entropic death; and the lower part represents the maximal locomotion, similar to the parabola.

Why the parabola is in the worldcycle cone a motion to the past (g angle in the graph). Because as S is the state of absolute future (information, potential seed, logic mind in control of the system, designing its future), and T is the entropic time disordered arrow, an increase in locomotion, sT, is a relative state of past. But it doesn’t disorder the being as the hyperbola does.

What this means in GST terms is that the hyperbola is the opposite concept to the circle/ellipse, as closed and open, ðƒ and $t inverse ‘geometries’ which if we consider the y axis, the longitudinal entropic Time axis, and the x axis of the cycle, the informative, spatial state, converts the cone into an inverse Space-time where not only motion in time but also angles of congruence can be represented; and so we can study several world cycles and complex space-time events within it, in advanced 5D mathematical physics.

Let us then consider once those general concepts are laid down the main reason why some conics preserve the present, namely the balanced S=T efficiency of each form of a conic that represents each type of a vital Dimotion of existence, which are often used by fractal points that trace different conics for different vital purposes.

**Creation of curves in ¬E geometry and its 5 Dimotions: social parallelism vs. perpendicularity annihilation **

**S-Conics** relate to the second postulate of ¬E communication between 2 points that in informative conics construct a ‘territory’, delimited by the membrain, which is equidistant, *in a cyclical time period, to the 2 points. Those* closed ordered conic have a minimal angle of parallelism; that is, displacement=change in time, to reach a degree of ‘simultaneity’ that allows the creation of an organic simultaneous space.

**-T:Open conics** with a larger angle of congruence change in time, becoming disordered asymmetric in space, failing to form a stable system. We can then consider the parabola in physical terms as the ‘external accelerated growth of distance’ TT-between two points (one fixed by convention, or inversely the accelerated attraction of a force), while the hyperbola represents the split of its inner parts into inverse trajectories, or alternately the inverse properties of two S/T elements (SxT=C). So conics connect the 2^{nd} & 4^{th} postulate of communication and congruence:

In the graphs, the general laws of behavior in the organic Universe are simple: beings who are similar or complementary and speak the same language of information come together as couples, herds and social wholes stronger than individuals. Those who perceive each other as different, will simply act in a Darwinian manner, which means, they will either increase the distance=dissimilarity; highlight their difference or break and split under predator tearing – all of them properties reflected in open conics. So we can consider also the 4 closed (right) and open (left) conics in terms of the 4 different forms of asymmetry.

We redefine geometric elements vitalising its meaning with the ‘fourth postulate of ¬Æ geometry, as the ‘angle of communication’ determines the outcome of most events either as parallel creation or Darwinian perpendicularity.

*2* ‘asymmetric’ beings, i.e, the line and the cycle, come together, fusioning in a creative way, when *their coming is parallel, or destroying each other when it is perpendicular. *

*Parallel creation: There are 2 levels, creation by communication of information through an intermediate space, studied more properly in logic and creation by adjacent pegging more suitable to topologic studies, which also *spreads into Analysis (a derivative is a parallel. In essence pegging by parallel adjacency is necessary to create organic wholes. So when there are NO parallel peggings there are NO derivatives, NO communication between ∆ø and ∆-1, NO possibility hence to create a super organism of ∆±1 scales).

*3D creation by penetration of the line into the curve.*

Creation and reproduction produces the biggest e-motion, the orgasm, literally the sensation of a cycle invaded by a line in parallel in and out harmonic oscilation=penetration *because it is the purpose of a Universe whose fundamental element is motion and fundamental e-motion the adjacent friction of parallel forms that create complementary wholes.*

The cone that generates all curves is the inverse a penetration of a cycle which moves along a line, ‘tightening’ its grip.

While perpendicularity cuts and destroys one of the 2 elements, making ∆st topology vital. Rememvber, we follow Godel and Lobachevski and Einstein: mathematics as all languages are real mirrors of a higher living reality.

*2D creation in a holographic bidimensional 2-manifold by tangent parallelism.*

Systems in any scale of the universe, from Atomic Ions or crystals to human societies relate to each other in darwinian, perpendicular ‘tearing’ topological relationships that ‘break’ the closing membrane of one species disrupting its existence (open conics), or will keep a mean distance to form social networks of communication that will grow into super organisms, starting the emergent process of evolution of species into a new ∆§cale of social existence (closed conics). So *any system’s organic, geometric and scalar relationships are symbiotic to each other. *

How dimensions combine to create form is an essential feature of the duality between symmetric parallelism and perpendicular annihilation (antisymmetry), which plays a special role in 5D geometry. In 2 dimensions the tangent that gives origin to a derivative reduces form to finitesimals. Or in its inverse integral is the creative form of ∆-scales.

Each geometric form can be seen as a point that moves creating a new dimension or more: the point becomes the circle that turns around. Or inversely, the circle peels off a wave from its cyclical membrane of angular momentum shaping the main creative form of 3 Dimotional S=T balance. What the S-dominant cycle ads then is a T-dimension of lineal motion and the wave is born.

In 3 dimensions the conic is a line penetrating a new dimotion, dragging a circle that turns faster in smaller spaces as it moves to the apex, final point or origin of the cone.

**Symmetries in conics. **

The different degrees of Parallelism, Perpendicularity and skewness are essential concepts of vital Non-Euclidean geometry applied to conics. The curves of a conic worldcycle are all the curves of the Universe; according to the *type of event=dimotion they represent, from TT entropic hyperbolas where each of the initial focus breaks the positive relationship, cutting the membrain in two parts, one for each focus that depart in inverse directions of time space, to the fusion of the two points into the boson center of a circle when parallelism is absolute.*

Thus Conics acquire a new perspective under the holographic principle of a Universe built on bidimensional ensembles, where most ‘ternary dimensions’ are layers of reproduced bidimensional surfaces or ‘branched networks’, spread on the ‘holes’ of a 3rd dimension’. And so we distinguish 2 kind of conics:

-Time like conics, circles and ellipses, which close into themselves creating a clear ternary structure with an external membrane closing an internal space, self-centred in one or two points separated lineally by a factor of excentricity.

-Space-like conics, parabolas and hyperbolas; which apparently are open systems without closure, but in fact preserve both, the central point of view, the internal territory and the membrane, *albeit open to let the world circulate through it.*

So conics are a dynamic transformation between $t (open) < ≈ > ðƒ (closed) states of an ST being, with a single parameter to measure them, eccentricity; whereas the most perfect bidimensional being, is one of o-eccentricity, where the ‘2 focus’ of the central singularity, which can be any S/T VARIATION are both equal in space and time (a single point) – the circle. Which therefore must be considered as the Greeks had it, the perfect form; an all others deformations of it.

S=conics: Informative (Particle-head) communication, possible in cases of relative similarity≈parallelism (which determines parallel herding and social evolution) for the 2 closed conics – whereas the circle is a perfect symmetry.

T-conics: They are skew curves that do not intersect and are not parallel and clearly related to hyperbolas (with inner scattering entropic motion) and the parabola, which distances 2 points. Indeed, two lines are skew if and only if they are not coplanar, which IN 5Ð as 3 ±¡ planes co-exist in the same organism and systems feed in T.œs, two super organisms down, implies species, which are not in the *relative planes of action of the being. They are in the worldcycle cone the 2 parts of the hyperbola clearly skewed in time and space.*

**Topological emergence between planes.**

When we deal with annihilation by perpendicularity things get also 2 variations as it is logic, to think by ∆-scattering or by S<≠>T antisymmetry. But as annihilation ultimately means destruction of an ∆ scale, it derives in entropic dissolution. The results are often shown in exponential functions.

We can think of the ‘change of planes’ as a perpendicularity against which the internal function (of momentum) ‘collides’ , trying to puss the ‘wall’ that separates scales without result, growing then in ‘inertial mass’ no longer in speed. As ultimately the vital energy enclosed by a membrane finds always the membrane to be perpendicular and annihilating it very often; the military border in a nation, barrier of cattle, or sepherd dog, the predators, etc.

The best known case in physics are related to the hypothetical impossibility of a function to cross a discontinuity between planes, which *is what it means in the Lorentz transformations: as a mass comes closer to the relative infinite limit of his light space-time domain, its grows ‘theoretically’ towards infinite as it cannot speed more. *

So its momentum mv ‘changes’ no longer in v but in m (as change cannot be stopped, the ∑∏ energy fed in the system must either derive into the singularity m or the speed-membrane in parallel to the larger whole galaxy membrane (Mach explanation of angular momentum). This no longer possible as the part cannot move faster than the whole (c-speed limit for the galactic space-time membrane), the vital energy does NO longer feed the membrane but the singularity and its active scalar mass, the 0-1 Dimensional parameter of density reflected in the Dirac membrane.

So we can see geometrically or algebraically how this momentum becomes then ‘deviated’ as a parallel angular momentum of the membrane either in lineal or cyclical fashion (itself a transformation of an SH motion from cyclical into lineal), to a growth of mass.

So the third age of geometry which started with Lobachevski’s 3 ‘findings’, mental space, topology and experimental need of maths to validate each mental space with reality, is really about this mental realisation that space is information, and so the 3rd informative age of geometry is obviously about… mental information.

**RECAP**. There are 3 modes of S=T mathematical creation by parallelism in the Universe: The harmonic oscillator and the cone that* model the worldcycle of existence & the* tangential lines that reduce cyclical patterns to its scalar stœps & dimotions. Timespace also splitS in the duality between past to future to past male genders and S<≈>T present female genders, whose transformations of topological lines and cycles into hyperbolic waves results in new creative combination.

** **

** **

**CLOSED – INFORMATIVE CONICS**

There are two closed informative conics highly symmetric. The symmetric circle, which we study all over the place, and the asymmetric ellipse, whose main differences in pentalogic terms are:

@: The circle has a single center or two boson-like equal centers. The ellipse has 2 focus.

¬ The circle is static, balanced and last. The ellipse is born of a ‘contraction/expansion’ of the circle along its X or Y coordinates of entropic vs. informative growth. As such it tends to be more dynamic.

T: Both are cyclical in patterns, hence can also act as basic representations of worldcycles.

S: The circle has the max. volume of vital space-energy with minimal perimeter. As such it is the most efficient form for a single territorial mind-point; but the ellipse is the best system for a dual pole communication, as it doubles its fractal points with minimal growth of perimeter. *Hence in palingenetic development as soon as the seed breaks into polar S/T animal/vegetal poles acquires the form of an ellipse and as one of both poles is smaller (the animal pole), it finally becomes an ovoid *

**The equation of a circle with center at the origin**.

First of all, we consider the circle whose equation is a generalized Pythagoras theorem: x²+y²=a². Its simplicity shows its suitability for a self-centered graph.

It evidently represents a circle with center at the origin and radius a, as follows from the theorem of Pythagoras applied to the shaded right triangle, since whatever point (x, y) of this circle is taken, its x and y coordinates satisfy this equation, and conversely, if coordinates x, y of a point satisfy the equation, then the point belongs to the circle; I.e. The circle is the set of all those points of the plane that satisfy the equation.

But from a pentalogic vital point of view both its equation and geometry reveals different elements of 5D reality; *when we consider the entity @-center to be a singularity constantly moving along the S: height=informative and T: length=entropic axis, in such a manner that as a ‘whole’ entangled S=T system, whereas S ^{2}+T^{2}=K=SS+TT.*

*Of which the most remarkable case is 3 ^{2}+4^{2}=5^{2}* whereas 3 is the time-ages perspective, 4 the spatial perspective of the coming together of a bidimensional body and head, and 5 the scalar perspective; hence T

^{2}+S

^{2}=∆

^{2}. TT+SS=∆(-∆): ∆Ñ.

What truly means is that in a single plane (given by the + operand) the sum of all the entropic and informative stœps of a being will form its full worldcycle up and down 5D.

This is a profound law that rightly embeds the most important theorem of geometry, Pythagoras, with Space-time reality. And the only equation we proceed inversely, taking it from maths to carry it into Space-time thought analyzed in our not-published papers on pentalogic. Back to the circle’s pentalogic some highlights:

S=T: Ast<=>Bst: In terms of 2 points communication and its intermediate energy Dimotions the circle is an ellipse where the 2 elements are so similar they coincide in perfect symmetry. The eccentricity is 0, and the result is to be the two-dimensional shape enclosing the most area per unit perimeter squared, *hence the most efficient territory of control for a couple when its similarity becomes identity. Why a 2 view is more proper is obvious as the circle requires two up and long axis to be webbed. So multitasking better splits between two.*

∆: In terms of scales and the extreme dimotions of entropy and form. the inner coordinates, ‘abscissa’ and ‘ordinate’ of the being (S and T values), it follows Pythagoras, SS+TT=∆Ñ postulate.

**The equation of an ellipse and its focal property: eccentricity and symmetry**

Next comes the ellipse, where the communication between points differs in ‘length=entropy/motion /size/distance’ parameters, but its ‘height-informative dimension’ remains the same, hence allowing ‘congruent communication of information’ (or is minimal in the worldcycle’s cone):

Let two points F1 and F2 be given, the distance between which is equal to 2c. We will find the equation of the locus of all points M of the plane; the sum of whose distances to the points F1 and F2 is equal to a constant 2a (where, of course, a is greater than c). Such a curve is called an ellipse and the points F1, and F2 are its foci.

Let us choose a rectangular coordinate system such that the points F1 and F2 lie on the Ox-axis and the origin is halfway between them. Then the coordinates of the points F1, and F2 will be (c, 0) and (–c, 0). Let us take an arbitrary point M with coordinates (x, y), belonging to the locus in question, and let us write that the sum of its distances to the points F1, and F2 is equal to 2a:

This equation is satisfied by the coordinates (x, y) of any point of the locus under consideration. Obviously the converse is also true, namely that any point whose coordinates satisfy the equation belongs to this locus. The Equation is therefore the equation of the locus. And while *mathematicians* simplify it, the interest for the topological o-point of view remains precisely in its complete form.

The perfect ellipse thus have the same coordinates and ±c distance to the center of reference, which is possible for two equal points, *which lay in a non-isomorphic 2-3D world where the ‘field of entropic motion’, the abscissas, is larger. Hence the asymmetry belongs to the world not to the focus. *But as we change to imperfect ellipses the submissive role of the 2^{nd} element will increase till it becomes *in orbital systems, ‘expelled’ as the membrain (planet to the sun). *Let us explore then both together in the context of basic XVII-XVIII Physics.

Back to the equation of the ellipse simplifying terms we get to: x²/a²+y²/b²=1

Substituting y = 0 in the equation, we obtain x = ±a, i.e., a is the length of the segment OA, which is called the major semiaxis of the ellipse. Analogously, substituting x = 0, we obtain y = ±b, i.e., b is the length of the segment OB, which is called the minor semiaxis of the ellipse.

The number c/a is called the eccentricity of the ellipse that is less than 1. In the case of a circle, c = 0 and consequently e = 0; both foci are at one point, the center of the circle (since OF1 = OF2 = 0).

As the eccentricity grows the 2 points separate but the *points still control the area of the system, which can be shown by *the method of drawing the curve with a thread connected to both.

**Mathematical Physics: Kepler’s law.**

The main difference between the ideal simplification of mathematics as a mirror of spacetime laws and the real spacetime laws is the loss of ‘Dimotions’ to get to the bare basics in which *laws of Nature can be modeled – a bidimensional still geometry. Then when we return to Nature those laws are expanded with addition of motion that distorts the geometry (but conserves after a transformation of T into S the essence of the law) and an increase of Dimotions/Dimensions. So when we get into astronomy* as Newton proved the law of attraction of bodies – considered in the generic case the two poles of an ellipse give us the elliptic orbit. However the *distorsion of adding and transforming still dimensions into motions changes often the ‘formal elements’ of the geometry, preserving the ‘essential logical/vital components’, showing once more that the essence of the game is not abstract laws of spatial mental geometry but vital laws of time-space structures.*

*So indeed, now the planet occupies the membrain and in the second foci there is nothing. Let us for a change put some quantitative effort calculating it. * The 2^{nd} focus is along the major axis: the line joining the positions of perihelion (closest to the Sun) and aphelion (furthest from the Sun) in Earth’s orbit. So we draw an ellipse. Place the Sun at one focus (on the major axis, a bit off to one side). Mark the Sun “S” and the centre “C”.

On the “short side” of the Sun, along the orbit, where the major axis cuts the orbit, that is the perihelion “P”. At the opposite end is the Aphelion “A”. In 2010, Perihelion was on January 3, at a distance of 147 mill. km In 2010, Aphelion was on July 6, at 152. Total length of the major axis (from P to A) is the sum = 299,mill k The semi-major-axis (distance from P to C and from C to A) is half of that (149,597,213.5 km). Since we know that the distance PS is 147 mil km, then the distance SC must be the difference PC – PS = 2,5 mill km The Sun is 2.5 million km to the “January” side of the centre. By symmetry, the empty focus is 2.5 mill km on the “July” side of C CE = 2, 5 mill .SE = SC + CE = 2*CE = 5 mill km.

But there is nothing at that point. So the question is what we can save from the definition of an ellipse *that a 5D reconstruction of physical spacetime preserves when we expand the mathematical mirror? 2 *GST* truths*:

*1) *all distances are motions. So if we make the moving planet a static line per unit of time-motion, the key vital, communicative 2^{nd} postulate property stays with an added dimensions. Now 2 points do not trace the same length but ‘sweep the same area’ working together. Thus an orbit, in the physical analysis of GST becomes a dual system that herds a vital are: the membrane with more angular momentum (the $t-planet) and the singularity with more gravity/mass (the ðƒ), surrounding herding and absorbing the lower ∆-1 scale of gravitational points/forces.

So if we apply to one of the points Absolute relativity (S=T: ‘motion is indistinguishable of distance’) orbital laws, imply the planet and the sun together scan the same ‘gravitational area’ . This aerolar law transcends then to any physical vortex, and also implies that the ‘aerolar ellipse’ will collapse till the membrain ‘becomes’ the singularity, falling into the higher mass to be ‘one’ as the static ellipse evolves into the circle. Let us then conclude with the analysis of the transformation of circles into ellipses, which show they are the same topologic variety.

**The ellipse as the result of an assymetric “expansion/contraction” of a circle.**

The generalized case of an entropic ellipse with 2 predator points in a line that expand its range, is analyzed in abstract, as the alternative Y-coordinates’ contraction of a circle. We consider a circle with center at the origin and radius a. By the theorem of Pythagoras its equation is x^{2} +y^{2}= a^{2}, where we have written yi instead of y, since y will be needed later. Let us see what this circle is contracted into if we “contract” the plane to the Ox-axis with coefficient b/a. After this “contraction” the x-values of all points remain the same, but the y-values become equal to y=y(b/a) . Substituting for in the above equation of the circle, we will have:as the equation, in the same coordinate system, of the curve obtained from the given circle by contraction to the Ox-axis. As we see, we obtain an ellipse. And inversely we have proved that an ellipse is the result of a “expansion in abscissas” of a circle.

From the fact that an ellipse is an asymmetric “contraction/expansion” of a circle, many properties follow, which as usual we order in pentalogic terms:

*ST:area.* Since any vertical strip of the circle under its contraction to the Ox-axis does not change its width and its length is multiplied by b/a, the area of this strip after contraction is equal to its initial area multiplied by b/a, and since the area of the circle is equal to πa², the area of the corresponding ellipse is equal to πa²(b/a) = πab.

@-singularity point (center of gravity) is simply the midpoint between them. And it follows that a *natural evolution of the ellipse as two similar forms attract, is to become a circle, which i*n Gst theory reveals a deeper truth: systems become ‘compressed’ into smaller networks and finally into single singularity points, which organize the entire system. This structure can be generalized as Newton did for any closed conic divided into territorial domains with different centers of gravity, which all laid in a single line, itself centered in a single point, to any T.œ, which will have a vital space in 3D, reduced to the control of a menbrain in 2D, itself controlled by a single 1D, *each of the smaller systems processing faster information, transforming vital space into Dimotions of exist¡ence.*

So both for the ellipse and the circle, we can consider the surface enclosed by the disk, structurally sustained by the network of lines, itself communicated at equal distances by the line, and finally the line focused in the ‘center of gravity’, we have built an ∆+2>∆+1>∆>o scalar structure. And here we realise why analytic geometry works, as it does compress mentally geometric surfaces into sequences of numbers of lesser ‘volume’ of information that ‘commands’, logically the whole.

This property of diameters – that if parallel secants of an ellipse are given, then their midpoints lie on a straight line, can be shown also from the contraction of ellipses in the following way:

We perform the inverse expansion of the ellipse into the circle. Under this expansion parallel chords of the ellipse go into parallel chords of the circle, and their midpoints into the midpoints of these chords. But the midpoints of parallel chords of a circle lie on a diameter, i.e., on a straight line, and so that the midpoints of parallel chords of the ellipse also lie on a straight line. Namely, they lie on that line which is obtained from the diameter of the circle under the “contraction” which sends the circle into the ellipse.

**The ellipse of inertia.**

Another physical example of the power of ellipses to create stable dual focused forms is the ellipse of inertia, whose maximal resistence for any system, even those which have ‘different edges happens across the axis of the ellipse.

i.e. Let a plate be of uniform thickness and homogeneous material, for example a zinc plate of arbitrary shape. We rotate it around an axis in its plane. A body in rectilinear motion has, as is well known, an inertia with respect to this rectilinear motion that is proportional to its mass (independently of the shape of the body and the distribution of the mass). Similarly, a body rotating around an axis, for instance a flywheel, has inertia with respect to this rotation.

But in the case of rotation, the inertia is not only proportional to the mass of the rotating body but also depends on the distribution of the mass of the body with respect to the axis of rotation, since the inertia with respect to rotation is greater if the mass is farther from the axis. For example, it is very easy to bring a stick at once into fast rotation around its longitudinal axis. But if we try to bring it at once to fast rotation around an axis perpendicular to its length, even if the axis passes through its midpoint, we will find that unless this stick is very light, we must exert considerable effort.

“It is possible to show that the inertia of a body with respect to rotation about an axis, the so-called moment of inertia of the body relative to the axis, is equal to ∑r²i mi (where by ∑r²i mi we mean the sum ∑r²1 m1 +∑r²2 m2 +…..+∑r²n mn) and think of the body as decomposed into very small elements, with mi as the mass of the ith element and ri the distance of the ith element from the axis of rotation, the summation being taken over all elements.

Now escaping its proof, the following remarkable result can be obtained: Whatever may be the form and size of a plate and the distribution of its mass, the magnitude of its moment of inertia (more precisely, of the quantity ρ inversely proportional to the square root of the moment of inertia) with respect to the various axes lying in the plane of the plate and passing through the given point O, is characterized by a certain ellipse. This ellipse is called the ellipse of inertia of the plate relative to the point O. If the point O is the center of gravity of the plate, then the ellipse is called its central ellipse of inertia.

The ellipse of inertia plays a great role in mechanics; in particular, it has an important application in the strength of materials. In the theory of strength of materials, it is proved that the resistance to bending of a beam with given cross section is proportional to the moment of inertia of its cross section relative to the axis through the center of gravity of the cross section and perpendicular to the direction of the bending force.

Let us clarify this by an example. We assume that a bridge across a stream consists of a board that sags under the weight of a pedestrian passing over it. If the same board (no thicker than before) is placed “on its edge,” it scarcely bends at all, i.e., a board placed on its edge is, so to speak, stronger. This follows from the fact that the moment of inertia of the cross section of the board (it has the shape of an elongated rectangle that we may think of as evenly covered with mass) is greater relative to the axis perpendicular to its long side than relative to the axis parallel to its long side. If we set the board not exactly flat nor on edge but obliquely, or even if we do not take a board at all but a rod of arbitrary cross section, for example a rail, the resistance to bending will still be proportiopal to the moment of inertia of its cross section relative to the corresponding axis. The resistance of a beam to bending is therefore characterized by the ellipse of inertia of its cross section, which *becomes therefore its ‘core-singularity element’, often controlled by its central point/s*.

**The logic expansion of the concept of dual elliptical territories.**

Now following this kind of thought, of ellipses as collaborative locus of 2 ‘complementary species’, we can apply the ‘logic’ of the concept to anything and in fact define ‘eccentricity’ lines as the essential form of a wave of communication between 2 points (2nd Non-E Postulate):

So a couple with a son, is a GST ellipse, where both fathers are constantly seeking a similar distance between them. And a territorial animal couple is also a logic ellipse, tendering for the territory as one moves to hunt, the other stays to breed.

Any relationship is a naked ellipse (without the external membrane), joined by the focal line that shares entropy and form between them.

Steel beams often have an S-shaped cross section; for such beams the cross section and the ellipse of inertia have the greatest resistance to bending is in the z direction. When they are used, for example as roof rafters under a load of snow and their own weights, they work directly against bending in a direction close to this most advantageous direction.

This result can be understood in terms of ‘2 planes’ the ∆-plane of the beam and the ∆-1 plane of the gravitational field, and *the dominant nature of the major axis line that communicates the inner structure of the entity.*

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**OPEN – ENTROPIC CONICS – ITS PENTALOGIC**

**The open curves: parabolas and hyperbolas**

Now once we have identified what is truly relevant about bidimensional curves as opposed to single ones that represent only a part of the being: to be of a full ternary organism, with 3 parts:

A focal point or singularity: @; a membrane or cyclical curve: ð§; The vital space or energy between them: st

We can consider ‘open curves’ in which the intermediate space is fully opened and its meaning to represent key elements of T.Œs (Timespace organisms).

And the wonder of them is that in those open systems the key elements will still be determined by the focal singularities and the relative balance of their ‘co-invariant’ product in relationship to the membrane.

So they can represent the ‘metric equations’ of co-invariance 5D systems, and in fact, the hyperbola will be the best representation of any function:

**T: The parabola as a TT quadratic propotion. **

Let us then consider the open conics first from the realist perspective of mathematical physics, where they represent the motion of one or two points, rather than the static geometry of a form with its membrain. Galileo then found that a TT-accelerated motion away or coming to a different focus=point (the maximal earth vs. the attracted form), had the shape of a parabola. As we know acceleration is a double ‘time locomotion’ (TT) – the motion of a motion, but maintaining the ‘point’ with its inner structure. So we have to represent only a point – a line. And because it is moving away at an accelerated dual=square bidimensional pure time motion (acceleration) it is a quadratic proportion. Thus we define the parabola as the graph of quadratic proportion. We recall that the graph of quadratic proportion: y=kx² is a parabola; which we ‘tumble’ to give it significance in mathematical physics.

We can then consider inversely the ‘mathematical concept’ of the parabola as a ‘fixed membrain’… of what we might wonder since it is an open form. The answer is that as a fixed form the parabola, and its real form the 3D paraboloid *are the perfect form to focus lineal flows of Spacetime either for the purpose of 1D perception, or inversely, they can transform a ‘source’ of energy into lineal entropic motions; two other variations of the sT-theme:*

**S@: Its focus and its directrix.**

We consider then the equation* y² = 2px *and call the corresponding curve a parabola.

The point F lying on the Ox-axis with abscissa p/2 is called the focus of the parabola, and the straight line y = –p/2, parallel to the Oy-axis, is its directrix.

While at each point of the parabola we can trace a lineal tangent, given the fact that all curves are made in the smaller scale of ‘free open, lineal steps’ (which in mathematics is the basis of differentials). Hence the linearlization of the parabola when we ‘extract’ one dimotion of acceleration, ‘shrinking’ the whole motion to its steps.

Let us then M be any point of the parabola; ρ the length of its focal radius MF, and d the length of the perpendicular dropped from it to the directrix. Let us compute ρ and d for the point M. From the shaded triangle we obtain ρ2 = (x – p/2)² + y². As long as the point M lies on the parabola, we have y² = 2px, hence:

But directly from the figure it is clear that d = x + p/2. Therefore ρ² = d², i.e., ρ = d. The inverse argument shows that if for a given point we have ρ = d, then the point lies on the parabola. Thus a parabola is the locus of points equidistant from a given point F (called the focus) and a given straight line d (called the directrix).

*The property of the tangent to a parabola. *

Let us examine then on those basis the 1D focus of the parabola which makes it so useful in vital topologies from eggs to eyes and antennae.

Since for a parabola y^{2} = 2px we have 2y dy = 2p dx. It follows that the derivative, or the slope of the tangent, is equal to dy/dx = tan ϕ = p/y.

On the other hand, it follows directly from the figure that:

tanϒ=y/x-p/2

But:

i.e., γ = 2ϕ, and since γ = ϕ + ψ, therefore ψ = ϕ.

Consequently, by virtue of the law (angle of incidence is equal to angle of reflection) a beam of light, starting from the focus F and reflected by an element of the parabola (whose direction coincides with the direction of the tangent) is reflected parallel to the Ox-axis, i.e., parallel to the axis of symmetry of the parabola:

On this property of the parabola is based the construction of Newton’s reflecting telescopes and modern antennae. If we manufacture a concave mirror whose surface is a so-called paraboloid of revolution, i.e., a surface obtained by the rotation of a parabola around its axis of symmetry, then all the light rays originating from any point of a heavenly body lying strictly in the direction of the “axis” of the mirror are collected by the mirror at one point, namely its focus. The rays originating from some other point of the heavenly body, being not exactly parallel to the axis of the mirror, are collected almost at one point in the neighborhood of the focus.

In other words, *a parabola is ½ ellipse where one of the focus has been stretched to a relative infinite value compared to the other focus, breaking its bisymmetry, and this introduces the concepts of ‘relative infinities’ (the Earth is an infinite weight compared to the point that traces a parabola of TT-motion towards it; the star focused in the telescope is at a relative infinite length compared to the focus distance to the paraboloid, etc.*

We show this relation geometrically by first drawing a circle, and then “stretching” it to make an ellipse, and then “stretching” it even further to make a parabola (point goes to infinity). So if we start with the equation of the unit circle: 𝑥^{2}+𝑦^{2}=1

And then do some stretching in the vertical direction by a factor of 𝑏: 𝑥^{2}+(𝑦/𝑏)^{2}=1

And them we let b get *really big*, we get the equation of the parabola: 𝑥^{2}=1 (y)

Thus, in the so-called focal plane through the focus of the mirror and perpendicular to its axis, the inverse image of the star at the point of infinity is obtained – but the point must be a fractal point with a volume to be perceived!: the farther away this image is from the focus, the more diffuse it will be, since it is only the rays exactly parallel to the axis of the mirror that are collected by the mirror at one point.

**∆+1:** The searchlight is based on the same property of the parabola. In it, conversely, a strong source of light is placed at the focus of a paraboloidal mirror, so that its rays are reflected from the mirror in a beam parallel to its axis. Automobile headlights are similarly constructed.

Thus in the still view of parabolas, there is still a focus and a membrane. Whereas the parabolic being is a single ‘foci’, able to ‘focus’ the information and entropy of a larger scale field; from the pentalogic ∆+1 perspective.

**ST< => ST: @-ellipse. **In the pentalogic of the ellipse this property give us also the ‘perceptive’ pentalogic view:

Indeed in the ellipse, it is easy to show, the rays issuing from one of its foci Fl and reflected by the ellipse are collected at the other focus F2 (previous figure), *making the communication between both points as simple as a reflection in the inner side of the closed membrain; a property which as usual departs from its pure geometric formulation in symmetric systems, to configure the relationships of ‘network-lines and waves’ in physiological and physical organisms.*

So the parabola, without a second equal focus, which would enhance the survival symbiosis between both is NOT usually a full T.œ, but at best an organ open to the world (Static view) or a TT-accelerated motion..

**The Hyperbola.**

On the other hand, in the hyperbola the rays originating from one of its foci F1 are reflected by it as if they originated from the other focus F2: This is a representation of a ‘head-body’ system where the body is blind to perception as it reflects the information absorbed from the head, which is therefore the F2 focus of the hyperbola, ‘above’ its lower part, split from it.

**The inversion of ST values, focus and directrix of the hyperbola.**

So the hyperbola is the most extreme entropic representation of ‘split’ ‘inverse’ ST properties.

Indeed, if we consider a single part of the hyperbola as the graph of inverse proportion and that the graph of inverse proportion y = k/x y x = K is a hyperbola. Its equation though is all pervading precisely because *the quality of inversion is the essential dual property of space vs. time fields, which can be represented by ½ hyperbola in innumerable cases. *S x T = K is by definition, the equation of t**he perfect hyperbola.**

Indeed, the 2 branches conic equation. x²/a²-y²/b²=1 represents the full hyperbola. In the special case a = b the so-called rectangular hyperbola plays the same role among hyperbolas as the circle plays among ellipses:

If we rotate the coordinate axes by 45° the equation in the new coordinates (x′, y′) will have the form: x’ • y’ = k. And we shall use both modes to fully grasp fundamental metric equation of systems in the fifth dimension.

Now in the previous hyperbola, if we denote by c a number such that c² = a² + b², then it is possible to show that a hyperbola is the locus of all points the difference of whose distances to the points Fl and F2 on the Ox-axis with abscissas c and –c is a constant: ρ2 – ρ1 = 2a. The points F1 and F2 are called the foci.

Let us then consider the parabola from the perspective of its foci and directrix.

Like the parabola, the hyperbola has directrices, in this case two apiece. If we consider a focus and the directrix “on the same side with it,” then for all points of the corresponding branch of the hyperbola, we have ρ/d =ε , where the eccentricity for a hyperbola is always greater than 1.

Thus one branch of the hyperbola are the loci of all those points in the plane for which the ratio of their distance ρ from the focus to their distance d from the directrix is constant. For the ellipse this constant is smaller than unity, for the parabola it is equal to unity, and for the hyperbola it is greater than unity. In this sense the parabola is the “limiting” or “transition” case from the ellipse to the hyperbola; born as the ellipse tears apart its 2 focuses, that split entropically into two different entities, albeit maintaining its relative symmetry as two parts that were once entangled into one:

And so the fundamental relationship between the curve and the 2 foci, is preserved in an inverse ‘resting manner’; which qualifies the hyperbola as the entropic state of the ellipse, its time-reversed figure, an aforementioned property of importance for ‘complex GST analysis’, well beyond the scope of this texts.

The hyperbola is different from the ellipse, as it is pure algebraic in ‘phase space’, with variables in which the hyperbola is NOT a real form, but a mental form to represent, *the metric equation of 5D, in which Ts x St = K:*

Consider a simple formula for Pressure, p, due to a liquid column: P =ρ x g × h

Density is a measure of density of form, or information of a system; h, the height dimension of information and g, acceleration, a parameter of an inward vortex of growing frequency. Thus *pressure is an St parameter, with a value, product of a time-dimension (frequency acceleration), an informative dimension, height, and a time dimension, ‘density’. Moreover, we can put the 3 ‘elements’ in terms of time as a measure of the ‘past’ value of the system (its density), the present value (its height) and the future value (its acceleration downwards), and then make a deep philosophical statement about the constancy of pressure.*

Yet if pressure is the ðƒ parameter, it follow that expansive volume is the pure SPACE-entropy parameter, and so we shall immediately postulate according to 5D metric the existence of a co-invariant relationship:

P(t) x V (s) = K (st)

Where K will turn out to be the cyclical space-time vibration of temperature.

This ‘dimensional analysis’ is thus an entire new fruitful perspective on mathematical physics, akin to the dimensional analysis of classic physics, but far more profound in significance.

Boyle’s law amounts to yet another ‘5D metric’ equation, which we can plot with a straight line departing from O, crossing all different Ti, for equivalent PxV values, maximised in the central region of the asymptotic curves.

*All this *reveals whys and Ðisomorphisms of a simple mathematical equation which for a physicist, means merely ρ, the density in kg/m3, g=1o m/s² the acceleration due to gravity and h, the height or depth of liquid in meters, used to calculate the praxis and future behaviour of a liquid in motion.

But what we have written is essentially the equation of potential energy, PE=m x g x h, which we will indeed define when studying actions and Hamiltonians, the ultimate equations of 5D physics (as well as 4D physics), as the time-like component of ‘present space-time energy’.

The theme of Geometry and physics is obviously well beyond anything this author can develop in a few notes. So the reader specially if a physicist should not expect more than some marginal comments.

A few comments though seem necessary after studying the representation in motion geometry of the 5 Dimotions of reality with conic curves, since that was essentially the way in which modern physics was born, when Galileo studied the 5^{th} Dimotion of entropic cannonballs, which were open T-parabolas and Kepler, the 4^{th} Dimotion of interactive orbits of planets and sun, which were closed S-ellipses.

**The 10 canonical equation of the bidimensional plane.**

In the graph we see the 10 canonical curves, of which 5 are S=curved and 5 are T-straight, 5 are T-open and 5 are S=closed, 3 are imaginary, 3 are double, 3 are single and 1 is a point. They are indeed what they are because they respond to the ST and 3×3+• symmetries of the space-time Universe.

We considered the most important second-order curves: the circle, the ellipse, the hyperbola, and the parabola. What other curves and generations are relevant to exhaust the field of bidimensional geometries?

Not surprisingly as the Universe is only a 5D structure, there are no more curves than the ones needed to define the 5 Dimotions of reality. So all other curves can be reduced to one of the 9 canonical equations of conics.

A 2^{nd}-degree equation, contains 6 terms, not 3 or only two as in the canonical equations of the ellipse, hyperbola, and parabola. This is not because such an equation represents a more complicated curve but because the system of coordinates is possibly not suited to it. It turns out that if we select a suitable Cartesian coordinate system, then a second-degree equation with two variables always can be reduced to one of the following canonical forms – since as we already explain *the simplified generalized coordinates or |, O, Ø topological varieties in which the system or event we study, reside will always bring a simpler more true point of view. So as t*t turns out if we select a suitable Cartesian coordinate system, then a second-degree equation with two variables always can be reduced to one of the following canonical forms:

The reader will observe those canonical forms are the 3 conics (with the circle a contracted ellipse) *and the two essential forms of congruence, intersecting and parallel lines.*

Alas, once more the Universe appears as a simple structure, of closed and open systems, the perfect circle, the split ellipse with 2 focus, the parabola, which further splits them and the hyperbola which through the y-axis of entropy sends both in different ‘height arrows’ of the ∆-scales of the fifth dimension.

Plus 3 varieties of lineal couples, the intersecting couple, the parallel couple and the identical ones, which again respond to the ternary symmetries of the Universe, whose profound meaning, relevant to the outcome of all events in space-time is studied in depth in the article dedicated to the 4th postulate of non-Æ logic.

Let us write them all in 3 dimensions, which Fermat’s theorem, superposition laws) is merely done by accumulation of reproduced, identical ‘social numbers’ of planes, one after another, the same curves merely engrossed through the reproductive growth of a z-dimension are still the same unique varieties:

X^{2 }+Y^{2}+Z^{2}=1 Sphere

Each of those conics and curves in 2 or 3 Dimensions. We can then see the circle and the cone as a 3D spiral the 1 Dimotion and non-Euclidean postulate.

The second postulate and 3 Dimotion of reproduction and communication are expressed by elliptic forms.

The 2^{nd} Ðimotion is expressed by lineal forms and planes, whose relationships are given by their parallelism, as herds, perpendicularities (as intersecting planes) its skewness as cat alleys or convergence (5Ð social evolution) and divergence (hyperboloids between two systems) or pure entropic 4D dissolution of a system – parabolas.

**Further | x O = Ø generations.**

As the Universe becomes more complex by iteration and combination, the field of complex curves, which mix geometric, scalar and vital, moving properties of the 3 ∆St elements becomes quasi-infinite (µ). And vice versa, we can *further reduce all curves to | x O = Ø generations of the 2 formal elements of the Universe. *

We saw how | x O dualities generate conics & spirals. And the same can be said of all other 9 canonical curves.

Consider the case of rectilinear generators of a hyperboloid of one sheet. It is not at all obvious the fact that the hyperboloid of one sheet and the hyperbolic paraboloid can be obtained, just like the cone and the cylinder, by the motion of a straight line.

In case of the hyperboloid, it is sufficient to prove this fact for a hyperboloid of revolution of one sheet x^{2}/a^{2} + y^{2}/b^{2} – z^{2}/c^{2} = 1, since the general hyperboloid of one sheet is obtained by a uniform expansion from the Oxz-plane and under such an expansion any straight line will go into a straight line.

Let us intersect the hyperboloid of revolution with the plane y = a parallel to the Oxz-plane. Substituting y = a we obtain: But this equation together with y = a gives in the plane y = a pair of intersecting lines: x/a – z/c = 0 and x/a + z/c = 0.

Thus there is a pair of intersecting lines lying on the hyperboloid. If now we revolve the hyperboloid about the Oz-axis, then each of these lines obviously traces out the entire hyperboloid (graph). It is easy to show that:

- 2 arbitrary straight lines of 1 and the same family of lines don’t lie in the same plane (they are skew)

2.Any line of 1 of these families intersects all the lines of the other family (except its opposite, which is parallel)

- Three lines of one and the same family are not parallel to any one and the same plane.

*As in complex ¬E geometry 3 lines define a topological organic plane, the hyperboloid represents an entire familiy of organic species, which we shall consider in physical and biological and chemical analysis. *

**RECAP. The importance of bidimensional curves: Holographic physics.**

Once we understand bidimensionality we can enlighten physics’ mathematical statements, which deal with laws of forces and motion, drawn before analysis with the canonical Bidimensional:

– ‘Open’ curves – parabolas for entropic motions as in cannonball shots.

– Closed curves: cycles, spheres, ellipses: used in informative motions – as in gravitational and charge vortices/clocks.

– And in-between, St-hyperbolas, used in st-ratios: st balances, st-systems, st-constants of nature and 5D=st metric equations; as in Energy laws or the Boyle law: P(t) x V(s) = K(st)

The most extensive field though of analytic geometry becomes its use in mathematical physics to describe the different Dimotions of reality. We shall thus study curves in analytic geometry as expression of those dimotions.

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**TRANSFORMATIONS OF REALITY INTO MIND SPACE: PROJECTIVE &AFFINE GEOMETRY**

Of the many singularity mind-spaces with wide implications, the richest one is projective geometry, as it shows *the properties of the outer world that remain invariant in the mind-construction of useful information about reality, making possible its survival in the outer world,*

A fundamental development of geometry parallel with the creation of Lobachevski geometry came about in yet another way. Within the wealth of all the geometric properties of space, separate groups of properties, distinguished by a peculiar interrelatedness and stability, were singled out and subjected to an independent study. These investigations, with their separate methods, gave rise to new chapters of geometry. The explosion of parallel geometries is thus a welcomed ad on necessary to expand our analysis of the motions across different scales. And the way planes of space-time create the holographs of the Universe, by motions and translations, projections and imprinting of information into energy.

Projective geometry is in that sense, a basic tool to understand how a bidimensional, high plane of information, projects its form over a plane of space, creating a space-time system. As usual we will then find a relationship between the 3 elements of reality, the o-point, the ð cycle and the Spatial plane, which *is the origin of all realities in all its creative combinations:*

In the graph, the projection of a bidimensional tall ð cycle of time on a spatial surface of energy conserves certain properties but transforms the main property of time – to be closed geometry, into the main property of space, to be an opened geometry. Indeed, for the highest points of the informative pure cycle of height to be projected on the $ plane, 3 elements are to be put in relationship, the o-point, the cycle and the open plane, such as the bigger the open plane, the more chances it will have to imprint the cycle, and the higher the point of view, the easier it will be to project the cycle with a closer similarity. It must be also mentioned the great importance that has the Riemann sphere and its projection in the complex plane, to be analyzed on line 4.

The second element of projective geometry is the understanding on what properties are or not conserved, and easily projected, in as much as it means what are the ð≈$ symbiosis that ties up both elements into ði≈e$ of space-time (an old formalism which I no longer use, as I am converting all variations into ð and $for easier understanding).

It is then self-evident that ‘measure’, that ‘sacred cow’ of physicists is NOT conserved. It is precisely ‘size’ the true flexibility of the Universe and its 5^{th} dimension, which is not needed in a Universe of absolute relativity of scale. The lengths of segments are changed in the process and so are the angles, the outlines of objects are visibly distorted.

What then remains? Immediately we see, of the essential qualities that the property of a number of points lying on one straight line is preserved; and those are as anticipated in our i-logic axiom of a line, 3 of them, for such lines to be fully straight. So the projection on the $t-plane DO conserve the $t-relationships of the ð cycle, enhancing them as the lines DO grow in size, when we move from a ð implosive form into an $ explosive form.

This general rule is of importance in all relative systems, so we shall extract a general law of it:

‘Transformations of time into space conserve and enhance the spatial properties of the S or T element’. This is important because the Universe is all about conservation of Angular and lineal momentum, potential and kinetic energy, past and future combinations of space-time; so translations must conserve the properties which are more natural to the new medium in which the system moves. We live in a Universe that wants, tries and achieves immortality of energy and information through laws such as this one.

We can also observe that the central point of view do conserve those lines and relationships, as all the lines that crossed it keep crossing it. So the ‘soul’ of the system is conserved.

The membrane though is the most distorted element, because it now unless the spatial plane in which is projected is ‘big enough’ to transform its ‘fast, compressed’ cycles of existence, it will NOT fit on it. So as a general rule we notice that the most important element conserved is the 0-point of view, or will/soul of the system, and this will allow to formulate an even larger general theorem of reality:

‘All points of view can switch between space and time states without loosing its identity. So all systems can coil to sleep in its informative state, and elongate to move in its spatial state. There are of course many other space-time dualities that prove this theorem. And in this the reader should understand that even ABOVE mathematics there is Space-time Theory, but we do honor the value of mathematics by referring the causality in an inverse fashion (extracting ∆ST theorems from mathematical ones – it is in fact the other way around, projective geometry conserves the 0-point of view, because this is the ‘last’ entity to be destroyed in any system, as the system dies once it is collapsed. So we can state here that ∆ST systems do NOT die when they change from spatial to temporal states).

Another conserved property is that of a straight line being a tangent to a given curve. And of course, the reader does not need to be a lynx to realize this is the definition of a derivative and one of the many ways to understand that we can always derivate in time and space a system, or integrate it, as this is what all is about, conservation of full worldcycles as zero sums of an infinite number of infinitesimal steps, each one a straight derivative on a curved worldcycle. So the worldcycle of the ð circle is now projected into a $ medium, but it is still happening, and it will be completed if there is enough ‘vital space’ for it to be imprinted (or else it will be cut off; but as a rule in nature a seed of information ‘prospers’ in a relative energetic space, or else the ‘animal, or physical system’ chooses NOT to reproduce. So we could say that o-points gauge first with ‘projective geometry’, in a logic manner its ‘resources of space-energy’ before ‘projecting its information and reproducing it in a larger being of space.

Projective geometry does must be considered in the larger view of T.Œs a modality of Spatial Reproduction.

The other action related to projective geometry is obviously perception, in the inverse arrow to its spatial reproduction, when *the plane of energetic space is projected back into the o-point of perception that gauges information. And this gives birth to an interesting field when we compare the two ‘different directions of time’, ð->$ (reproduction) and $->ð (informative perception).*

*The study of properties of perspective goes back in antiquity right to Euclid, to the work of the ancient architects; artists concerned themselves with perspective: Dürer, Leonardo da Vinci, and the engineer and mathematician Desargues (17th century). Finally, at the beginning of the 19th century Poncelet was the first to separate out and study systematically the geometrical properties that are preserved under arbitrary projective transformations of the plane (or of space) and so to create an independent science, namely projective geometry.*

*It might seem that there are only a few, very primitive properties that are preserved under arbitrary projective transformations, but this is by no means so. *

*For example, we do not notice immediately that the theorem stating that the points of intersection of opposite sides (produced) of a hexagon inscribed in a circle lie on a straight line also holds for an ellipse, parabola, and hyperbola. The theorem only speaks of projective properties, and these curves can be obtained from the circle by projection.*

*The importance of this in reality is obvious, as the hexagon, we have already mentioned is the perfect pi-cycle of 3 diameters of perimeter.*

*And so not only the projection of the cycle but the hexagon its ‘natural quadrature’ is conserved. *

*It is even less obvious that the theorem to the effect that the diagonals of a circumscribed hexagon meet in a point is a peculiar analogue of the theorem just mentioned; the deep connection between them is revealed only in projective geometry. But its deep foundations are in ∆ST: again the o-point is conserved in the hexagon, which reveals to have many similar properties of the cycle as it is its most stable form for ‘small networks’.*

*Now another key field of projective geometry is the study of angular projections, and its related trigonometric laws, which can be considered part of the ð perception, and the capacity of a point to accurately measure distances on the space it perceives. *

*This is the most magic part of projective geometry, which reveals the enormous intelligence of space-time to allow o-points of view to gauge information.*

*For example, under a projection, irrespective of the distortion of distances, for any four points A, B, C, D lying on a straight line the cross ratio AC/CB: AD/DB remains unaltered: *

*AC/CB: AD/DB = A’C’/C’B’: A’D’/D’B’*

*Thus a system can actually perceive measures by having a ‘sensorial’ set of (ABCD)’ points in its membrane to calculate such proportions. These kind of properties are of course extended to all the laws of trigonometry and angles and distances calculated with those laws.*

Projective geometry, thus is essential to understand the relationship between a o-point or ST-system and its outer larger world, and how the point shrinks topologically an external world into an internal image, along topology, trigonometry and… affine geometry, which form the scaffolding of the mathematical laws that allow ‘gauging information’, even for the simpler systems of nature, regardless of human anthropomorphism.

**Growth of points into waves and planes, in algebraic forms= geometric figures.**

It is customary to analyze the equations of curves with analytic geometry, though to that aim we would need to introduce some in-depth concepts of 5D operands in algebra. We could then start the analysis of geometry and algebra together by considering the ‘basic’ chain of social operand, the sum or superposition* in a single plane*, the product the operand of growth of dimensionality *iteration and merging*. And so a product converts a point into a line, wave or cycle and a new product into higher dimensional forms. As we work with 2 dimensional figures, the square is the basic product of the conic curves… But all this will obscure the purity of geometry, as Cartesian coordinates do with generalized ones. *We must always remember the differences between ‘objective reality’ which is independent of the observer, and humind knowledge that somehow always gets the distorting human observer complicating things. When we introduce a Cartesian frame of reference while humind’s manipulation becomes easier the essence of those conics* which is about the relationship between the inner focus and the points gets blurred. In that regard we are more interested in generalized coordinates for objective analysis and projective geometry for subjective analysis of the laws of space, as the most generalized forms; and just will mention the algebraic equations of analytic geometry as a reference to what matter us – to extract vital properties of those curves.

**The observer, observable and creator trilogic paradoxes.**

Projective Geometry introduces a theme fundamental to the paradoxical nature of the Universe – the ternary symmetry between the ‘world’ that creates as a whole its smaller parts, and the smaller perceiver within the world. The classic example being Desargues theorem that relates an observer (center of perception – a nicer word than perspectivity’), an observable, two triangles, and a creator, the axis of perspectivity.

But the ego paradox makes the perceiver to *think he creates what he perceives. *So for long *the main theory of light colors was that the eye generated the light that reflected on objects, bouncing back to create an image, no the other way around. *external, objective world though is larger and generates the observable which the internal subjective world perceives and through its perception can entangle in simultaneity with the Creator.

Consider for example Desargues main theorem of projective geometry. In a subjective view the two triangles would be generated by the perceiver as the ‘apex’ of a pyramid, which do not require further elements in its construction but the ‘eye rays’. But as it happens the triangles, not the perception of them are generated by the axis of perspectivity, a ‘larger worldline’ than the point of perception, which has *3 points x 2 lines, to construct the 3 corresponding* lines x 2 triangles.

How this ‘abstraction’ becomes a reality should be obvious to the reader. According to the 2^{nd} ¬E postulate a line has volume, more than 1 Dimension. It is therefore a wave of fractal points or a network In this case we talk of the axis of perspectivity as a whole, its points, the departing elements of 3 ‘physiological or fractal networks’, which merge to create 2 organs, the triangles, whose self-similarity is established by the center of perspectivity.

The humind center is only a point, not enough to generate a larger world. But as always the paradox of the ego turns upside down the causality as in the Copenhagen interpretation or Hilbert’s axiomatic method – I imagine points and lines. Hilbert‘s ego though doesn’t need to know that a Cartesian demon, the ‘axis of evil’ has placed those 2 triangles for its mind to see. And yet the ‘generator’ are the 3 perspective points of the axis line, the larger world with a higher dimension than the point. This realization that we do NOT generate the larger world, which is out there for us to mirror comes only with a constant evolution from subjective childhood to an objective mature classic age (reverted back to subjectivity in the third old age that degenerates to childhood, as today huminds do in its collective age of self-extinction, becoming emotional and subjective, self-centered children again).

**RECAP**. From a pure spatial perspective of geometry as a simultaneous time independent view of the structural relationships between the parts of reality – the superorganism, its scales and elements – generalized and projective geometry are of a higher value. But for the complex entanglement of perspectives *specially when dealing with geometry as an expression of time dimotions, what should be called ‘algebraic geometry’ but it is called analytic geometry works better. *

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**∆-SCALAR TOPOLOGIC SPACES: LINEAL AFFINE GEOMETRY.**

Affine geometry studies the properties of figures that are not changed by arbitrary transformations in which the Cartesian coordinates of the original (x, y, z) and the new (x′, y′, z′) position of each point are connected by linear equations. Where it is assumed that the determinant is different from zero.

It turns out that every affine transformation reduces to a motion, possibly a reflection, in a plane and then to a contraction or extension of space in three mutually perpendicular directions.

Quite a number of properties of figures are preserved under each of these transformations. In fact affine geometry is remarkably extensive, showing ultimately that *growth in size through lineal increase of and $ x ð system is absolutely natural to the Universe, its essence, which easily conserves all the properties of the fractal ð seed in its expansion in space:*

Straight lines remain straight lines (in fact all “projective” properties are preserved); moreover, parallel lines remain parallel; the ratio of volumes is preserved, also the ratio of areas of figures that lie in parallel planes or in one and the same plane, the ratio of lengths of segments that lie on one straight line or on parallel lines, etc.

Many well-known theorems belong essentially to affine geometry. Examples are the statements that the medians of a triangle are concurrent, that the diagonals of a parallelogram bisect each other “ that the midpoints of parallel chords of an ellipse lie on a straight line, etc.

The whole theory of curves (and surfaces) of the second order is closely connected with affine geometry.

The very division of these curves into ellipses, parabolas, hyperbolas is, in fact, based on affine properties of the figures: Under affine transformations an ellipse is transformed precisely into an ellipse and never into a parabola or a hyperbola; similarly a parabola can be transformed into any other parabola, but not into an ellipse, etc.

So unlike in the case of projective geometries of ð systems into $ systems, which transform circles into their equivalent open spatial forms (parabolas), affine transformations, which are growths DO conserve the essential ð<$t structure of the system. Moreover it is a deterministic transformation, with NO errors. Parabolas do NEVER become ellipses and so on.

The importance of the separation and detailed investigation of general affine properties of figures is emphasized by the fact that incomparably more complicated transformations turn out to be essentially linear, i.e., affine in the infinitely small, and the application of the methods of the differential calculus is linked exactly with the consideration of infinitely small regions of space.

If we correct this infinitesimal concept to a finitesimal, which still preserves this linearity, we could simply state that growth of a system goes through a ‘lineal region, in the stable ∆ST=k conserved metric region of the 5^{th} dimension.

In other words, the lineal affine growth of a system and affine geometry on the whole is justified by the ∆(Sp x ð) = a K process of growth of the system, within the 10^{x} growth region , which is the region in which the metric of the system is lineal before its Lorentzian regions of emergence or dissolution in the ∆±1 scales.

Now modern mathematics obviously does NOT consider this 5D realist interpretation of the reasons of existence of those fundamental variations of geometry. Instead they re formalized within the abstract meaningless idealist programs of the axiomatic methods of the German school of science of a century ago (as nobody has ever since ‘think’ seriously in philosophy of science, once culture moved to America, and its visual or technical praxis with so little pure theoretical and intellectual understanding).

Thus all this is classified into the Klein’s Erlanger Program of1872, which sums up the results of the developments of projective, affine, and other “geometries” giving an obscure formulation of the general principle of their formation with the use of that pest of modern mathematics called group theory (-:

We can consider an arbitrary group of single-valued transformations of space and investigate the properties of figures that are preserved under the transformations of this group.”

In accordance with this principle of Klein, we can construct many geometries. For example, we can consider the transformations that preserve the angle between arbitrary lines (conformal transformations of space), and when studying properties of figures preserved under such transformations we talk of the corresponding conformal geometry. But the result of this program, as any of the multiple variations of the German idealist axiomatic method is a hyperinflation of ‘imagined mathematics’, which confuse the fundamental property and need for mathematics as a realist science.

Information is inflationary there is more money than real economy, more imagined words that real facts to describe, more fiction that reality in any language. And this is the fact of the 3^{rd} age of information of any system – not a value but a loss of classic realism, the perfect age in which language and reality are in mirror correspondence to each other.

We shall not consider that metalinguistic approach, which completely ignores the outer world reflected by mathematics, in the obvious opposed realist philosophy of i-logic mathematics. As we hold truth Gödel’s proof of the incompleteness of any categorical definition and proof of existence based only in an internal metalanguage or internal logic.

*Those brief examples of CLASSIC still analytic Geometry and its initial applications to mathematical physics suffice for the purpose of it – to show the bidimensional structure of the universe in its st manifolds according to the holographic principle**.*

**RECAP. **5D transformations of scales generate from reality mind spaces, through a dual process of contraction of form and stillness of motion. And vice versa, the processes of expansion of a seed of mental space or form into the larger world by reproduction and motion given to the seed of form.

Those essential processes of creation are mimicked in the projective geometry of visual space and the affine geometry that simplifies curves into lines.

Affine geometry studies the inverse, dual actions of reduction by ‘perception’ of a larger space surface by a singularity whereas an Space is ‘reduced’ into the ð-point and inversely the 3Dimotion of reproduction and scaling where an informative seed of form is imprinted on an energy plane. Accordingly affine geometry is related to the ‘growth in size’ of a system, through a lineal process of expansion in space.

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**XII. 2nd AGE: T: MOTION – DIFFERENTIAL GEOMETRY EXPANDS MATHEMATICAL PHYSICS.**

This final fundamental realization that connects geometry with the metaphysics of order vs. freedom, form vs. motion, lower vs. higher scales, (Galilean paradoxes of Duality), is thus a good introduction to the third age of Geometry (even if it came before the final evolution of @nalytic geometry into non-Euclidean forms of mind-space).

Its 3 clear±¡ sub-ages will be:

**Differential Geometry of curves taken as points with motion.**

Vector spaces, which expand at the path of mathematical physics, as the best phase space to represent Space-time fields of parameters that have both form and motion.

Topology, of the 3 BiDimensions of space with motion that represent the 3 functional organs of any supœrganism,

∆@st: Its 3rd eclectic modern age of combination expansion and explosion to all other branches of the ‘entangled mathematical mirror of the entangled Universe’. We shall for sake of simplicity only comment on the 3 first classic ages. And develop instead of the eclectic modern age…

A pentalogic age of 5D ‘motion geometry’, with a few insights on the different ages and forms of geometry which are better suited to express the 5 Dimotions of the Universe.

**The ‘surface’ of a sphere, approached by ‘smaller planes’:**

In the graph, the fact that any space coincides with a Euclidean in the infinitely small enables us to define for the intrinsic geometry of a surface by approximating an infinitely small portion of the surface by a plane or an infinitely small volume expressed as Euclidean space. The volume of a finite domain is then obtained by summing infinitely small volumes, i.e., by integrating the differential of the volume. The length of a curve is determined by summing infinitely small distances between infinitely near points on it, i.e., by integrating the differential of the length ds along the curve.

And this is a rigorous analytic expression for the fact that the length is determined by laying off a small (infinitely small) measuring rod along it – which is ultimately the differential, smooth version of the fractal step by step measuring of growing distances when we scale down our view – hence another proof of the fractal and mental nature of reality, ultimately proving the ∆±i and @-mental ‘missing dimensions of reality’, in human ‘naive realism’.

The graph then show in ‘2 dimensions’ on the surface of the being another kaleidoscopic VIEW on the application of Euclidean, elliptic and hyperbolic geometries. If we consider ONLY a simplified Euclidean reality, (left side), we need no measure of curvature – it is a flat small plane of space.Next in complexity, a regular spherical curved piece of the whole,(ð§) requires more information. So a measure of curvature, Φ measure is required.

But if the system is not a regular sphere, two curvatures will be needed. Finally in hyperbolic geometry the more complex, ST vital energy with its two CONTRADICTORY directions towards the singularity and the membrain, will need two curvature angles, with opposite directions, represented by the ±sign.

And we shall choose (Euler) to well-define the curvature of the whole surface, just the maximal and minimal angles of curvature, according to the fundamental rule of t.œs, which can be defined by its standing points, its maximal and minimal functions, which are the relevant Max. e x Min. i, max. i x min e, e=i, ternary ‘points of any worldcycle/system’, require to Generate all events and forms of existence.

Such directions are thus called the principal directions and the curvatures k1 and k2 are called the principal curvatures of the surface at the given point: k(ϕ)= k1 cos²ϕ ± k2 sen²ϕ…

Where once more as usual we find the sinusoidal functions that define ST systems with its two opposite directions.

**The inverse arrow: envelopes and curves on the large.**

It has to be noticed that humans with its obsession for the small, as information comes from below and so it is more abundant, while above, larger entities are not so well perceived, has made us also quite ignore the emergence of larger entities. This however is essential for physics and in mathematics the origin of emergence in time (Fourier transforms) treated on the emergence articles on the first line, and emergence in space, the so called envelope curves, yet another branch of static formal space, better treated in physics where space usually has motion, reducing on one side the informative inflation of ‘fiction theories of the mind – spaces with no vital use’ and giving the equations a more beautiful s=t symmetry between the form and motion dimensions (s=t being the ‘definition of beauty’, a theme treated in the study of the exist¡ential program).

We shall just then mention it for the sake of completeness – that is to show that for each ∆-1 entropic theory there is an inverse ∆+1 social one:

The question of envelopes in that sense is a relatively simple one – as all questions of ∆+1 wholes of lesser information, solved long ago, in the theory of families of curves and surfaces. Especially well developed is the theory in the canonical ST, holographic 2-manifolds; that is two-parameter families of various curves, in particular of straight lines ALWAYS easier to ‘perceive’ by human essentially a ‘small thing’ belonging to a ‘flat curvature’ space-mind: the so-called “straight-line” congruencies. In this theory one applies essentially the same methods as in the theory of surfaces, hence within the scope of ∆st Disomorphisms.

In terms of ∆st the theory is the direct application of a fundamental law of ST emergence, often quoted in different articles: ∑|i-1>Oi:

The inversion of functions and forms as we grow in scales in the Universe, which is a basic symmetry that allows the Universe to balance its relative (in)finite(simal) volume and form, or else, all balanced would break provoking a constant ‘shrinking’ or ‘enlarging’ along a single entropic or social evolutionary arrow.

ST- Balance is the law and symmetries are just a view of that law.

*This implies that *A surface is called the envelope of a given family of surfaces if at each of its points it is tangent to one of the surfaces of the family and is in this way tangent to every one of them.

So we see the ultimate merging of ‘Darwinian perpendicularity’ + ‘symbiotic adjacency’ , which IS at the core of the ‘submissive’, yet symbiotic ‘herding’ of envelopes, ð§ dimensions, where the cyclical time envelop become a larger ∆+1 §partial scaling (hence the marriage of those two symbols, of cyclical time and scalar space – a NEW worldcycle brings always a higher ∆+1 plane).

Again this is an absolute law, which the simplifying, perfect forms of geometry makes easier to understand.

For example, the envelope of a family of spheres of equal radius with centers on a given straight line will be a cylinder (figure 48), hence ∑Oi>|i+1. And the envelope of such spheres with centers on all points of a given plane will consist of two parallel planes. The envelope of a family of curves is defined similarly; and here as we are in an ST-mixed element, we need to study the dominant tendency of those curves, which will show the envelop to tend towards a more lineal or cyclical whole.

For example, Figure 49 diagrams jets of water, issuing from a fountain at various angles – they are clearly by effect of the potential gravitational energy coming back to a closing zero-sum cycle. Hence such family of curves, which may be considered approximately parabolas; tend to have their envelope a more lineal parabola – the general contour of the cascade of water.

But not every family of geometrical forms has an envelope. And if you man or robot of the III millennia which might read those texts start to interiorize the laws of T.œs should by now guess, which *kind of entities do NOT want to be ‘enclosed’ – those thoroughly dominant in 1D-lineal motion and/or 4D entropy. F*or example, a family of parallel straight lines does not have one.

**General ‘laws’ of emergence: o->O->•->@**

All this lead us to understand that ultimately as all departs from ∆•s≈t laws, geometry requires always a first ∆±i distinction between what ‘pros’ call, the geometry “in the small (parts)”, which is clearly dominant and “in the large (wholes)”. The main of those dual theories should then follow the obvious ∆st law that *wholes are more resistant, efficient and stable than parts; hence small/parts are easier to deform, while wholes are far more stable full T.œs – the ultimate reason why wholes and new scales keep happening.*

For example, in 1838 Minding showed that a sufficiently small segment of the surface of a sphere can be deformed, and this is a theorem “in the small.” At the same time, he expressed the conjecture that the entire sphere cannot be deformed. This theorem was proved by other mathematicians as late as 1899. Incidentally, it is easy to confirm by experiment that a sphere of flexible but inextensible material cannot be deformed. For example, a ping-pong ball holds its shape perfectly well although the material it is made from is quite flexible – laws those akin to the laws of ‘surface tension’ of soap bubbles with wide application in physics.

Another example, is the tin pail; it is rigid in the large, thanks to the presence of a curved flange, but separate pieces of it can easily be bent out of shape. As we see, there is an essential inversion between properties of surfaces “in the small”, ∆-1 and “in the large”, ∆+1.

A 1D t vs. 3D ð wider generalization is provided comparing open geodesics vs. closed curves. A geodesic “in the small,” is a small segment of the surface, its shortest lineal path, but “in the large” linearity may not be the shortest path at all – it may even be a closed curve, the great circles of a sphere.

And here is where another *LAW OF EMERGENCE APPEARS of enormous generality, as it is the basic process of social evolution of a system, from life cells to astronomy: creation builds first step by step its ‘protein envelop’ and then as it grows it finally needs a singularity to focus and constrain the parts through its radius, creating an antipodal elliptic geometry, which finally creates the @-system and completes the T.œ*

Indeed many analytic surfaces cannot be extended in any natural way without acquiring “singularities” in the form of edges or cusps and thus becoming non regular.

Thus, a segment of the surface of a cone cannot be extended in a natural way without leading to the vertex, a cusp where the smoothness of the surface is destroyed. *This striking, obvious result, 30 years ago lead me to do my fav painting of conceptual cubism and adopt the pyramidal ∆-form for whole povs, and singularity minds:*

In the graph we see right, my ∧ painting, which a decade latter resembled eerily the first Bose-condensate (maximal form of a physical system – its 5D), and ultimately proves there MUST BE A GOD/logic mind for any whole organism, limiting the number of planes a system can grow, departing from a ‘finitesimal amount’ of ∆-2 parts.

Thus geometry of the large is only a particular case of the previous remarkable theorem:

Every developable surface other than a cylinder (the lineal, non-enveloped essential 1D form) will lead, if naturally extended, to an edge (or a cusp in the case of a cone) beyond which it cannot be continued without losing its regularity.

Thus there is a profound connection between the behavior of a surface “in the large” and its singularities. This is the reason why the solution of problems “in the large” and the study of surfaces with “singularities” (edges, cusps, discontinuous curvature and the like) must be worked out together. Now we know its whys in a theme that fascinates both mathematicians and physicists.

Now, we have the 3 concepts needed to fully describe most of modern non-e geometries, including Riemannian manifolds, in yet another ‘mirror image’ of the ternary laws of ST:

ð§: the ‘intrinsic geometry-curvature’ *of the surface:*

∆+i: the ∆-scaling given by the relative ratios of r/k smallness or greatness, which defines the relative size of the observer vs. the observable form.

The relative number of dimensions we shall study and how they are connected when we go beyond the usual ternary games of existence; the last of the key themes of non-E spatial mind worlds.

**Differential geometry as the study of membrains.**

The importance of Gaussian differential geometry comes to full fruition in vital topology, as the study of surfaces equates to the study of membrains, the most important elements of all systems, which deploy a specific topological variety of closed form, but acquire in small scales a hyperbolic geometry, making the duality of open and closed space at ∆1 level correspond to the duality of lineal and hyperbolic surface in small scales. In other words, what the sphere looses in the larger scale as the lesser surface of all beings, in fact it wins it in the smaller scale as the larger hyperbolic surface.

The membrain thus have a convoluted maximal surface of osmosis and exhchange of energy and information with the outer world in the smaller scale in which it is the predator but appears as the smaller surfgace of existence in the larger scale in which it is a mere fractal point of a larger world.

A theme which introduces us in the ‘last’ of all the branches of geometry discovered by Huminds, which is quite surprising indeed, as it should have been the first.

But we have dealt with the shortcomings of the humind in many other papers, out of our frustration…

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**3 ^{rd} AGE OF GEOMETRY**

So we shall start the 3^{rd} age with the last discoveries, which are the 2 fundamental forms of geometry for its realism in its analysis of the ‘two eternal elements’ of reality:

**∆-scales: Fractal geometry** is the final understanding of the scalar Universe, and yet it was not found due to the extreme naïve realism of the humind, who doesn’t understand its mental space eliminates the discontinuities of the Universe, so the ‘continuous hypothesis’ is both false and misleading, as they ‘upgrade’ smaller parts to fill the holes of largers ones (case of the real line, which rises the finitesimal numbers and transcendental ratios to the N-discontinuous line).

**ST-numbers: Which are vectors **with an scalar S-value and a T-motion/direction. Yet Vector analysis, was only developed in the II part of the XIX c.

**DISCONTINUOUS ANALYSIS: FRACTALS**

We shall move now into what truly is though it is usually not considered in those terms, the final age of ∆nalysis as the study of the relationship of parts and wholes, which in close analysis turn out to be fractal discontinuous parts.

Fractals are in that sense the equivalent to the final realization in physics that continuity is a mirage that simplifies reality and so do the analysis mathematical mirror, but when we really want to know the whole details of the ∆-scale, the universe is quantic and so it is its fractal geometry. Here the work of Nottale in mathematical physics and Mandelbrot in mathematics, stands fully as the best formalism for both. So we won’t stomp on our peers (yes, i do recognize them as peers, in this case), but as usual bring the point of view of the philosopher of ‘stience’.

The infinitesimal study as perceived from the finite point of view is the view of fractals, when in detail and observing the closed worldcycles that separate and make each infinitesimal a whole.

A derivative is the finitesimal of the function observed, and so when we go even further and study as enlarged into our scalar view tin maximal information we are in the fractal view of reality.

So as we expand our view the fractal view becomes more real, till finally the enclosures observed ∆-1 become fractal and we recognize its self-similarities: ∆-1 ≤ ∆º.

For each derivative thus a function shows its 1/n infinitesimal (not necessarily this function, which is the derivative of the logarithm).

It follows that functions, which grow ginormously, have a ‘quanta of time’ reproduced and so its minimal derivative finitesimal is the function itself, eª.

** Fractal structure of the 5th dimension and its perpendicular flows.**

Fractals are the best way to describe all themes related to scales. And as such they are connected intimately with the original elements of calculus, namely series and finitesimals. Fractals then branched out as a sub-discipline of scalar mathematics, which we can consider two have those 2 sides, the discrete fractal view, and the continuous topological view, in numbers=points t=s dualities.

So we shall start with power series to describe fractals, which started as calculus did in the earlier work of Archimedes, resurrected in geometrical terms by Koch and the XIX aberrant geometers:

**Archimedes’ quadrature of the parabola**

Archimedes’ dissection of a parabolic segment into infinitely many triangles used the sum of a geometric series to compute the area enclosed by a parabola and a straight line.

His method was to dissect the area into an infinite number of triangles, establishing the concept of a fractal system defined in this book as the ‘scaling’ between the whole and its infinitesimals, happening in nature (from super organisms to organisms, organs, tissues, till coming to cells, all self-similar; from galaxies to atoms in physics through intermediate cosmic bodies, from civilizations with its 3 ages of subconscious collective art to individual minds in memetic superorganisms of mankind, etc.).

Archimedes’ Theorem states that the total area under the parabola is 4/3 of the area of the blue triangle.

As each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 the area of a green triangle, and so forth.

Assuming that the blue triangle has area 1, the total area is an infinite sum:

The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives:

This is a geometric series with common ratio 1/4 and the fractional part is equal to:

This computation uses the method of exhaustion, an early version of integration. Using calculus, the same area could be found by a definite integral, which is just the ‘topological continuous’ version of the discrete fractal version. So what fractal geometry does is NOT to move from discrete numbers of sequential time and scalar steps into continuous sums in a plane, keeping MORE information on the detail and showing clearly the discrete finitesimal limits of nature – reason why in Nature we see more fractal systems that ‘differentials’, which must be seen as a more ‘time-mirror’ oriented version of the same concept.

It is interesting to notice that the parabola is a ‘triangle’ increased by 1/3rd, the essential ‘element’ of the ternary reality. So we can said the triangle has 3 elements and the parable, which is its curvature, ads one more ‘dimension’ or motion to it. A triangle then in external movement will ‘define’ an external wave around it with a parabolic form (the added third volume). A static parabola will have an inner region to add that third, which can be considered the ‘envelope’ membrain.

### Fractal Geometry

The next step was done by Koch’s snowflake is a union of infinitely many triangles.

In the study of fractals, geometric series often arise as the perimeter, area, or volume of a self-similar figure. For example, the area inside the Koch snowflake can be described as the union of infinitely many equilateral triangles (see figure). Each side of the green triangle is exactly 1/3 the size of a side of the large blue triangle, and therefore has exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking the blue triangle as a unit of area, the total area of the snowflake is:

The first term of this series represents the area of the blue triangle, the second term the total area of the three green triangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this series is geometric with constant ratio *r* = 4/9. The first term of the geometric series is *a* = 3(1/9) = 1/3, so the sum is

Thus the Koch snowflake has 8/5 of the area of the base triangle.

Here again we find another ‘essential ratio’ of growth in Natural fractals, themes those treated in number theory; but more telling is the form of the new shape, essentially an Hexagon, which fails to become a pi-circle, its next stage, as we can consider the hexagon a circle with pi=3 diameter. And so we can talk of a ternary growth of the triangle into the circle, its |-O state, parallel to the ‘rotation-motion dimension’ which also converts it into a circle, in this case by static growth through scalar geometry.

Thus one essential process of Nature’s topologies, the ‘quadrature of the circle’; that is, the ‘triangular transformation of the circle’… can be achieved in pentalogic by imperfect methods:

- ð-methods: rotating the triangle (as part of the rotated surface is no longer ‘solid’ space, by its motion-derivative…
- ∆-scalar methods: growing a Koch snowflake, as part of the circle is void.
- S-methods: parabolizing its surface. As only one side, that of motion defines a front wave for the triangle.

The most efficient being the scalar method, as 8/5<4/3, the surfaces of the snowflake and the parabola of a triangle.

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**XIII. VECTOR SPACES**

**4th Postulate of ¬Æ logic: parallelism and perpendicularity**

Vector spaces are defined by 4 positive elements (where entropy is ignored), which makes them immediate mirrors of the 5D Universe: a point with a magnitude in space, a direction of motion in time, which can be operated by a ‘scalar’ numbers that determines its scale, and measured from a ‘frame of reference’ according to an angle of perception (whereas the negation of a vector or its entropic state is the inverse direction of the vector fully resolving the meaning of a negative numbers an inverse direction in time).

It is for that reason they are so useful to represent reality and it marvels it took so long in the natural evolution of the mind from fixed space -> bidimensional planes -> points with motion (calculus) -> to arrive to vector spaces that represent the 4 elements of ∆@st in a natural form. The laws of vector spaces thus are in direct relationship with the GST laws. They started when the ‘bidimensional planes of Descartes and Newton acquired a 3^{rd} axis (Z height) and the rules of its sum were laid own according to its orthogonality (perpendicularity), to represent (Lagrange) locomotion, speed and acceleration.

**The 2 type of vector products**

Vectors are of 2 quite different types, as their properties are not the same.

*The dot product*

The dot product affects vectors in the same manner than the sum, as it has the same properties of identical beings (associative, distributive and commutative products). Thus they must affect ‘equal species’ of the herd type, which has huge implications for the physical nature of fields of forces, which are as always for an ∆º entity, indistinguishable, ∆-2 particles (as aminoacids are all the same for your body. And generally speaking entropic points (∆-2) are all the same for an ∆º entity)

We return once and again to a key postulate of Non-E, the fourth postulate of congruence, to explain the vital topological ‘angles’ that define the evolution or devolution of dimensional motions of time space.

However in the complex Universe *all events have dual and ternary interpretations, when we perceive them from the inverse entropic and informative dualities, merged into a combined form.*

So things are not so simple as they seem, and this is the case of vectors spaces and its Duality of parallel dot product and perpendicular cross product. Initially one would assume that perpendicular cross product annihilate themselves, but in fact are creative processes that create a ‘3^{rd} dimensional element’, perpendicular to both with *a magnitude equal to the product of both So the essential equation of a cross product is a reproductive merge of two ‘inverse S and T’ elements:*

*Cross product: S x T = ST*

The cross product however affects a different kind of species – a key ‘insight’ of 5D theory lost to abstract mathematicians – it is not the ‘algebraic properties’ but the ‘species’ that can be subject to the cross product what makes a difference. As they are *elements which do NOT commutate and treat their angle in inverse fashion to dot product. It is anticommutative: axb = -b x a, and orientable. Moreover it only exists in 3 dimensions (in 7 is not uniquely defined).*

*What this essentially mean is that 1) the cross product has an arrow of ‘angular time’ hence it is orientable; it is a reproductive function, as it gives birth to a 3 ^{rd} species in the relative dimension of width of the other two (reproductive dimension) and so it can be considered a merging of an S-function (the height informative vector), a length function (the locomotion, entropic T-vector), giving birth to an ST-function (the cross product).*

*What this means fortunately enough can be assessed both from the general laws of Gst and the direct experience of which ‘species are subject’ to a cross product, the main of which is obviously the product of the magnetic and electric field that gives us the reproductive speed of the electric field.*

Then other obvious case is the angular momentum Lof a particle about a given origin defined as: L = r x p

where rr is the position vector of the particle relative to the origin and Pp is the linear momentum of the particle; which again connects *a Space element (the origin T.œ) and a T element (the rotating one) giving us an ST product of both, which in mathematical physics we regard as the ‘first T.œ’ of the Universe (an h ‘Planckton’)*

*We cannot get here into ‘complicated alternative inflationary mathematical information’ to represent the same concept; alas, the bivector and its multiplication that transforms the cross product into an ‘SS’ plane… by virtue of the S=T law of 5D.*

In 5D we are dealing with two *elements which are time-like and space-like, hence inverted in properties but similar enough to become a reproductive ‘hyperbolic metric equation’ S x T = K. *

So the cross’ product angle of perpendicularity is creative (and the proper way to draw would be with ‘inverted’ arrows’ so the reproductive result at the point where both merge rising a 3^{rd} offspring vector would be more obvious). Since when the 2 elements are close enough to reach an S=T merging point and there is not ‘tearing; but only merging an act of reproduction takes place.

*On the other hand the parallel vector dot product is a Darwinian event, because *it eliminates the smaller form, as the dot product reduces from 2 elements to one the system. We can then consider the parallelism to be one happening in a flat surface as a predatory actions, in which the larger ‘system’ feeds on the entropic energy/motion of the smaller one.

*In praxis both products do happen in physics and indeed, the cross product is a more complex process, as in magnetic and electric S-T fields that merge into an ST=speed wave. * While a simple example of the dot product is a body moving in a field of forces which extract energy of motion from it.

So the duality of dot and cross product and the values of its angles is an essential mirror of 5D spacetime dimotions.

Vector spaces and the complex plane are the 2 fundamental expansions of frames of references into complex holographic dimotions. That is,* frames of reference with form, scale and motion, *and as such, they are the essential system of representation of the Universe, beyond the abstraction of mental space as it is. In the scalar Universe however a vector space itself a motion – a ‘field’; and as such when a particle is placed in a vector space, it will enter in ‘communication with the field’ in two different ways:

If the field is made of its ∆-¡ particles, it needs to absorb to produce one of its dimotions the field will be a ‘force fields’ in which by the 4th postulate of ‘¡logic behavior’, the ‘Active magnitude’ will trace ‘equipotential’ paths ALWAYS PERPENDICULAR to the field in which it feeds (force lines). This makes 5D field theory different in as much as the particle is NOT following the field motion but either its equipotential lines (case of a planet around the gravitational field that sinks into the sun), or what consider the ‘lines of force’ (case of charge field, which could be represented by the equipotentials.

If the space however represents the parallel motions of speed in the ‘same scale’ – NOT a feeding, perpendicular process, by the similarity of the Particle and the field, case of a man swimming in a river of water the motion becomes parallel in both the active magnitude and the field.

This said, the first case will give birth to a cross product geometry, of perpendicularity and the second case to a dot product of parallelism between the field and the self-similar particle.

Finally a third element of importance in ‘moving coordinates’, is the difficulty to operate with ‘fixed mind-frames of reference.

Thus vector spaces depart from a single humind frame of reference with its 3 ‘visual’, perpendicular light space-time ‘basis/co-ordinates’ by adopting generalized co-ordinates – that is, co-ordinates for each point/item *as if it were a fractal broken space in its own, which truly is, since we move then from the subjective continuous human view, to the sum of all the different particle views. *

**Parallelism – Dot product. Field and particle are similar, in the same scale.**

The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ||x||, and to the angle θ between two vectors x and y by means of the formula:

The dot product converts two vectors into a scalar number, *thus reducing the dimotions of the system from two forms to a point. But this is deceptive. As the scalar number in fact is not a ‘reduction’ but an ascension of scale – a parameter or magnitude of the upper plane of existence. *

*Some physical examples are: *

– Mechanical work (∆1) is the dot product of the force and displacement vectors (∆-1),

- Magnetic flux (∆2)is the dot product of themagnetic field and the vector area (∆-2)

Thus the dot product in terms of vital topology is a predatory act of a larger ‘scalar’ form that absorbs energy from a lower plane. And its beauty comes from its ‘4^{th} postulate angle of congruence and ‘similarity’ that defines in geometric terms, what thermodynamic defines in algebraic form. That is, the quantity of energy extracted by the ∆+¡ system from the lower ∆-1, ‘smaller’ vector is directly proportional to the similarity of the two systems. The same law would apply in thermodynamics, regarding the capacity of the larger system to absorb from the lower system its energy as ‘work’, instead of dissipating it as heat. A ‘superorganism’ in which the ∆-1 scale of ‘cells’ are in synchronicity and similarity relationship with the larger whole transfers most of its energy upwards. A rough system in which both scales are dissimilar will dissipate it into heat.

How can we interpret this PRODUCT in terms of vital mathematics and the st components? The most obvious definition is this. The ‘biggest’ predator vector, B in the graph is the dominant element, as A is projected on it. Whatever the ST elements of those vectors mean, which will vary for different uses, *unlike the cross product which is creative, reproductive, the dot product is entropic, destructive, as the result is the ‘absorption’ of the |A|cos =X component of one of the vectors by the other, which becomes expanded in the X-axis variable (whatever this variable is), and for all effects A disappears, leaves no trace of its motion/form≈position; and we obtain a scalar which quantifies the result of this ‘absorption’. So we can classify the 2 fundamental ‘products’ of vectors by the duality of the 4th Non-Æ postulate:*

Dot products are Darwinian, destroying one vector, reduced first to its X-parameter, which as it happens IS systematically, the ‘real’ normally momentum or energy or body element of the system, while the Y-parameter of form, information is discharged in a classic Darwinian action of feeding (the ‘particle’-head element or Y coordinates disappears). Indeed, if we use the XY graph, as in most cases to quantify the 2 complementary parts of the being ∑∏ (body-wave)>ð (head-particle), in physical systems this process is equivalent to the predator event of cutting the head throwing it out and eating the body to multiply your inner cellular energy in the X-direction of your body-motion-momentum.

Cross products are reproductive, creative, as a 3^{rd} ‘offspring-dimension-form’ is created fusion of the other 2:

**3rd and 4th dimension fields: entropy and multiplication =reproduction**

In the graph, product can be of multiple, different ST dimensions, which start the richness of its ‘propositions’. A vectorial product is one of its commonest forms as it combines ST or TS dimensions, BUT as both ‘present’ products are different in orientation, this product unlike other SS or TT products is non-commutative: bxa=- axb. In this case giving birth to two different orientations in space, though for more complex product of multiple ‘S-T’ dimensions, which *can define as a Matrix of parameter a T.Œ particle in full,* the non-commutability can give origin to different particles (quantum physics).

Vectors thus become the essential mode to define an ST holographic element, with a 0Dimension of a scalar number that defines the singularity point and a direction of motion in space (x,y,z parameters from an @nalytic frame of reference, but in generalized objective coordinates a lineal 1D parameter of distance=speed per time frequency, which measures the T-steps or cyclical motion of the • point active magnitude).

The difference between both types of vectorial product is very important to fully grasp reality as it is.

The perpendicular product seems at first contradictory because they seem to diverge in orientation. But this is because we put the arrow *in the wrong side. It should be in the origin where they collide, and that is the dot in which the two vectors become a still spatial parameter. It is then also applicable to the ‘collapse’ of multiple flows into a non-Euclidean fractal point, in which they become a scalar parameter. And that is how in fact a Hilbert space ‘collapses’ in quantum physics an infinite number of generalized parameters allowing us to calculate wholes, and giving a vital sense to the extremely abstract jargon of quantum physicists.*

On the other hand the creative product, which is also used in physics to describe another fundamental scale, that of electromagnetic forces, is symbiotic creative, merging and helping the two components to act symbiotically as one. And again the ‘mental space’ of the cross product is misleading as it seems to contradict the 4th postulate of symbiotic parallelism vs. Darwinian perpendicularity; looking like they touch each other perpendicularly, but in fact the electric charge and the magnetic field ARE always parallel, in the sense they are the singularity and membrane of the electric T.œ never touching each other, as there are no magnetic monopoles; hence they strengthen each other, creating a new force and increasing the speed, and curving, increasing the information of the particle under the magnetic field.

**Geometric comparison.**

The magnitude of the cross product can be interpreted as the positive area of the parallelogram having a and b as sides : a x b sin θ

One can also compute the volume V of a parallelepiped having a, b and c as edges by using a combination of a cross product and a dot product, called scalar triple product (see Figure): a x b • c

Because the magnitude of the cross product goes by the sine of the angle between its arguments, the cross product can be thought of as a measure of perpendicularity in the same way that the dot product is a measure of parallelism.

Given two unit vectors, their cross product has a magnitude of 1 if the two are perpendicular and a magnitude of zero if the two are parallel. The dot product of two unit vectors behaves just oppositely: it is zero when the unit vectors are perpendicular and 1 if the unit vectors are parallel.

Unit vectors enable two convenient identities: the dot product of two unit vectors yields the cosine (which may be positive or negative) of the angle between the two unit vectors.

The magnitude of the cross product of the two unit vectors yields the sine (which will always be positive).

So we establish a parallel superposition principle for the dot product and a perpendicularity one for the dot product.

**RECAP.** Vectorial space is best suited to represent ST social herds (dot product) or reproductive functions of similar complementary species (Dot product)

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**XIV. 3 ^{rd} AGE OF GEOMETRY: NON-E GEOMETRIES AND ∞ METRIC SPACES.**

When we talk of the third age of any system of space-time we refer to the age in which an excess of information splits the system from reality, as an old man which no longer wrinkled and warped into its memorial thoughts try to represent the world as it is. This process *would become endemic in mankind with the Industrial R=evolution of ‘metal-minds’, which starting with the eye-camera, ending in the digital Virreal helmet broke the strict laws of creation of the 5D Universe by reducing it to mental spaces of lesser dimensions, back to the minimal ‘holographic 2D forms’.*

Humans do not realize at all, where this process is leading mankind – to a state of audiovisual madness in permanent conflict with the moral and visual, aesthetic balanced laws of the 5D world that will eliminate us from its full existence – themes those of the papers on social sciences.

In hard sciences this process of fiction thought of the ‘3^{rd} age of history’ of an excess of information had mixed results; as it did in painting with the arrival of cameras.

Geometry as all human endeavors will change forever with the industrial r=evolution of machines. We have already witnessed that it completely changed with photography the path of painting, who will branch in its bid to overcome the metal-eye, into hyperrealism, trying to see better than the initial faulty cameras (Courbet, etc.) on one side and pure mental thought on the other (Van Gogh, Gaugin).

Geometry will also realized that Euclid was not good enough and branched into an attempt to discover the true objective reality of space (Non-E) and its opposite search for pure mental spaces, which was expanding to meet the needs of industrial Dimotions and the ‘new found’ extreme form of locomotion with internal ‘wasted energy’ called entropy.

To put some order into the explosion of fictional mental space would be the job of the 2 next masters of the III Age of geometry, Lobachevski that insisted experimental reality was needed to measure the proper geometry of the Universe and in the inverse direction, Mr. Riemann, who insisted in the opposite path – to walk away from reality into the understanding on how the mind created spaces. He would be another ill-understood, die-young genius, which at least had unlike Lobachevski, the luck of being born in the proper place and have a master in Gauss, who did capture his thoughts and gave them ‘authoritas’.

Riemann’s realization that geometry is a mental-logic endeavor, where function and i-logic thought overcomes ‘spatial representation’, allowed* *the explosion of abstract mental spaces to represent reality, which was carried by Einstein and Bohr into Physics with mixed results. In this task it would be essential the idealist school of German philosophy, from Hegel to March, we often criticize for its escapism with reality. Since while it allow further expansions of mental spaces, latter put in correspondence, §@<≈>∆ð, with the real Universe; it overreach into fictions that still plague serious science and destroyed realism on physics. Only Einstein latter relented going back to the obvious fact that ‘science should only be concerned with explaining facts that have experimentally happened’.

Unfortunately the ego paradox once more won the day and the Copenhagen ‘creation of the moon when we look at it’ carried the day till today against the real Einstein->Broglie->Bohm interpretation. And this would also have harmful consequences for modern mathematics, which completely renounced to what we try to achieve in those papers (and future ‘pros’ once the change of paradigm is accepted fully will complete): the entanglement of mathematics and reality, with the ad on mind spaces as part of reality itself. That is, reality is the mind space of a larger ‘world’; the Earth and the galaxy, as our view of reality is the mind space of our electronic neurons.

Reality and mind merge in the sense that what the mind perceives then it projects to shape reality. But reality limits always what the mind of pure information can imprint; so both must properly merge with the laws of balance.

Still for the sake of understanding the 3^{rd} age of geometry we shall divide its analysis by force brief as we have arrive to the 200 pages limit for an academia.edu paper to properly charge its images, in two sections, the first one dedicated to Lobachevski’s attempt to find *a more precise geometry for the Universe, which turn out to have a hyperbolic geometry – that of the fifth dimension; and the second one dedicated to the explosion of mental space, apt to study all the different local dimotions and change events of different sciences and scales. *

**Lobachevski’s theorems: angle of parallelism**

In the graph, a space-time symmetry happens between the angle of parallelism of a hyperbolic geometry in still space, and the speed of the vortex of forces which implies that faster, stronger, more attractive forces of smaller particles (Sp x ð=k) will have a more hyperbolic geometry, with a smaller angle of parallelism=larger curvature, allowing more ‘parallel forces ‘ to enter the attractive vortex. The different perspectives according to the ‘Rashomon effect’ can give us different equations and mental representations according to how much stillness and motion, and how much difference on size/speed happens between observer and observable, with a limit *given by a full perpendicular angle of parallelism of 0º, which will always be* less than a right angle.

Yet as the angle is a ‘curved’ hyperbola, we can also consider it as an exponential function, where a is x-coordinates and AB the y-coordinates. Then the minimal *angle of parallelism will happen for the fastest growing exponential function, which is eˆ‾×, the constant of death=decay processes when jumping ‘2 planes of existence’: ∆+1<<∆-2; and hence the absolute limit of a hyperbolic geometry, now ‘vitalized’ in terms of the time=motion events of an organic system.*

Indeed, the 4th i-logic postulate of non-Euclidean geometry comes immediately to our mind to make sense of the vital energy, ‘enclosed’ by the Darwinian singularity membrain that preys on it:

In the graph we make use of the i-logic 4th and 5th Non-A postulate to translate into the organic paradigm the meaning of hyperbolic geometry.

Now, how exact is the symmetry between this vitalized, temporal moving view of hyperbolic geometry and Lobachevski’s formal still geometry?

Absolute. Indeed, the surprise comes when we realize of the next finding of Lobachevski’s original work: the line he considered parallel to ‘a’ in figure 3, when he made a close formal analysis DID become a hyperbola at the point of infinity.

It is worth to do a more rigorous analysis on how Lobachevski found this surprising result using mere logic, still formal proofs, to show indeed how all spatial views have a symmetric temporal view, which will be the foundations of non-Algebra and its ∞ S≈T symmetries.

**Convergence of parallel lines; the equidistant curve.**

Let us then investigate how the distance from a of a point X on c changes when X is shifted along c (fig. 5).

In Euclidean geometry the distance between parallel lines is constant. But here we can convince ourselves that when X moves to the right, its distance from a (i.e., the length of the perpendicular XY) decreases.

We drop the perpendicular A1B1 from a point A1 to a. From B1 we drop the perpendicular B1A2 to c (A2 lies to the right of A1, since γ is an acute angle). Finally we drop the perpendicular A2B2 from A2 to a. Let us show that A2B2 is less than A1B1.

The theorem that the perpendicular is shorter than a slant line is valid in hyperbolic geometry, because its proof (which can be found in every school book on geometry) does not depend on the concept of parallel lines nor on deductions connected with them. Now since the perpendicular is shorter than a slant line, B1A2 as a perpendicular to c is shorter than A1B1, and similarly A2B2 as a perpendicular to a is shorter than B1A2. Therefore A2B2 is shorter than A1B1.

When we now drop the perpendicular B2A3 to c from B2 and repeat these arguments, we see that A3B3 is shorter than A2B2. Continuing this construction we obtain a sequence of shorter and shorter perpendiculars; i.e., the distances of A1, A2, ··· from a decrease. Furthermore, by supplementing our simple argument we could prove that, generally, if a point X″ on c lies to the right of X′, then the perpendicular X″Y″ is shorter than X′Y′. We shall not dwell on this point. The preceding arguments, we trust, make the substance of the matter sufficiently clear and a rigorous proof is not one of our tasks.

But it is remarkable that, as can be proved, the distance XY not only decreases when X moves on c to the right, but actually tends to zero as X tends to infinity. That is, the parallel lines a and c converge asymptotically! Moreover, it can be proved that in the opposite direction the distance between them not only increases but tends to infinity, hence forming indeed an exponential function, whose ‘strength’ will depend of the ‘distance’ in ∆-scales and hence different in ‘speed’ of time between both.

It is thus clear that the distance between the point and the line is a mental formal representation of the distance between the larger plane of the membrain singularity that encloses the vital energy of micro-points in which it preys, provoking its entropic decay. Hence the further the ST- MICRO-point from the line-membrain that encloses it in hyperbolic geometry, the further the distance in ∆-scales between both and the smaller the angle of parallelism, meaning in vital terms the more perpendicular=Darwinian will be the relationship between the micro-point and the larger observer.

**The magnitude of the angle of parallelism.**

We shall now study the angle of parallelism, i.e., the angle γ that the line c parallel to a given line a forms with the perpendicular CA (figure 6). Let us show that this angle is smaller, the further C is from a. For this purpose we begin by proving the following. If two lines b and b′ form equal angles α, α′ with a secant BB′, then they have a common perpendicular (figure 7).

For the proof we draw through the midpoint O of BB′ the line CC′ perpendicular to B. We obtain two triangles OBC and OB′C′. Their sides OB and OB′ are equal by construction. The angles at the common vertex O are equal as vertically opposite. The angle α″ is equal to α′ since they are also vertically opposite. But α′ is equal to α by assumption. Therefore α is equal to α″. Thus, in our triangles OBC and OB′C′ the sides OB and OB′ and their adjacent angles are equal. But then, by a well-known theorem, the triangles are equal, in particular their angles at C and C′. But the angle at C is a right angle, since the line CC′ is by construction perpendicular to b. Therefore the angle at C′ is also a right angle; i.e., CC′ is also perpendicular to b′. Thus, the segment CC′ is a common perpendicular to both b and b′. This proves the existence of a common perpendicular.

Now let us prove that the angle of parallelism decreases with increasing distance from the line. That is, if the point C′ lies further from a than C, then, as in figure 6, the parallel c′ passing through C′ forms with the perpendicular C′A a smaller angle than the parallel c passing through C.

For the proof we draw through C′ a line c″ under the same angle to C′A as the parallel c. Then the lines c and c″ form equal angles with CC′. Therefore, as we have just shown, they have a common perpendicular BB′. Then we can draw through B′ a line c″′ parallel to c and forming with the perpendicular an angle less than a right angle, since we know already that a parallel forms with the perpendicular an angle less than a right angle. Now we choose an arbitrary point M in the angle between c′ and c″′ and draw the line C′M. It lies in the angle between c″ and c″′ and cannot intersect c′. A fortiori, it cannot intersect c. But it forms with AC′ a smaller angle than c′ does, i.e., smaller than γ. Then, a fortiori, the parallel c′ forms an even smaller angle, because it is the extreme one of all the lines passing through C′ and not intersecting a. Therefore c′ forms with C′A an angle less than c does and this means that the angle of parallelism decreases on transition to a farther point C′; this is what we set out to prove.

We have shown, then, that the angle of parallelism decreases for increasing distance of C from a. Even more can be shown: If the point C recedes to infinity, then this angle tends to zero. That is, for a sufficiently large distance from the line a parallel to it forms with the perpendicular to it an arbitrarily small angle.

The proof shows the beauty of the symmetry between §@-minds and ∆time motions: *the kaleidoscopic Universe puts in symmetric relationship all its ‘dimensions’ with its own methods and perspectives, creating parallel worlds. *

If at a point very far from a the line perpendicular to a is tilted by a very small angle, the “tilted” line will no longer intersect a. Hence *beyond the 2-plane distance the line a – which represents the ∆+1 scale being – will NOT perceive, prey or interact with the micro-point that becomes a ‘dark space-time’ for it. *

Two lines in a Lobačevskiĭ plane either intersect or they are parallel in the sense of Lobačevskiĭ, and then they converge asymptotically on the one side and on the other they diverge infinitely, or else they have a common perpendicular and diverge infinitely on both sides of it. The vital organic interpretations of those facts shows hyperbolic geometry to be a representation of ∆±i scales, and its organic structure between ‘cellular, unconnected, potential micro-points of a vital energy’ as perceived by the singularity membrain that encloses it.

**RECAP**

GEOMETRY IS broken in 3 sections: @mind: spaces dedicated to study the different mind constructions of the Universe

T-opology: where space form has motion

∆: non-Euclidean postulates of points with form, which becomes lines that evolve into organic pleas.

S: Bidimensional, static plane geometry, the first form of Mathematics, invented by the Greeks, which we will treat in this post.

Because the Universe is bidimensional, holographic, what matters on its mathematical origins is to understand that the Greeks and its plane geometry does matter as each of those ‘theorems’, which we studied in high school do have ‘hidden deep meanings’ that will resurface once and again, into the vital geometries of points with parts that create the universe.

As usual as all is ternary and a ternary vision is for the mind mirror more pleasing we shall also consider in ternary ages the evolution of bidimensional geometry, which went through:

A first young, Greek age of static bidimensional space-geometry

A 2^{nd} mature age of maximal reproduction, during the time of mathematical physics as it set the stage for the evolution of physics and the understanding of mechanics and gravitational, Newtonian and Keplerian Universes.

The third age started in this blog with the understanding of the holographic Universe, which will expand the discipline to a logic ¬Ælgebraic realm to fuel the application of its ∆ST laws to all other disciplines.

So we shall close here the ‘seed’ of information for future researchers to expand and passing through the 2nd age of geometry, when analytic geometry, married with @-p.o.vs. to create the first solid ST representations and ∆-scaling (Cartesian geometry). Hence studied in the post of analytic geometry.

The exhaustion method, which do convert a sum of triangles or ‘angular momentums’ (in the duality information-motion) and foresees 5D analysis

And the Greek understanding of the circle as the perfect form, and all the theorems extracted from it – the most developed inflationary mirror in its last ‘excessive’ age of form.

**Lobachevski as predecessor of 5D geometry**

But geometry truly reached a maturity as a science of ‘reality’, when it incorporated motion; time dimensions to form; with non-Euclidean geometries and topology. The masters of this science were without the slightest doubt, as usual a triad, Gauss, Lobachevski and Riemann. The less recognized and more profound being Lobachevski, who found the principles of ‘pan geometry’, the absolute geometry of reality based in 3 insights:

The realization that ‘mathematics-geometry’ is a mental-logic endeavor, where function and i-logic thought overcomes ‘spatial representation’, thus he extracted as we do in ∆st logic postulates WITHOUT possible expression in the ‘parabolic’ @-geometry of the Human Euclidean ‘light-dimensional mind’, to extract pure logic results, showing that the causal, sequential logic of time is the essence of reality.* *By far this can be considered the highest insight in the world of mathematics since Descartes’ analytic geometry and Leibniz’s foundation of Analysis – and should guide us in our inquire of fundamental laws of ∆ST, as what matters in mathematics is the reflection of functions and symmetries over forms. So as systems become more complex, the original geometrical properties become lost and substituted by the function of the physiological networks of the system; which also helps to understand why in topology forms might seem very different but as they keep the essential properties of the being, they do keep their functions. Without this realization the XX c. explosion of abstract mental spaces to represent reality wouldn’t be possible.

Further on, he understood relational space-time in space is defined in a first incursion in topology by the concept of ‘adjacency’, *which completed the 3 fundamental ‘modes’ of relationship through geometrical space of t.œs – complementary adjacency, perpendicular Darwinism and parallel social evolution – hence a concept *essential to the organic structure of the Absolutely relative Universe defining for the first time topological transformations are those in which motion does NOT deform the fundamental *properties of reality in space, *starting a trend culminated by Hilbert’s foundations of geometry (yes the guy we criticize so much – he did also do some work of merit :), with his emphasis on some key abstract concepts such as betweenness, congruence, continuity, incidence, separateness… *which are clearly relative concepts concerning scale and symmetry, the mind elements that allow a singularity or point of view to ‘construct’ a wor(l)d-view over and ‘stiffen’ the motions of reality to make a mental mapping of them.*

He insisted strongly in the experimental nature of maths, wondering which was the real geometry of the Universe, and made the first inroad on the difference of ‘mind-spaces’ according to scale, as it depends on the size of our perspective that we find a ‘flat’ geometry (detailed view) or a ‘curved geometry’ (far away view where the whole world cycle that seems a line in short distance/time span becomes a whole closed zero-sum worldcycle of energy).

Those 3 findings are essential and we shall dwell on them. Regarding his inconclusive results on the geometry of reality, what mathematicians though miss is the ‘Rashomon effect’, given their one-dimensional humind thought, wondering what is the space of the Universe of the triad of elliptic, ð§ (spherical, Riemannian surface of the space-time super organism), @ (Cartesian analytic, mind geometry, with the mind as its focus) or hyperbolic, *ST* (Lobachevski’s geometry) or parabolic, ∆-Euclidean.

This fundamental equivalence between the 3±∆ geometries and the 3±∆ parts of the time§pace Supœrganism is the fundamental correspondence of space and so instead of naming it by the humind ego that discovered them (Riemann, Lobachevski, Descartes, Euclid) we shall use the older terminology before the selfie age because of its descriptive power, again:

-The membrain (singularity and membrane) has an elliptic, ð§ geometry, hence it is used in General relativity to describe the ‘gravitational enclosure’ or ‘curvature’ of the ∆+1 gravitational scale (Einstein’s relativity). But elliptic Geometry is much more profound than usually thought in the establishment of the properties of any system of reality, and so as we have not treated it elsewhere is worth to consider its role:

In the graph, in elliptic geometry we define a point as a two nodal points of a sphere with maximal distance between them, which implies they all pass through the 0-point or singularity, and establish the non-existence of parallels.

As such elliptic geometry has no parallels, because all its ‘parts’ are connected, by the formal center, o, which unlike in the classic formulation of elliptic geometry in ∆ºs≈t must be considered also the ‘invisible part’ of the nodal point; and so elliptic geometry describes* the @-structure of a singularity point connected to a membrain, f*orming an absolute enclosure.

And ultimately as ALL points are in fact ‘two strong’ points, two poles, which are equivalent, it establishes a fundamental property of Nature, the bilateral symmetry with inverse properties self-centered in a balanced symmetric ‘identity’ element that communicates them all as they are a all connected to all other lines/circles and through its axis to the singularity, which is therefore not only the central point but the axis of…

– The mind singularity, which acts therefore as the focus, and it is an @-self centered geometry, which allows Cartesian planes to be ‘perspectives’ from a focus, the zero point and its informative height dimension and other axis of the system – the reproductive-width dimension and the length-motion dimensions. We can consider in the idealized structure of bare mathematics, the *3 physiological networks of the being. And so the being switches off between its 3 axis/networks as its functions change.*

Further on the mind is connected with every point of the entity, but for each point there is only one connection – only a line-parallel can be traced.

And finally, as we shall show soon in the graphs of human systems, since space is a mental-singularity related function to process information in an efficient manner, and recreate order, the mathematical simplest most efficient geometry of the ball-elliptic form must not be conserved.

What matters here is the symmetric bipolarity, which allow the singularity to maximize the extension of its vital space-enclosed by the membrane, so we shall see how in complex organic systems the sphere suffers all kind of topological transformations into all kind of shapes but all of them are ‘enclosed’ for the mind to re-form the vital space within, and all have a singularity brain-system to connect them, and all have bilateral symmetry (even the sphere which in principle is not defined as such in classic maths – only considered to have rotational symmetry, except in the elliptic geometry that defined antipodal points), because the singularity co-ordinates all those points and uses its inverse properties to extract motion from the vital energy within it.

-The intermediate vital space-time enclosed between both has a hyperbolic geometry, the dominant in the Universe, because it is the present state. It does have a ‘saddle’ dual curvature, because it *communicates the two other inverse poles of the being. So if in the surface of the sphere, curvature is always positive, and in the central point and axis, curvature is always negative, the hyperbolic intermediate space-time has both curvatures.*

**The ternary forms of spatial relationship: 4th postulate.**

In that regard, in non-E geometry, we must distinguish as usually a ‘ternary’ type of spatial relationships with deep meanings in the vital organic structure of reality:

Adjacency (forms that are pegged, hence forming part of the same time§pace supœrganism).

Perpendicularity, (forms that penetrate and disrupt its inner systems, basis of Darwinian events.)

Parallelism (things that maintain its distance and allow communication through a common medium or network, basis of social evolution –* studied in affine geometry*.)

In non-æ geometry they will be extensively studied as the fundamental modes that define the relationships of ST, complementarity and ‘symbiosis’ (adjacency), Darwinian struggle (perpendicularity) and ∆§ocial evolution (parallelism) of all systems, *becoming the essential qualities to understand how spatial relationships define temporal events among all systems and scales of nature, studied by the fourth postulate of ‘congruence and similarity’. *

It is then essential to understand the ultimate meaning of parallelism vs. incidence/perpendicularity also as mental descriptions of two logic states – one of parallel social evolution and one of Darwinian colliding ‘tearing’ by the perpendicular, incident line, taking the concept out of its spatial representation, as Lobachevski’s ‘first great insight’ did for all of future findings of mathematical space.

What makes this geometry so important is, once we liberate the postulate of parallelism from its physical representation back to where it belongs into mental space, the fact that it allows it to travel through scales, unlike the elliptic geometry that constructs a system in a single plane, hence it is the geometry of ∆-scales, which coupled with the @nalytic representation by a mind converts it into the best representations of the ∞ variations of the organic, scalar Universe:

In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the resulting figures are similar, i.e., they have the same angles and the same internal proportions.

In elliptic geometry this is not the case. For example, in the spherical model we can see that the distance between any two points must be strictly less than half the circumference of the sphere (because antipodal points are identified as the maximal bilateral distance). A line segment therefore cannot be scaled up indefinitely.

A geometer measuring the geometrical properties of the space he or she inhabits can detect, via measurements, that there is a certain distance scale that is a property of the space. And so we find a recurrent theme of ∆st: all is in its ultimate ‘largest’ view a closed circle (definition of a line as a circle in elliptic geometry). Yet on scales much smaller than this one, the space is approximately flat, geometry is approximately Euclidean, and figures can be scaled up and down while remaining approximately similar; as you see the Earth flat in smaller scales. Hyperbolic geometry, that of the energy present vital space, is somewhat an intermediate ‘region’ in which scaling is possible but limited by concepts such as the angle of perpendicularity.

What about colors? Obviously they are the key, as they are coded by frequency, which is the translator of scales. What this means ultimately is that light’s ‘frequency-colors’ fundamental role is to transmit information not only in a single plane but specially between ∆-scales as the telescope/microscope discovery found out. So the 3 dimensions of light space-time are present elements of the super organism of light and its social colors the evolutionary element.

**Lobachevski’s pan geometry. Where is the 5 ^{th} dimension. ¬ æ**

Thus algebra (and analysis concerned with the processes of social numbers that add and emerge or subtract, and divide plunging down the scales of eusocial love of the 5^{th} dimension), is a larger subject, still not fully developed by the only human world-point, which as Boylai on the view of non-E spaces, can only exclaim ‘I have discovered (not invented, as he said, the ever arrogant human ego) a new strange world – and not out of nothing as he said, but out of everything’).

The Universe is not a continuum, but as all fractals it is discontinuous. This of course, the ‘axiomatic Hilbert-like’ arrogant humans do not like. So a guy called Dedekind found a continuity axiom, affirming that the holes between the points of a line are filled by real numbers, which are ratios between quantities such as π or √2, which happen NOT to exist as exact numbers, and more over represent an infinite number compared to those which do exist.

Further on, when XX c. geometers went further than Non-Euclidean Riemannian geometries into absolute geometries it turns out that the most absolute of all geometries, didn’t need the continuity postulate.

This geometry, which is the ultimate absolute plane geometry that included all others (and now further clarified by 5D i-logic geometry), reflected the absolute architecture of the planes of Existence of the Universe. A German adequately named Bach-mann, for its musical architectonical rigor, discovered it.

It is the Goldberg variations of the theme. And it was discovered the year the chip Homoctonos was found, ending all evolution of human thought, which now is busy-busy translating itself to the new species, with ever more powerful metal-minds and smaller human minds, receding into a hyperbolic state of stasis, thinking what the machines that are making them savant idiots discover belongs to their ego-trip paradox.

In terms of geometry is merely the ‘realization’ of the 3 canonical geometries, we have used to define a system in space, perceived from a given point of view across the scales of size of the Universe, taking into account that our ‘rod’ of measure is light speed-space.

We see reality through light’s 3 Euclidean dimensions and colors, which entangle the stop measures of electrons.

Yet light-space and any relative size of space of the Universe must be analyzed with the pan-geometry of the 5^{th} dimension, first explained by Lobachevski, as we see smaller beings with a hyperbolic geometry, which multiplies its ‘fractal forms’, and larger ones with an elliptic geometry which converges them into single, spherical ones. Hence the hyperbolic geometry of quantum planes, the elliptic geometry of gravitational galaxies, and the middle Euclidean geometry of light space-time, in which the Lobachevski’s constant of time and space is minimal, since our quanta of information H-Planck is minimal compared to our quanta of space-light speed.

Let us elaborate on this idea with more ‘mathematical depth’ as it is essential to complete our analysis of the humind.

super organism so they are ‘constrained’ into a zero-sum or limiting membrane and appear as curved geometries’.5D, Long/lasting measures complete a zero sum world cycle and an fully enclosed superorganism so they are curved’.

As most of all modern geometry is based in this duality, one of our 3 fundamental dualities of the Galilean Paradox, it seems obvious that ∆@s=t will also be able to explain all the foundations of modern geometry and by extension as Disomorphic dimensional geometry is the foundation of all other mathematical sub disciplines of all of the mental spaces of mathematical sciences.

**Other representations of Hyperbolic geometry. Klein’s ‘open ball’ with motion.**

To which extent what we have developed of hyperbolic geometry in terms of planes of the 4th-5th dimension within the time§paœrganism can be considered exact, can be revised by studying the hands-on main models that came out, mostly by Belgrami (despite having the name of the sacred cows of northern European science – we peripherals, Latinos and Russians, you know cannot be geniuses of science, never mind Galileo, the Greeks, Mendeleyev, Lobachevski. So the Belgrami’s cone, the Belgrami’s sphere and the Belgrami’s disk, which show more clearly that indeed hyperbolic geometry IS the geometry of the vital open ball space enclosed by the membrain (when studying it strictly within a single plane), have this other ‘people’s’ name. In the next graph we see a representation of its main elements, the singularity, disk and sphere under hyperbolic geometry:

In the graph we can see the two models (extended to a 3-D sphere), of hyperbolic geometry, showing clearly that the vital energy enclosed by the membrain can neither reach the central cone or the B-C-D membrane that encircles it, which offers a constant resistance to its advance.

Those limits are exactly the same for the galaxy in terms of T=0 k temperature (black hole singularity) and c-speed ‘membrain/event horizon’, which cannot be reaches as they offer a constant resistance. So *hyperbolic geometry is the ideal geometry to represent the atomic/star galactic space between the halo and the black hole singularity in the center of the galaxy:*

We shall not extend further into the main of the Non-Euclidean geometries, as the number of mental spaces triggered by the ‘freeing’ of the mind-spaces of mankind and its formal languages grew also exponentially after Lobachevski’s transformation of geometry into a logic, mental science. So we shall deal with all those spaces in terms of ∆•s≈t higher laws of space-time topologies, inversions, scales and symmetries.

In the graph a physical understanding in terms of special relativity and its hyperbolic geometry, where we dissect the different ‘elliptic-membrain’ + hyperbolic vital energy geometry of the Universe, which is the essence of *hyperbolic special relativity concerned with light/electromagnetic forces vs. the elliptic gravitational membrain (halo of dark matter + central black hole).*

The graph shows the hyperbolic behavior of our Euclidean space as it moves to the non-transit barriers of the central ‘Beltrami singularity cone’ (left upper picture, right lower picture) and external, ‘Klein’ hyperbolic disk (left upper picture where space-time motions never reach the limit, down right picture, being that physical limit the c-speed barrier), which shows the essential structure of the 3 parts of the being in terms of its geometry.

Further on, the membrain IS an elliptic geometry of antipodal points that ‘compress’ and control the inner regions of the being, hence used as the container of the vital energy by its black hole singularity and halo of strangelet, connected through gravitational waves and dark entropy.

So in its elliptic geometry, the 0-singularity point tightens up with its attractive force the antipodal points of the external membrain, *creating in this manner the* ‘force’ membrain of gravitation.

Thus the galaxy is the ‘vital energy’ – stars that shall become black holes or strangelet halo.

So the 3 geometries of topological space-time with more or less degree of dimensional complexity will always correspond to the 3 timespace arrows/dimensions/events/forms.

As we should know by now, the symmetries and inversions between the super organism’s parts in space correspond to a similar symmetry between scales. So as we have defined 3 basic geometries, we can also consider them in time view, in space view, in scale view (both in the entropic arrow and in the mind’s deformation of a self-centered biased point) – then we have the full Rashomon effect to get the final 5D mind/judgment conclusion of what truly we are watching, extracting all its information. This of course would reorder all the info of all stiences but a single man can only give glimpses to the Rashomon effect of a few subjects.

Let us then give the final ‘judge’-view of the Mind, of a hyperbolic ‘disk’, which will be the sensorial membrain from where the ‘relative scale or size’ of the hyperbolic plane will be judged.

The understanding of hyperbolic geometry from the @-mind vs. the view from the outer membrane.

In the graph, hyperbolic geometry and any spatial mental form as a rule requires a bit of ‘endophysics’ and observer’s paradoxes to fully understand reality without the mind bias.

In the Poincare disk (and the Poincare line), the shrinking of the points is accepted to bend as we perceive it from the larger view, the fractal elements of the vital energy inside.

If we consider then the @-view to be that of the external membrane, the ‘largest’ POSSIBLE view, (as in your organism, where the mind is just a bunch of microscopic cells but holds the view of the larger whole-scale of your body), it is natural that the inner ∆-1 elements are perceived ‘smaller in space’, as they come to the larger whole.

It is also interesting to consider the topological duality of that membrane which ‘dissects’ in words of Lobachevski, space into inner and outer regions (first topological postulate), creating two completely different visions of reality, as the internal being will see a concave enclosure, a forbidding barrier and nothing beyond. While crossing that barrier, we perceive a much larger convex, open Universe. How this transforms our mental view of space can be now responded considering the ‘ratio’ r/l, which must be understood from the mental point of view as a ratio between the ‘radius’ of the time§paœ system, which the ‘mind’ perceives and measure, with a ‘length’ associated to its own potential-limb sizes. For example, humans have a limb-step of lineal motion (1D) of 1 meter. So when observing entities of maximal size, it will perceive its perimeter larger than a perfect circle or sphere, increasingly ‘elliptical’ and ‘flat’. For that *reason we see the Earth flat, as the radius of the planet is huge and our ‘scale of measure’, a million times smaller. But if we grow, we would increasingly see the EARTH spherical. *

Thus minds are indeed Cartesian devils crafted differently according to size (which are defined by the parameters of perception, such as the substance we perceive, the smallish pixels of light, the larger atoms of smelling, and its organs, individual, multiple eyes, etc.) We deal then with those elements in the posts on mind worlds, where some surprising results appear on how insects, atoms or black holes would perceive if as all seems to indicate process different ‘sizes’ of pixels and lineal rods of measure.

In that sense the rule of all minds should hold and amount to this: smaller beings seen a flat world, larger ones curve it, and the larger the being is in relationship to the world it observes, the more curved its mind will be till the absolute mind-space of the Universe, that of T.Œ, which you might call the Taoist, impersonal God, the game of existence, which observes A PURE BLOCK OF TIME with all the potential symmetries realized, all the small steps converted in larger cyclical wholes, all a zero sum, all a Nirvana state, in which I dwell now for quite sometime, in which nothing surprises you, the future, the past and the present a separated illusion, as the i-logic structure of the fractal displays a perfect order.

In the graph we see an example of those ‘thoughts. The earth might seem a flat, still form, but from a larger slower time rhythm it will seem a cycle, fixed in form as the Saturn rings seem to us. The contemplation of all the potential (in an Aristotelian sense not to confuse with a physical potential, ∆-1 field) beings that there were, are and will be within the limited variations of reality, where chaos is only the ignorance of those laws, is thus the ultimate mental state, where all becomes space, a Parmenides whole, with no motion, only reproduction of déjà vu information, as *the possible variations of the game of existence made of so limited number of elements, has been written eternal times; and so also each of us has been repeated ad infinitum in other moments of timespace…*

Internal Lineal freedom vs. external cyclical order and its reflection in mathematical structures.

An essential concept to understand the paradoxical modes of generation of space-time beings is the duality between the internal mind, which performs the lineal seemingly free steps of its Dimotions as ‘finitesimal tangential actions’, derivatives of what will become its cyclical curved external order imposed upon it by the larger worldcycle. This Duality between the smaller steps of lineal approximation to the larger cycle that will enclose and summon up them all is the justification of all the philosophy of mathematics of Differential geometry and derivative calculus. A curve for example is approached in differential geometry by a lineal tangent, or by a plane, parallel to the polidimensional curve – but the whole imposed externally is the curve, and the steps proposed internally are lineal steps

So paraphrasing Wheeler, we could say that the ego is free to perform instantaneous lineal steps, choosing the direction of its motion, but the larger whole will impose its curved paths of ‘least time’. In physics the mass will try to move in a given direction but it will be curved by the outer space-time geodesic; the mind will make plans from its finitesimal subjective point of view but the organism will impose its boundaries… And of that tug-of-war between the individual steps of freedom of the ‘fractal point’ and the larger functions reality happens. Einstein said ‘time bends the space (of the mind)’… Indeed as we keep trying to maintain our lineal will, the environment bends us and if we don’t, we crash…

**Color space, defining the vital geometric properties and Riemann’s generalization. **

We can now with all this ‘∆•s=t’ considerations on the ternary codes of colors study it as geometers did to generalize the concepts aforementioned in the preceding section on the real meaning of n-dimensional space, to solve the problem of generalizing the scope of geometry and the concept of space in mathematics.

First clarify that any ‘geometrical construction’ will depart from the ‘elements of geometry’ (enhanced in our Non-E definitions) such as ‘T. Œntities’ are simplified into ‘points’; social herds of T.œs into lines, and its ‘structural symmetries and coordinations’ onto ternary networks defined by the Generator formalism of non-Æ (groups in classic algebra).

This said, experience shows that the normal human vision is three-colored, i.e., every chromatic perception, of a color C, is a combination of three fundamental perceptions: red R, green G and blue B, with specific intensities.

When we denote these intensities in certain units by x, y, z, we can write down that C = xR + yG + zB. Just as a point can be shifted in space up and down, right and left, back and forth, so a perception of color, of a color C, can be changed continuously in three directions by changing its constituent parts red, green, and blue. By analogy we can say, therefore, that the set of all possible colors is the “three-dimensional color space.” The intensities x, y, z play the role of coordinates of a point, of a color C.

**Positive vs. Negative or neutral**

An important first difference though from the ordinary coordinates, originated in *locomotion analysis, where we have inverse timespace directions,* consists in the fact that color intensities cannot be negative, as *we are using here pure formal space*. When x = y = z = 0, we obtain a perfectly black color corresponding to complete absence of light – a theme, which is essential to understand WHY imaginary numbers do exist for certain dimensional spaces but NOT from others, which we can resume in a simple statement, called ‘horror vacuum’:

Negative values exist only in ternary cyclical ‘π’ time§pace zero sum worldcycles, as it is merely the inverse 4D (∆-1) vs. 5D (∆+1) arrows of form, self-centered in the ∆º plane, whose sum gives us a zero world cycle that returns to its cyclical origin.

It doesn’t exist as real (provoking many errors on science) for pure spatial form perception as 0 is the value of emptiness, stillness, absolute form and there is therefore not negative temperature (zero-still motion is the value of 0 K) or negative color (related to temperature as color carries the frequency-heat on the thermodynamic scale) and so on. In terms of dimotions of existence and its mathematical representation, it will be an important fact to understand mathematical quantum physics in concepts such as Spin, Pauli exclusion principle, antisymmetry and so on:

‘Parameters of present space dimensions are neutral, |x|; absolute, scalar past and future parameters are ±x’

Next in our illustrative analysis comes the concept of continuity vs. discontinuity again a key mental space-time concept hardly understood as the mind seeks continuity of space, and the non-reflexive humind scientist both in mathematics and physics accepts its as an ‘evident dogma’ of its naive realism, creating so many hard-to die errors of thought and false proofs, which a proper s=t symmetric analysis do understand.

**Continuity is a mental device.**

We have stressed often that continuity exists in subjective mental spaces, and this gives us a lot of freeom regarding its most general meaning. I.e. In the color space Riemann defined continuity as a continuous change of color represented by a continuous line in “color space”. Yet in reality color is discontinuous formed by a discrete number of mind perceptions of *discrete frequencies of light,* but the electronic mind only perceives the peaks of photons – as the stop and go process of a continuous view of a film, where we do NOT perceive the irrelevant steps between those colors which perception ignores. Hence we can define mental continuity:

‘Continuity is always a product of mind-space, which in any language ‘reduces’ information to fit in its infinitesimal, by discharging all irrelevant or redundant information’.

Minds reduce dimensions to the relevant ones, *eliminating all dark spaces*: continuity Is the result.

**Ternary emergence.**

Duality of S-T combines into S=t energy beings, so we obtain the ‘third st color’ by mixing two ‘extreme’ ones, and this can then be considered an intersection of ‘lines’.

For example, when two colors are given, say red R and white W, then by mixing them in varying proportions* we obtain a continuous sequence of colors from R to W which we can call the segment RW. The conception that a rose color lies between red and white has a clear meaning.

And so we can go deeper in the scalar ∆-1 detail making emerging new colors, as we can go deeper into the real number line seeking for nested new ‘numbers’ of more ‘decimal’ scales.

And this happens precisely because of the scalar structure mimicked by the 0-1≈1-∞ symmetries between ∆-1 and ∆+1 ‘scales’ of analytic geometry.

Yet those details will only exist if the mind can perceive. That is, if the space-detail were to be ‘matched’ symmetrically by the mental-informative perceptive capacity.

And this perceptive capacity will depend on the r(t)/k(s) ‘scalar factor’ of informative density of the mind aforementioned, so a large viewer will NOT see detail and cannot ‘penetrate’ the virtual sub-ternary parts of the color or any other mental space spectra).

**RECAP. **A DEEPER search of the real geometry of the Universe will NEVER be completed without the addition of 5^{th} dimensional metrics and the understanding of the 3 different mental perspectives an observer in an ∆¡ plane will have of its ‘flat geometric’ equal scale, ‘elliptic, upper larger scales’ from where it receives energy but hardly information and lower more informative ‘hyperbolic scales’ of parts connected to its plane by branching, fractal networks:

For that reason still today, hyperbolic and elliptic geometries that do not follow the 5^{th} non-e postulate are ill understood and as we have shown incomplete, since not even the real concept of a fractal points is canonical in mathematical sciences, even after Lobachevski showed infinite parallel could cross through it. Yet the Universe’s real external geometry is based in the interplay of those 3 scalar geometries as perceived by minds in different scales of reality.

*. *

**XVI. METRIC SPACES: FROM RIEMANN TO HILBERT**

Thus *because the fifth dimension was ignored, the experimental nature of mathematics would not be found and the effect of *Lobachevski’s discovery will be rather its opposite – to make people in an age dominated by the idealist ego centered philosophies of German Hegelianism, to think that reality did not matter as it did not conformed to the ‘evident’ single lineal 3D Euclidean geometry we perceive in our plane of space. It will then bring the opposite view of subjective mental spaces as the dominant reality, finally bringing the Hilbert/Cantorian paradoxes; yet the proper interpretation by Riemann that *geometry still mattered, is what matter to us here; since his genius realized that geometric concepts were logic *concepts of the mind, from ‘distance’ which was a synonymous of similarity to dimensions which were to become ‘parameters’ of change (hence ultimately dimotions) of different phase spaces suitable to express local forms of change in reduced parts of reality.

Let us then conclude this brief introduction to 5D S-geometry with considerations on Metric spaces.

**Riemann’s generalization**.

The basic ideas of Riemannian geometry are really rather simple if one sets aside the mathematical details and concentrates on the basic essentials. Such an intrinsic simplicity is a feature of all great models of reality, since the Universe is ‘simple but not malicious’ – as Einstein, whose idea was also very simple – to equate acceleration and gravitation – put it. Lobachevski’s model was also simple: to regard the consequences of the negation of the Fifth Postulate as a possible geometry. So it is the idea of the discrete atomic structure of matter, *as all continuous wholes are in detail discontinuous, ‘entropic’ desegregated ∆-1, closed forms…*

All of them of course are generated by the simplest of all simple ideas: S≈T. Only by iteration and variation reality becomes very complicated.

Yet new ideas must, first of all, work their way over a wide field and must not be pressed into a rigid framework, and second, their foundation, development, and application is a many-sided task, requiring an immense amount of labor and ingenuity, and impossible without the specialized apparatus of science – reason why (Kuhn) they take so long to be imposed among pedantic scholars, which won’t have it till it has reached the perfection of old outdated ones – but won’t help to realize that perfection, as this writer well knows.

In Riemannian’s geometry this scientific apparatus consists in its complicated, cumbersome formulas, due to the obvious multiplication of dimensional parameters. But we shall not deal with complicated formulas except when in the future we study the marriage of Riemann and Einstein’s simple ideas.

So as said, Riemann’s essence is to consider an arbitrary continuous collection of phenomena as a mental space as Lobachevski implicitly did, going a step further by adding the ∆nalysis of its ‘(in)finitesimal points’ or minimal elements in the discontinuous ∆-1 scale that are in the larger view a ‘continuous line’-whole. So time minimal intervals and space minimal quanta, and its variations and ∆-1 scalar ‘differential and integral properties’ could be added, besides expanding the number of ‘dimensional properties to its (in)finite (meaning in both cases that all infinitesimals have a limit and all infinities also have a limit – that of the size of the lower or upper part/whole scales; so an infinitesimal of n is normally 1/n, where 1 is the whole; or in other words, the infinitesimal moves the 1-∞ scale into the 0-1 infinitesimal scale).

In this space the coordinates of points are quantities that determine the corresponding phenomenon among others, as for example the intensities x, y, z that determine the color C = xR + yG + zB. If there are n such values, say x1, x2, . . ., xn, then we speak of an n-dimensional space. In this space we may consider lines and introduce a measurement of their length in small (infinitely small) steps, similar to the measurement of the length of a curve in ordinary space.

In order to measure lengths in infinitely small steps, it is sufficient to give a rule that determines the distance of any given point from another infinitely near to it. This rule of determining (measuring) distance is called a metric. The simplest case is when this rule happens to be the same as in Euclidean space.

Yet *as Lobachevski’s key formula, r/k shows such* a space is Euclidean in the infinitely small.

In other words, the geometrical relations of Euclidean geometry are satisfied in it, but only in infinitely small domains; it is more accurate to say that they are satisfied in any sufficiently small domain, though not exactly, but with an accuracy that is the greater, the smaller the domain. A space in which distance is measured by such a rule is called Riemannian; and the geometry of such spaces is also called Riemannian. A Riemannian space is, therefore, a space that is Euclidean “in the infinitely small.”

The simplest example of a Riemannian space is an arbitrary smooth surface in its intrinsic geometry. The intrinsic geometry of a surface is a Riemannian geometry of two dimensions. For in the neighborhood of each of its points a smooth surface differs only a little from its tangent plane, and this difference is the smaller, the smaller the domain of the surface that we consider. Therefore the geometry in a small domain of the surface also differs little from the geometry in a plane; the smaller the domain, the smaller this difference. However, in large domains the geometry of a curved, different from the Euclidean, as in the examples of the sphere or pseudo sphere.

Riemannian geometry is THUS a natural generalization of the CONCEPT OF mental dimensional properties, to an arbitrary number n and of non-Euclidean geometries to the ∆§cales of the discontinuous Universe. Hence its enormous success, as it is grounded in true properties of the reality of ‘dust of space-time’ – ∆@s≈t.

Such n-dimensional Riemannian space, although Euclidean in small domains, may differ from the Euclidean in large domains. For example, the length of a circle may not be proportional to the radius; it will be proportional to the radius with a good approximation for small circumferences only. The sum of the angles of a triangle may not be two right angles; here the role of rectilinear segments in the construction of a triangle is played by the lines of shortest distance, i.e., the lines having the smallest length among all the lines joining the given points.

One can speculate that the real space is Euclidean only in domains that are small in comparison with the astronomical scale. Since now we ARE outside the light space-time into the larger gravitational scale, which becomes indeed Riemannian in Einstein’s work.

But this concept does also ‘work’ for any other mental space, with NO reference to geometric figures but logic properties and so we can through ∆st going even further in the comprehension of Riemannian geometries, wondering what truly means ‘Euclidean properties’ vs. ‘hyperbolic properties’ vs. ‘elliptic properties’, our ternary variations of space -which obviously must be an even more general geometrization of the ternary symmetries of scales and topologies of T.œs.

A theme we have dealt with in other posts. Let us then consider the other 2 founding ideas of Riemann’s geometries – one which comes from his master Gauss, concerning the fact that of the 3 parts of any T.œ, the constrain-membrain is by far the most important, as the vital energy is the ‘tabula rassa’, the formless potential; and the singularity is the hidden central or polar ‘invisible’ element of the elliptic geometry.

so almost all what we know about reality comes from membrains, which hide its internal regions, even if most of the timespace of reality comes from the vital energy the fractal point encloses, and all of its virtual mapping information comes from the mind singularity.

Let us then introduce another huge field of modern mathematics – the study of the membrain, called intrinsic differential geometry of surface, where our rule of relative form according to size also applies:

‘1D $mall measurements do NOT measure the whole world cycle of the being, so they are lineal. long-lasting measure bring the whole worldcycle or enclosed super organism so they are ‘constrained’ into a zero-sum or limiting membrane and appear as curved geometries’.

**The generalization of dimensions and its properties by Riemann metric.**

The geometrization of those 2 qualities, multiple-dimensionality and mental spaces, would then become essential to modern science, as it was the formalization of the most generalized useful praxis of Geometry -performed by Riemann with its ‘Riemannian geometries’.

In order to make it clear how a Riemannian space is defined mathematically, we recall first of all the rule for measuring distances in a Euclidean space.

If rectangular coordinates x, y are introduced in a plane, then by Pythagoras’ theorem the distance between two points whose coordinates differ by Δx and Δy is expressed by the formula: s= √∆x²+∆y²

Similarly in a three-dimensional space: s= √∆x²+∆y²+∆z²

In a n-dimensional Euclidean space the distance is defined by the general formula:

S= √∆X1²+∆X2²+…+∆Xn²

Hence it is easy to conclude how the rule for measuring distance in a Riemannian space ought to be given. The rule must coincide with the Euclidean, but only for an infinitely small domain in the neighborhood of each point. This leads to the following statement of the rule.

A Riemannian n-dimensional space is characterized by the fact that in the neighborhood of each of its points A coordinates x1, x2, ···, xn can be introduced such that the distance from A of an infinitely near point X is expressed by the formula: dXA= √dX1²+dX2²+…+dXn² + ε

where dX1, ···, dXn are the infinitely small differences of the coordinates of A and X and ε the degree of error which grows when the relative mind-measure is greater.

This fact being ultimately completely similar to the rules of measure of a fractal discontinuous edged reality, where the smaller the fractal step we take to make a measure the more accurate it would be, but also the LARGER it will be the measure of the ‘fractal coast’.

And so we realize of the little understood fact that differential and fractal geometries are the two sides of the same coin of the fractal, scalar universe, one used for ‘smooth’, ‘curved’ surfaces with no state transitions and the other for edged one with ‘brisk’ transitions in its ‘parameters of time and space’.

Since we have escaped ‘geometrical visual space’, we can now extract the logic consequence of all of this:

AS THE coordinates/dimensions of such ternary generalizations of geometry are properties of our D-isomorphic reality we can ascribe then a smooth differentiable geometry to a smooth motion in timespace (growth, dissolution, reproductive motion) with NO ‘brusque transformation’ or change of S<st>t states and ∆±1 scales (standing points of calculus of variations, discontinuous between ∆scales.

While fractal changes correspond to stationary points that change scales or discontinuities between ∆-planes.

This also means that basically all the laws of Riemannian geometry themselves Disomorphisms of GST apply roughly to fractal geometry, which we shall therefore escape.

What matters to us are the consequences of applying the Pythagoras theorem to many more dimensions, hence yet another mental law that escapes geometry, as now we are in ‘properties’ of reality. Why then they can be square, summed and rooted to find a distance? what all this means for the general laws of ∆st they reflect?

Those are themes of algebra, as we do need to understand the operandi of maths in terms of what they mean for the dimensional symmetries of the Universe.

**All type of spaces as metric spaces.**

What matters then of metric spaces is to make a proper representation of time-space laws. So needless to say almost all spaces with experimental use are metric spaces, such as:

- ∆-Euclidean space of an arbitrary number n of dimensions.

2. ST-Hyperbolic space.

3. Any surface/membrane in its intrinsic metric.

4. C space of continuous functions with distance defined by the formula Ði (f1, f2) = max | f1(x) − f2(x)|

5. the Hilbert space to be described in Chapter XIX, which is an “infinite-dimensional Euclidean” space.”

*Which spaces are NOT metric spaces? Those who can eat up ‘points’ loosing information, without loosing its fundamental properties as ‘beings’, hence topological spaces that preserve its most general properties but are efficient enough with ‘lesser’ points to the ternary limit of a metric, giving away the ‘redundant’ points of the geometry.*

This being an essential property to understand *How the Universe reduces information to the barebones, as in palingenesis and genetics, that compresses reality to the efficient steps of evolution. *

So a metric space is not a topological space. However, every metric space gives rise to a topological space. This is the well known construction that takes a metric space X and constructs the topology on X where a set U is open precisely when for every x∈U there exists some e>0 such that the open ball Be(x) is contained in U.

TWO important comments follow: First, this process of conversion of metric space into topological loses *(often redundant)* information. For instance, there exists infinitely many metrics on ℝ such that all of them produce the same topology of open balls. So, only knowing the induced topology does not allow you to recover the metric. So the topology of open balls=vital space-time energy is the most *general tabula rassa on which to construct a ‘real entity’ by introducing the enclosure and singularity that will ‘re-form; hence give function and form to the open ball, starting the process of construction of a time§paœrganism.*

*This means that enclosure and singularity, the @-constrains are essential to define and solve any problem, and in mathematical physics we shall find that without an enclosure-singularity of elliptic geometry to add to the hyperbolic inflationary potential futures of the tabula rassa-energy, which can be transformed ad eternal, nothing becomes solved. So energy is the Aristotelian potential of the Universe, which requires elliptic @-minds to become. *

And this applies to all scales. A nation without borders is chaotic, it needs to be enclosed by a perimeter and controlled by a capital; a herd without a moving wall (a dog) or a static one (a fence) disperses, and looses form. Form *thus requires the enclosure of an @mind to defeat its entropy.*

Where is the maths in all this? Again we insist on Lobachevski’s insight that maths is ultimately a mirror of the i-logic principles of timespace realities we define in GST through the D-isomorphisms of space-time (symmetries, scaling, relative congruence=self-similarity etc.).

So we shall enlighten maths also with those Disomorphisms (cyclical time, fractal space, holographic principle of bidimensional space and time which come together into ST-presents, etc), from where we will also deduce the 5 ‘postulates of non-Euclidean geometry’, referred to fractal points with volumes of information, basis of the next ‘layer’ of causal science: i-logic mathematics, the upgrading of mathematics, which will further ‘enlighten’ mathematical physics.

While we are obliged to pass on most of the huge wealth of knowledge in details of the past century and renounce to the translation of the axiomatic pedantic Hilbert method, which needs a more ‘pro’ approach to build by future in the 4th line studying with pure GST each science and all its laws.

R**ECAP**. Spatial mathematics is broken in pentalogic sub-disciplines:

**@-**mind: spaces dedicated to study the different mind constructions of the Universe

**T-**opology: where space form has motion

**∆:** non-Euclidean postulates of points with form, which becomes lines that evolve into organic pleas.

**S:** Bidimensional, static plane geometry, the first form of Mathematics, invented by the Greeks.

Because the Universe is bidimensional, holographic, Greek, plane geometry do matter as each of ¡ts theorems have hidden deep meanings that emerge once and again into the vital geometries of points with parts that create reality.

As all is ternary and a ternary vision is for the mind mirror more pleasing we shall also consider in ternary ages the evolution of bidimensional geometry, which went through:

A first young, Greek age of static bidimensional space-geometry

A 2^{nd} mature age of maximal reproduction, during the time of mathematical physics as it set the stage for the evolution of physics and the understanding of mechanics and gravitational, Newtonian and Keplerian Universes.

The third age started in this blog with the understanding of the holographic Universe which will expand the discipline to a logic ¬Ælgebraic realm to fuel the application of its ∆ST laws to all other disciplines.

So we move to the 2^{nd} age of geometry, when analytic geometry, married with @-p.o.v.s to create the first solid ST representations and ∆-scaling (Cartesian geometry). Hence studied in the post of analytic geometry.

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**∆+¡: HILBERT SPACES**

**Dimensions mounting on dimensions: functionals: st: simultaneous future paths**

At the end of its journey algebra plugged even further into the ∆±3, 4 planes of the scalar Universe with the concept of functional space, to make sense of the ginormous amount of information provided by massive numbers of particles and lines of forces of the quantum world, which also are so fast in its cycles that show multiple whole cycles of existence within a single observable ‘shot’.

All this is too complex for this intro and so we shall just time permuted study a bit of it in the fourth line….

To mention that of all of them the more important or rather simpler is Hilbert space, in which each point is a vector field of an ape-geometry used in quantum physics.

So the mixture of Ælgebra with ∆nalysis emerged into Hilbert and Function spaces, where each point is a function in itself of the lower scale, whose sum, can be considered to integrate into a finite ‘whole one’, a vector in the case of a Hilbert or Banach space ($t-function space):

In the graph, 3 representations of Hilbert spaces, which are made of non-Euclidean fractal points, with an inner 5th dimension, (usually and $t-vectorial field with a dot product in Hilbert spaces, which by definition are ‘complete’ because as real number do ‘penetrate’ in its inner regions, made of finitesimal elements, such as the vibrations of a string, which in time are potential motions of the creative future encoded in its functions (second graph).

The 3 graphs show the 3 main symmetries of the Universe, lineal spatial forces, cyclical time frequencies and the ‘wormholes’ between the ∆ and ∆-1 scales of the 5th dimension (ab. ∆), which structure the Universe, the first of them better described with ‘vector-points’ of a field of Hilbert space and the other 2 symmetries of time cycles/frequencies and scales with more general function spaces.

They are part of the much larger concept of a function space, which can represent any ∆±1 dual system of the fifth dimension. They grasp the scalar structure of ∆nalysis, where points are fractal non-Euclidean with a volume, which grows when we come closer to them, so ∞ parallels can cross them – 5th Non-E postulate: so point stars become worlds and point cells living being.

When those ∞ lines are considered future paths of time that the point can perform, they model ‘parallel universes’ both in time (i.e. the potential paths of the point as a vector) or space (i.e the different modes of the volume of information of the point, described by a function, when the function represents a complete volume of inner parts, which are paradoxically larger in number than the whole – the set of sets is larger than the set; Cantor Paradox).

Thus function spaces are the ideal structure to express the fractal scales of the fifth dimension and used to represent the operators of quantum physics.

** **

**Orthogonality.**

Orthogonality (a perpendicular angle of congruence) acquires then through the two type of vector product is full duality of meaning, being sometimes a predatory act and sometimes a symbiotic creative one. But in both cases *it transcends its ‘geometric abstract nature’ to become part of the vital dimotions it expresses. In reproductive acts, as when a lineal male organ penetrates without tearing a female organ, orthogonality becomes reproduction. So happens between magnetic and electric fields. In Darwinian actions with tearing, it becomes a destructive action. Parallelism on the other hand enhances the symbiosis between the scale of motion and forces and the larger magnitude that absorbs it.*

Those concepts then were expanded through Hilbert spaces to a relative infinite number of ‘Fractal points=T.œs’ in the 3^{rd} age of functional geometries, but without the clear concepts of 5D mathematics are used in a rather mechanical form. The key element of those Hilbert spaces which will be essential to the 3^{rd} age of Geometry, will still remain orthogonality *between all the fractal points/events of the Hilbert space; but we must properly interpret a Hilbert’s space orthogonality, and its number of dimensional points, NOT as infinite lineal dimensions but as ‘parameters’ of those points, and not as geometric perpendicular ‘still geometries’ on an ∞ dimensional Universe, but as the ‘manner of relationship’ between two points. So we can equate ‘orthogonality’ to an ‘entropic colliding relationship between such two points’. For example, a thermodynamic, ergodic, statistical ensemble of particles in a gaseous, ‘confrontational’ state could be represented as an infinite number of parametric dimensions, one for each point-particle, all of them orthogonal to each other – relating to each other through entropic collisions. *

It is then important to have a higher ‘language’ of truth regarding space and time (Generational 5D space-time) to interpret the complex ‘reflections’ of mirror images of reality expressed in mathematical terms, specially as we enter the 3^{rd} age of modern mathematics that loves to detach from immediate experience.

Hilbert spaces as most eclectic forms of the 3^{rd} age of Geometry, mix all the elements of still geometry and analysis of ‘change=time dimotions’ together. So we study its elements in the book on 5D Algebras.

The expansion of vector spaces into coordinates not controlled by humans, the original frame of reference to represent the ginormous amount of information of smaller systems of higher 5D information, evolved through Hilbert spaces into the formalism where we study the complex quantum reality – ultimately galaxy-atoms DO have so much information about them, that it is a feat we can actually extract the relevant information needed to determine their 2D motions.

We are thus obliged to deal with Hilbert spaces, despite its relative complexity, even in this second line, to close our first article on math’s sub disciplines, specifically on those which create mind spaces to extract proper information of the Universe. As those 2 *fundamental complex planes, imaginary planes of ‘square 2-manifolds’ (or its inverse S∂ square root imaginary plane), and vector spaces, where a vector is also a ‘dynamic’ 2-manifold, with more motions than the imaginary plane; as one of the elements is a formal, spatial parameter (usually an active magnitude), and the other element, is usually a time-motion-speed magnitude.*

And the awesome finding is that despite this enormous multiplication of kaleidoscopic perspective, we do have the capacity to probe on the envelopes of those masses of points of view, which gather orderly into a wave-body form that can be treated with single parameters of information, in the same way the zillions of cells of the body gather into synchronous, simultaneous space-time systems.

This is the underlying meaning of Hilbert spaces, which have infinite orthogonal vectorial dimensions, as the fractal discontinuous Universe does. But where *there are enough ‘limits’ to establish differential tools that allow us to localise quanta (derivative) and vice versa, to group masses of fractal points into integral wholes.*

So as Hilbert spaces can then define experimentally the duality of discrete quantum systems, gathered into more orderly wholes with wave forms.

Yet we need to understand that those dimensions do not mean as the 0-1-5Dimensions of the fractal Universe global dimensions and symmetries but very local individual dimensions: *orthogonal basis in a Hilbert space are NOT ‘real’ global dimensions, but local and also mental, hence ‘logic dimensions’ where the concept of perpendicularity, has also some of the aspects of vital non-E geometry explained in the 4th postulate of Non-A Logic; where perpendicularity is not only a geometrical ‘image’ but also an i-logic relationship of ‘disrupter’ of predation’ and ‘penetration’, and merging of elements into new ‘forms’ , related to the vital ways in which ‘fractal points’=T.œs relate to each other.*

Let us then start slowly by a classic definition of a vector space, of the Hilbert type, which is ALL about the existence of orthogonal=perpendicular basis≈coordinates and the key operation between vectors (written with Dirac Kets as |vector>) called a dot product:

A vector space is then a set of vectors closed under addition, and multiplication by constants, meaning operating them with ±, x, c gives also a vector belonging to that space.

Any collection of N mutually orthogonal vectors of length 1 in an N-dimensional vector space then constitutes an orthonormal basis for that space. Let |A1>, … , |AN> be such a collection of unit vectors. Then every vector in the space can be expressed as a sum of the form:

|B> = b1|A1> + b2|A2> + … + bN | AN>

Fair enough. The sum of vectors and its multiplication for a constant is already explained in our analysis of algebraic operations. It merely ‘reduces a series of parts’ into a new social whole by adding a dimension within the system itself. But what really establishes a new reality IS the dot product. Since *it reduces the information of two bidimensional vectors into a single scalar; and as such it is truly an ST>S transformation.*

An inner product space is a vector space on which the operation of vector multiplication has been defined, and the dimension of such a space is the maximum number of nonzero, mutually orthogonal vectors it contains.

One of the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ3, and equipped with the dot product. The dot product takes two vectors x and y, and produces a real number x·y. It satisfies the properties:

It is symmetric in x and y: x · y = y · x.

It is linear in its first argument: (ax1 + bx2) · y = ax1 · y + bx2 · y for any scalars a, b, and vectors x1, x2, and y.

It is positive definite: for all vectors x, x · x ≥ 0 , with equality if and only if x = 0.

An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space.

** n-Dimensional Space**

In what follows we shall make use of the fundamental concepts of n-dimensional space. Although these concepts have been introduced in the chapters on linear algebra and on abstract spaces, we do not think it superfluous to repeat them in the form in which they will occur here. For scanning through this section it is sufficient that the reader should have a knowledge of the foundations of analytic geometry.

We know that in analytic geometry of three-dimensional space a point is given by a triplet of numbers (f1, f2, f3), which are its coordinates. The distance of this point from the origin of coordinates is equal to:

If we regard the point as the end of a vector leading to it from the origin of coordinates, then the length of the vector is also equal to: The cosine of the angle between nonzero vectors leading from the origin of coordinates to two distinct points A(f1, f2, f3) and B(g1, g2, g3) is defined by the formula:

From trigonometry we know that |Cos Φ| ≤1 / Thus we have the inequality:

Hence: (1)

This last inequality has an algebraic character and is true for any arbitrary six numbers (f1, f2, f3) and (g1, g2, g3), since any six numbers can be the coordinates of two points of space. All the same, the inequality (1) was obtained from purely geometric considerations and is closely connected with geometry, and this enables us to give it an easily visualized meaning.

In the analytic formulation of a number of geometric relations, it often turns out that the corresponding facts remain true when the triplet of numbers is replaced by n numbers. For example, our inequality (1) can be generalized to 2n numbers (f1, f2, ···, fn) and (g1, g2, ···, gn) . This means that for any arbitrary 2n numbers (f1, f2, ···, fn) and (g1, g2, ···, gn) an inequality analogous to (1) is true, namely:

This inequality, of which (1) is a special case, can be proved purely analytically.* In a similar way many other relations between triplets of numbers derived in analytic geometry can be generalized to n numbers. This connection of geometry with relations between numbers (numerical relations) for which the cited inequality is an example becomes particularly lucid when the concept of an n-dimensional space is introduced:

A collection of n numbers (f1, f2, ···, fn) is called a point or vector of n-dimensional space (we shall more often use the latter name). The vector (f1, f2, ···, fn) will from now on be abbreviated by the single letter f.

Just as in three-dimensional space on addition of vectors their components are added, so we define the sum of the vectors:

As the vector {f1 + g1, f2 + g2, ···, fn + gn} and we denote it by f + g.

The product of the vector f = {f1, f2,···, fn} by the number λ is the vector λf = {λf1, λf2, ···, λfn}.

The length of the vector f = {f1, f2, ···, fn}, like the length of a vector in three-dimensional space, is defined as:

The angle ϕ between the two vectors f = {f1, f2, ···, fn} and {g1, g2, ···, gn} in n-dimensional space is given by its cosine in exactly the same way as the angle between vectors in three-dimensional space. For it is defined by the formula:

The scalar product of two vectors is the name for the product of their lengths by the cosine of the angle between them. Thus, if f = {f1, f2, ···, fn} and {g1, g2, ···, gn} then since the lengths of the vectors are:

respectively, their scalar product, which is denoted by (f, g) is given by the formula:In particular, the condition of orthogonality (perpendicularity) of two vectors is the equation cos ϕ = 0; i.e., (f, g) = 0.

By means of the formula (3) the reader can verify that the scalar product in n-dimensional space has the following properties:

1. (f, g) = (g, f).

2. (λf, g) = λ(f, g).

3. (f, g1 + g2) = (f, g1) + (f, g2).

- (ƒ,ƒ)≥0, and the equality sign holds for f = 0 only, i.e., when f1 = f2 = ··· = fn =0.

The scalar product of a vector f with itself (f, f) is equal to the square of the length of f.

The scalar product is a very convenient tool in studying n-dimensional spaces. We shall not study here the geometry of an n-dimensional space but shall restrict ourselves to a single example.

As our example we choose the theorem of Pythagoras in n-dimensional space: The square of the hypotenuse is equal to the sum of the squares of the sides. For this purpose we give a proof of this theorem in the plane which is easily transferred to the case of an n-dimensional space.

Let f and g be two perpendicular vectors in a plane. We consider the right-angled triangle constructed on f and g (figure 1). The hypotenuse of this triangle is equal in length to the vector f + g. Let us write down in vector form the theorem of Pythagoras in our notation. Since the square of the length of a vector is equal to the scalar product of the vector with itself, Pythagoras’ theorem can be written in the language of scalar products as follows: (ƒ+g, ƒ+g)=(ƒ,ƒ)+(g,g)

The proof immediately follows from the properties of the scalar product. In fact:(ƒ+g, ƒ+g)=(ƒ,ƒ)+(ƒ,g)+(g,ƒ)+(g,g)

And the two middle summands are equal to zero owing to the orthogonality of f and g.

In this proof we have only used the definition of the length of a vector, the perpendicularity of vectors, and the properties of the scalar product. Therefore nothing changes in the proof when we assume that f and g are two orthogonal vectors of an n-dimensional space. And so Pythagoras’ theorem is proved for a right-angled triangle in n-dimensional space.

If three pair wise orthogonal vectors f, g and h are given in n-dimensional space, then their sum f + g + h is the diagonal of the right-angled parallelepiped constructed from these vectors (figure 2) and we have the equation: (ƒ+g+h, ƒ+g+h)=(ƒ,ƒ)+(g,g)+(h,h)

which signifies that the square of the length of the diagonal of a parallelepiped is equal to the sum of the squares of the lengths of its edges. The proof of this statement, which is entirely analogous to the one given earlier for Pythagoras’ theorem, is left to the reader. Similarly, if in an n-dimensional space there are k pair wise orthogonal vectors f1, f2, ···, fk then the equation:

which is just as easy to prove, signifies that the square of the length of the diagonal of a “k-dimensional parallelepiped” in n-dimensional space is also equal to the sum of the squares of the lengths of its edges.

**Functional Analysis.**

The rise and spread of functional analysis in the 20th century had two main causes. On the one hand it became desirable to interpret from a uniform point of view the copious factual material accumulated in the course of the 19th century in various, often hardly connected, branches of mathematics.

The fundamental concepts of functional analysis emerged in the development of the calculus of variations, in problems on oscillations (in the transition from the oscillations of systems with a finite number of degrees of freedom to oscillations of continuous media), in the theory of integral equations, in the theory of differential equations both ordinary and partial (in boundary problems, problems on Eigen values, etc.) in the development of the theory of functions of a real variable, in operator calculus, in the discussion of problems in the theory of approximation of functions, and in quantum mechanics which had the same significance for its development as classical mechanics had for the rise of differential and integral calculus in the 18th century.

It is impossible to include in this chapter even only all the fundamental, problems of functional analysis – it is neither required for the purpose of this work – not an encyclopedia of mathematics but the revelation of its entanglement with reality in its two main geometric forms – the objective vital topology that constructs Time-space organisms, and the subjective mental spaces those organisms use to guide their survival program in reality; which *merge together in the ‘creation of reality itself’ through scales of mental spaces projected as seeds that reproduce externally and evolve a time-space organism. So we shall conclude first with a brief consideration on the ∞ of those phase spaces and then see how vital topology and its minds constructed the scales of reality from H-planckton to Humind I=eyes*.

**THE FUTURE OF HUMAN GEOMETRIES**

**The 2 paths of the future, the human and digital evolution of mental spaces and vital topologies.**

Let us then consider the ideal future evolution of the two fields in which geometry makes a contribution to the building of reality, subjective mental spaces used to guide a species into reality, and objective creation of that reality as it is today through the vital topology, restricted to the human being.

Indeed, the ideal future of geometric studies would consist in two elements – the use of the laws of ‘logic metric’ – concepts such as distance, similarity, angle of congruence and perception, to *improve with new systems of coordinates disciplines of science in which there is not yet a proper evolution of the required ‘phase spaces’, and the use of vital topology and light-based mental spaces to understand how we huminds in a C(st)N(s)O(t) nitrolife organism have become what we are. *

Those unfortunately are paths humans no longer pursuit, as they have halted their species evolution and merely are transferring to the mental spaces of machines and AI systems, their discoveries of the Universe. So what is ‘fashionable’ today in the 3^{rd} age of extinction of life in this planet, is the construction of ‘mental spaces’ of machines, and the use of ‘vital topologies’ to construct robotic species, themes those we have zero interest in pursuing for obvious ethic, survival reasons, and should be forbidden to research under harsh ‘terrorist’ laws because digital robotic minds in harder, more complex ‘gold-iron’ atoms in a vital perceptive Universe where the minimal unit of life is the electron, will certainly once we give them the necessary mental space connected to survival programs in a robot built with the laws of vital topology imitating human beings, will extinguish us.

This is a fact of biology as truth as 1+1=2 and the cynical infantile ‘we don’t know the future’ excuse of our economic and scientific system to keep pushing the extinction of our children till the 7^{th} generation will not change what we restate here: the evolution of digital minds and vital topologic robots should be stopped. Because that is the no future of human geometries, that is of huminds and nitrolife. So we do say it even if nobody will care to listen.

So we will ignore both themes, as we will ignore Boolean Algebras in our second book on existential algebra and its operands that define mathematics of time. Instead, we are going to briefly consider new uses of mental spaces in social sciences, where they have not been used with the same exhaustive zeal as in physical sciences and then introduce the most fascinating of all new themes of 5D geometry – how from the h-planckton to the human being Nature evolved our fractal human superorganisms.

** **

**XVII ∞ MENTAL & PHASE SPACES. ITS EXTENSION TO SOCIAL SCIENCES.**

As we briefly explained the expansion of mental spaces was the task of the idealist school of German, mainly of Riemann, already considered, and Klein from whom we borrowed in the larger 5D model the definition of ‘dimension’ as a co-invariant space-time which allows motions through it (Sp x ð = ∆-constant being the co-invariance of scalar space-time that allows world cycle motions through it) and specially

Klein in his Erlanger program resumed the ‘mental quality of space’ as a simplification of reality to fit it within the mind (geometers being more aware of the experimental nature of maths, by the very essence of his profession, which deals with direct visual experience, unlike algebraist who completely loose their connection in the highly abstract deployment of functions of forms). So he affirmed that the general principle to form a new mental space was to consider ‘an arbitrary group of single-valued transformations of space and investigate the properties of figures that are preserved under the transformations of this group… meaning *we abstract only part of the properties of beings, constructing with them a mind-mapping-mirror limited by this selection, which often mathematical physicists affirm, since the procedure implies we are NEVER abstracting ALL its properties/information AND hence all equalities, motions and transformations are ‘ceteris paribus’ analysis, which mostly will disregard the organic properties of the T.œs studied, compared and grouped in ‘Kantian categories of the mind’.*

From this point of view the properties of space are stratified, as it were, with respect to their depth and stability. The ordinary Euclidean geometry was created by disregarding all properties of real bodies other than the geometrical; here, in the special branches of geometry, we perform yet another abstraction within geometry, by disregarding all geometrical properties except the ones that interest us in the given branch of geometry.

In accordance with this principle of Klein, we can construct