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∆0 Mathematics: ¬E Space

±∞ ¬∆@ST:


INTRODUCTION: Geometry. Its pentalogic fields.

I Age: S: Bidimensional, still Greek Geometry.

II Age: S=T: Analytic Geometry.

3 Frames of reference as reflection of the 3 topologies of T.œs: flat, hyperbolic and cylindrical.

Canonical curves as reflection of the 5 Dimotions of reality.

III Age T: Differential Geometry. Motion of points. Vector spaces. Hilbert Spaces.

@: Geometry of Minds. Phase spaces.

±¡ Age: Non-Euclidean, Non-Aristotelian Geometry.


Foreword. The content and limits of those posts. 

As usual we recommend to read at least the introductory pages of the main post of the web, to understand what the 5Dimensional fractal Universe an its fundamental elements, space, time, scales and linguistic minds are.

The fundamental purpose of this and all other posts on maths is to UPGRADE the understanding of mathematics as a language mirror of the vital, organic, scalar properties of the space-time Universe. So we are not so much advancing maths beyond the upgrading of Non-Euclidean mathematics and its logic into pentalogic, but interpreting maths as an experimental science, that is as a mirror of the space=geometry and time=logic, algebraic and scalar=digital social properties of all what exists.

Abstract. Geometry is the study of the structures built departing from fractal points with inner parts and scalar content, and as such we have upgraded it with our post on ¬E topology.

Classic geometry though considers points to have no breath, no inner parts. And as such it must be viewed – within the principle of Correspondence – as the limit of ¬E topology when we ‘eliminate’ the 4th dimension of time-motion and the 5th dimension of fractal scales (in the classic view where the 5D ads a scalar dimension to 4D, even if a far more proper way is the study of all of them as dimensional motions – dimotions).

So how can we ‘transform’ classic geometry to 5D formalism? We can’t in its purest form as it represents a ‘simplified mirror’ in itself of the whole Universe, reduced in dimensions to 3 height-information, width-reproduction and length-motion, further on stripped of the ‘motion that gives functions to height, width and length, but as all languages are synoptic mirrors of the more complex fractal reality, consistent within its reduced view, classic bidimensional geometry IS a consistent mirror. And so what we can do is to ‘comment’ on its postulates.

The easiest way to do it is to consider then as we considered the ‘growth’ of a simple still seed of information or fractal point into a reproductive wave, then into physiological networks and finally into super organisms, to perceive geometry as the humind evolution of our comprehension of such complex Universe in ‘ages’ of increasing dimensionality and motion:

As a spatial, S@, language then we can see that geometry started without motion, in holographic bidimensional space, and yet it was a mirror good enough for trigonometry and geometry in the plane to develop to the heights of Greek Geometry. Ever since though as all mind-mirrors that try to become more complex to look more like reality is, with form and motion, mathematics evolved to acquire motion, and more dimensional form, into 3, 4 and many other unneeded spatial dimensions (an error of its still mental nature), which it finally acquired through differential geometry and topology motion, while the 3rd element of scalar depth came through networks and fractals, but still required the proper concept of a fractal point that grows in size as we come closer to it, to fully mirror the real universe.

So this advancement, which we we treat as the new beginning of geometry in other post, completes the journey of geometry in 3 ages to finally embody its perfection as a mathematical mirror of the simultaneous adjacent, ¬∆@st 5 elements of dust of space-time of a pentalogic 5D¡ reality of which we, as spacetime fractals are made of.

Finally to notice the fundamental duality of mathematics between static forms or points and its dynamic view as numbers:

The 5Сmotions of reality mirrored by all languages and systems define the main symmetries between the two ‘units’ of mathematics – the geometric spatial continuous points (S@) and the sequential, scalar, social numbers, ∆ð, both subject to ¬entropic limits beyond which any form of the Universe, including a mental linguistic form do NOT work (but mathematicians ignore, because of their ‘Idealist axiomiatic proof).

So let us study geometry, following that evolution, as a  mirror of the 5 ¬∆@ST elements of reality in mind space

The 3 ages of geometry.

We can in that sense consider Geometry to have evolved as all systems of Huminds into a growing awareness of the fractal scalar (∆), temporal, moving (T), mental (@) properties of the Universe, and its ‘¬’ entropic limits  beyond which any mirror, scale, topological super organism or time world cycle ends.

In a tendency we shall, time permitted map up for ALL languages, FORMS IN EVOLUTION, species, ages and cultures, the pattern of ‘evolution of a fractal image of the world cycle of the infinite Universe (only the whole of wholes is an infinite set), will be shown to have always 3 ages.

Yet the peculiarity of languages of the mind, which are in fact the last most formal ‘old age of information’ of reality, hence unable to go further in static, still form is to have the INVERSE motion to those life forms that start from a young age of maximal motion into a 3rd age of information.

Languages start with stiff, formal still nature and acquire motion only in its final age, to finally understand the more complex elements of the fifth dimension, scales and minds.

So Euclidean geometry started from:

  1. A deterministic, simple, young age of absolute beliefs, and mental spaces (bidimensional Greek Geometry), akin to the lineal kouroi of its sculptural thought.
  2. Into a second 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 ultimate 2 principles of reality (we identify with still languages of information or 5th dimotion and pure entropy of 4th Dimotion), merge together or age of @nalytic geometry, to:
  3. A third age in which lines became points in motion, or age of Differential Geometry.

+¡: to finally understand 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.

So this age, which is NOT completed will be the age we shall advance further, exploring beyond where the @-mind has never gone.

-¡. To finally die when all what the humind can say of mathematics is completed, feeding in its entropic end a new beginning, which is what it is just happening today as huminds become erased and transfer its intelligence to a new beginning of Boolean algebra, and computer chips. Our ethical sense as human beings though prevent us to explore that future, and so we shall escape in all our mathematical posts, the advance of computer thought. Good luck with it.

And so of the many ‘pentalogic forms’ of studying each species of the ¬∆@st




The sub disciplines of 5D mathematics.

‘I know when mathematics is truth but not when it is real’.

We study mathematics as ANY other T.œ (Time-space system) of nature, because LANGUAGES IN A UNIVERSE OF FRACTAL MIRRORS are as consistent with reality as any other system – reason why indeed the axiomatic method is consistent enough to prove mathematics without reference to reality – as ‘each point holds a world in itself’ (Leibniz). However IT MUST BE STRESSED that the purpose of languages IS to help the speaker to reflect reality within the language, and so all languages are ALSO experimental expressions of external realities, even the most abstract of them all Music, which expresses also e-motions and do have all the elements of reality, scales, temporal melodies, simultaneous spatial forms (tone of instruments) and @-linguistic nature.

Mathematics in that sense matters to mankind PRECISELY BECAUSE AS ALL MIRRORS IS AN IMAGE OF THE UNIVERSE in its syntax. But mankind ignores partially that correspondence between the deeper laws of space-time and those of mathematics. And certainly MATHEMATICIANS deny that experimental nature, due to the human egocy (we like to think the center of the universe, in this case the only who talk the only language of god); and specially what matters in praxis about mathematics – they DENY THE LIMITS OF MATHEMATICS, beyond which ‘the functions become entropic, chaotic’ disordered’ of little use. 3 examples will suffice: a differential equation in the complex plane has no solutions if it doesn’t obey the Cauchy ‘limits’; an exponential function of decay looses its periodical form after 10 scales-steps of social dissolution, according to the natural decametric scale of societies; Fermat’s final theorem shows there are not solution for a polynomial sum of ‘cubes’ because superposition in nature, indeed happens between flat bidimensional forms.

If those experimental profound elements of languages and its use were known of course, mathematics would still be an important language, but NOT the only one, and we would win a ginormous wealth of new knowledge about the WHYS of mathematical postulates, its limits, and inversely the laws of space-time reflected in mathematical mirrors.

Screen Shot 2018-05-08 at 5.46.42 PM

All this said t follows then from the definition of  mathematics as an experimental mirror of the 5 ‘entangled’ elements of all systems, an immediate classification of its five fundamental sub disciplines of mathematics specialised in the study of each of those 5 dimensions of space-time:

  1. S@: Geometry studies space. Some key ages/subfields are: Flat Euclidean Geometry, with no motion in a single plane. @nalytic geometry, which represents the different mental points of view, self-centred 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 and 2 Planes.  ¬E Geometry studies fractal points of simultaneous space, ∆-1, & its ∆º networks, within an ∆+1 world domain.
  2. ∆§: Number theory. Discontinuous numbers study time sequences. ∆-1 social numbers, which gather in ∆º functions, part of ∆+1 functionals; hence it is the first ‘stœp’ of:
  3. S≈T: ¬Ælgebra, which is the most important part, as it studies through operand the different Dimotions, from single S=T steps to larger associations of Dimotions,  in more complex ∆+1 structures (Functions) and further on IT DOES SIMULTANEOUS ANALYSIS OF super organisms in space, through the study of its a(nti)symmetries between its space and time dimensions (group theory)… So Algebra is first the science of operands that translate into mathematical mirrors the 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, mirrors for all those elements of the 5 D¡ universe.
  4. ∆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’, hence susceptible to be simplified with ‘social numbers’. We thus differentiate then 5 general applications of Analysis according to the Dimotion study, and the ‘level’ of analysis, from the minute STeps of a derivative, to larger social gatherings, and changes of entire planes (functionals). It is then not surprising that despite being analysis first derived of algebraic symmetries between numbers, it grew in complexity to study changes in functions (first derivatives/integrals), and then changes of changes of functions, as motions between scales of the fifth dimension (higher degree of ∫∂ functions, called functionals).
  5. ¬@-Humind:  Philosophy of mathematics studies the specific @-humind elements of mathematics (human biased mathematics) and its errors of æntropic comprehension of mathematics limited by our ego paradox. As such It is more concerned with those ‘entropy’ limits of mathematics as an inflationary mirror of information, which deviates from reality (limits of solubility of functions, etc.). And it puts in perspective the ‘selfie’ axiomatic methods of truth, which tries to ‘reduce’ the properties of the Universe to the limited description provided by the limited version of mathematics, based in 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. Instead we must develop a vital mathematics and its experimental philosophy.

So THOSE are the 5 sub-posts we shall slowly complete to translate the mathematical language to 5D.

Needless to say in a ginormous field as mathematics is, we shall just keep it simple. We cannot go into a full depth, as we build slowly this work from 30 years of notebooks, so the work will always be in progress, improved and as Descartes who had a quite adventurous, drifting life as this writer did, put it at the end of his ‘geometry’, we shall ‘stop here’ (in my finite point of my world cycle) for future generations to have something to do

So we have selected one the main principles of Geometry. We often copy as the background to add 5D insights some para+graphs from wikipedia and the 3 easy-to-understand volumes of “principles of mathematics and its applications’ by Aleksandrov, which if still alive would love to see as a member of the ‘dialectic=dualist’ soviet school of thought, the upgrading of his philosophy of science. Both they are also copyright friendly.

So far my intention is just to get to a level of work in which a well-intentioned, humble scholar soul realizes the discovery of a new gem of thought to improve with future researchers.

This post deals therefore with the way mathematics mirrors ‘space’, one of the fundamental elements of the entangled, ∆@st universe. We are going to interpret classic Geometry from the perspective of Non-Euclidean fractal points of vital Geometry.

So we first need to do an introduction to the 5 postulates of ® logic. Since then we shall study classic Geometry through its 3 ages, Greek Geometry, Cartesian geometry, analytic geometry and the 3rd age of differential topology, to be able to interpret them in vital terms.

Trilogic and pentalogic: the ages of Geometry

We switch the ¡logic depth level of truth of those posts, between ‘trilogic’ (a diachronic study in classic epistemology of the 3 ages of the system, or its 3 topologies) and ‘pentalogic’ (a synchronous study, of the 5 components, ¬∆@st of a super organism or its 5 scalar dimotions), those texts as usual start with a

TRILOGIC temporal analysis of the 3 ages of mathematics, the young spatial age of Greek Geometry, the balanced Age of Analytic Geometry which ads numbers, @frames of reference to still space, and the Time-motion dominant age of differential geometry and vector spaces.

And then we consider the more complex PENTALOGIC scalar ∆±¡ ‘elements’:

∆-1: Non-Æ geometry of fractal points, the true scalar units of geometry.

Ƽ: Phase mental spaces.

∆+1: Complex Hilbert spaces of multiple dimensions.

¬ But we will ignore the ‘entropic age’ of Boolean algebra (computer minds), which we can’t comment due to the D-element of the stientific method (ethical praxis of science that prevent us to advance AI, the future ‘chip homoctonos’).

As usually it will be done in simple terms, since the purpose of the blog is NOT to make an encyclopedia of sciences, but to show the path for future researchers to complete… by exposing the clear correspondence between different disciplines of science and the 5D ¬∆@st structures, demotions and spacetime Disomorphisms of all scientific systems.

In this context Geometry is ideally suited for af representation of Space, both mind space, and topologic scalar space.

Pentalogic applied to Geometry.

Pentalogic is then the fundamental tool to give deeper meanings to every element of mathematics and connect it as an experimental science with the Universe as it is, to be able to distinguish what of mathematics as an ‘Inflationary mirror’ of information Is truth and what is ‘Inflation, fiction’ proper of all languages, whose syntax allows to create combinations consistent with its inner structural grammar but with no reflections in the outer world.

Indeed, Pentalogic describes reality with 5 Dimensional motions in time parallel to its 5 Structural symmetric elements as an organic whole (Scales: ∆, Topologic space, S, Temporal actions, T and mind languages that organize the whole, @, and the entropic negation-death of the system, ¬), which give birth to ¬∆@st, Time-Space super organisms generated by ‘dust of space-time. And so when we apply pentalogic to any ‘system of Dust of space-time (¬∆@st), we can consider its scalar, spatial, temporal, linguistic parts. As we shall often ignore the ‘destructive’ entropic end of a system ¬, or consider it just as the ‘LIMITS’ of the system in space and time, this simple method of study of any structure of reality, including the languages that mirror reality, will also be applied to mathematics.

I.e. when we study a type of functions or elements or parts of mathematics, we shall discuss:
– its ¬ ‘limits’ in which the reflection of reality holds – which gives birth to fundamental truths of mathematics, such as the need for ‘Cauchy’ limits in the solution of differential equations and integrals.
– Both its algebraic, temporal expression and its ‘spatial, topologic elements’ and S=T symmetries and their use to describe the topology of the Universe – which gives birth to a fundamental law of mathematics, that all algebraic solutions have a similar topological way to demonstrate the model or theorem…
– And finally a temporal dynamic ‘analysis’, through calculus, based in the DEFINITION OF A DIMOTION, AS A DUAL SPACE-TIME element, by the Paradox of Relativity of Physics (paradox of Galileo: every form can be perceived both as a still spatial mind-form and as a motion in time, so we see the Earth still when it is moving, e pur is muove no move).
All this duly illustrated with the simpler findings of duality paradoxes caused by the Duality between wholes which are continuous and parts which are discontinuous, wholes that are closed, circular, curved and parts that are free, lineal (so all mathematical systems can be studied in differential geometry with lineal tangents and oscular planes, and in algebra, all curves can be approached with lineal tangential functions)… And a bit more complex ternary logic symmetries… of the 3 scales of any organism, its 3 topological varieties (the only ones needed to describe all forms, as Gauss proved) and its 3 temporal ages of a time worldcycle…

Pentalogic thus applies the laws of the fractal, scalar, cyclical timespace Universe to EVERY SCIENCE as they ARE THE UNDERLYING SYMMETRIES AND STRUCTURES OF REALITY COMPLETELY ignored by monologic man with its lineal time humind…

So while the data and equations we shall use are the same in any language, whose genetic structure IS INDEPENDENT OF MAN AS AN SPECIES OF ITS OWN in the vital organic mirror-making Universe (maths is an species of the fractal Universe as much as we are one, and so many species can have a fractal mathematical mind, and many might speak memetic linguistics and so on), we shall ground all ‘sciences’ in a far more profound philosophy of ‘stience’, based in the fractal, scalar properties of space, cyclical nature of time, and the organic, biological survival language derived from them. We shall always GROUND all realities in those topo-bio-logic PROPERTIES OF space, scales and time.

And the adventure of the mind that satisfied me for 30 years developing this Magna Opus, against the simplistic æntropic humind has been precisely to enjoy the perfect pentalogic grammar of all what exists, from biological species, to physical systems, from musical scores, to mathematical structures, from wo=men’s e-motions to physical forms, performing myself all kind of pentalogic exercises from the art of painting to the discovery of new laws of physics to the study of the life and death of civilizations, or the patterns of stock curves of reproduction of machines. When you know the pentalogic game of exist¡ence the mind holds not barriers in its perception of the perfection of the fractal Universe.

We have resumed the pentalogic of some disciplines of logic and mathematical languages in the next graph that shows what a structural analysis of a super organism in terms of the 5 Dimotions of pure causal logic and mathematical simultaneous space would look like:

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, this means by breaking its formless asymmetry into different spatial configurations according to that congruence (social parallel systems, complementary gender-mirror systems, darwinian perpendicular systems, or systems that are disymmetric and do not share any reality) , as it builds a casual pyramid of growth from a fractal point through waves of communication into social networks that become a super organism, ready to 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.

Each advanced language of reality thus CAN BE UPGRADED, AND IT WILL BE UPGRADED IN different POST, TO A PENTALOGIC ANALYSIS in its basic Grammar. And so very often we shall start a post commenting on the pentalogic different Dimotional views of the system we describe.

The post of actions and Dimotions and Non-Euclidean Geometry; most of the posts on physics and studies of different species and properties will be casted from a pentalogic point of view. In terms of the structural elements of reality, which reflect those 5 Dimotions, the method most used in the blog will be the study of ‘dust of space-time’, as made of ¬entropic destructive arrows that deny its 4 structural elements, the @-mind (1st Dimotion of perception), its scales (5th Dimotion), its spatial topologies (locomotion and organs) and its temporal ages and worldcycles. So we obtain a more concrete description of a ¬∆@st entity with reference to its organs and cycles of classic science.

Yet in this post we are more concerned with the @-mind languages and its ilogic mirrors. In the previous post we have upgraded the logic postulates of geometry, the basis of mathematics. Because mathematics is in its basic grammar, ‘Geometry’ of points and collections of points. So we have also defined numbers as social groups of identical beings, which in geometry mean they are its next social unit – regular polygons, which are the ONLY mathematical forms in which social points are all equal in Nature, hence numbers of collections of equal beings in pure number theory.

We have then upgraded (work in progress) each subdiscipline of abstract maths into vital mathematics of fractal points = numbers to make it look like the real vital, organic Space that GENERATES all entities in existence along the laws of pentalogic, which we shall try to build up in this post. So while the fractal point is the COMPLETITION OF THE concept first advanced by Non-Euclidean geometry of a point with parts, with volume through which infinite parallels can cross, from where we evolve the limited connection human mathematics has between its theorems and reality; the concept of a Dimensional motion, is the unit which will allow us to upgrade the even more limited command huminds have of the ¡logic processes of creation of futures and species of ∆@st in the highly intelligent Universe.

But we repeat ad nauseam, the truth purpose of logic mirror languages and mathematical geometries is NOT the language per se, but the vital survival of the entity that wishes to perform through its comprehension of the Universe with those languages, its PROGRAM OF EXISTENCE, its vital actions. And so as the point MUST grow in size to FIT multiple parallels, and so it is A FRACTAL POINT, A CONCEPT that substitutes Euclid’s axiom of definition of a point; the first Dimotion of perception will move, feed on energy, reproduce, and evolve socially generating a new plane of existence. So we can cast our original question as to how many Dimensional motions exist in terms on how many ACTIONS are needed to survive.

And it seems then we can answer the first question: 5 actions suffice, as we have explained in our posts on the meaning of life=action, since a system that perceives can then move to feed on energy and reproduce its forms, which will maintain then its 3 topological parts alive; but by doing so it will both shrink its mind system into a lower plane and rise order into the outer territorial larger world.

So the point-mind perceives, moves, feeds, reproduces and evolves socially into larger wholes. And yet even if 5 Dimotions are enough, the mind’s program will mostly assess those dimotions, exclusively from its own ‘temporal subjective perspective’ but GIVE TO THE EXTERNAL UNIVERSE IN WHICH THEY WILL ENTROPICALLY FEED ONLY THE DIMOTION OF ENTROPY which is useful to them. This is what we mean by a monist mind. Humans performs its 5 Dimotions from its selfish point of view and languages as the center of the Universe and so they DO have a complex pentalogic view of themselves, and shun off ‘one-dimensional men’, but when doing ‘external science’ they have a monist view. And that might suffice. In this blog though we shall do objective pentalogic science ALSO WITH all other entities of dust of space-time of the Universe.

Needless to say in a ginormous field as mathematics is, we shall just keep it simple. To that aim, we shall often use texts from one of my fav books of my learning years – the 3 volumes of “principles of mathematics and its applications’ by Aleksandrov, which if still alive would love to see as a member of the ‘dialectic=dualist’ soviet school of thought, the upgrading of his philosophy of science. But mostly from the wikipedia, because of the advantage of its format that can be copy-pasted to WordPress without loosing its formulae.

Plus the obvious fact we are trying as much as possible to make the blog respecting the new copyright laws (a clear form of censorship for the only thing the web is worth for – to evolve the subconscious collective mind of mankind), so wikipedia can be used without those problems and the oldest books of science used here from the Soviet school have also ‘Russian copyrights’ made available as they were state copyrights (so we use Aleksandrov’s for maths and Landau’s 11 books encyclopedia of Physics for physics, besides wikipedia, which were the books that in my teens back in the 70s I used to self-teach myself the essence of both disciplines, and I had annotated with my first insights on 5D, and will time permitted become little jewels as the last Ramanujan notebook (: that will be the day…



“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’ ‘Relativity’ Einstein, on the mental, focused nature of space.

Geometry is the first and most important language of ‘space’. As such we distinguish clearly as in the above quotes, the external, objective nature of fractal, topological space, as the ‘element’ put together to form super organisms, and 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.

Two themes that connect space with the body-waves of organisms and the particle-heads of its minds, to which we should add the scalar, lower ‘flat planes’ of open space, from where the potential-fields extract its motions.  This is the closer concept of space to present physics as defined in v=s/t and vacuum space.

We thus have to consider as usual the ‘multiple 5Dimensional perspectives’ on the concept of space as one of the two key parameters of reality.

As such the volume of knowledge to extract on space is ginormous and we can only treat here some introductory themes. Given its importance for 5D² models of reality we treat on a separate section the definitions of fractal points, the fundamental particle of reality, and its associations, waves, planes, similar, complementary or disimilar systems (parallel, adjacent and perpendicular forms).

Our aim here is to establish the mental nature of space, as a still-informative expression of reality.

Space=form in that sense is the essence of the mental ‘construction of reality’, which transforms cyclical time motions into simultaneous forms, both for an external mind observer, and for the internal cohesion of the Time§pace organism (ab. T.œ): ∆ð≥§@.

If i live longer than expected I will keep pouring in sub-posts specialised treatises in the fundamental types of geometry that matter to fully describe in detail the Universe (non-euclidean 3 varieties, projective and affine geometries, absolute, neutral, pan geometry, fractal differential, analytical – those treated though on analysis and @-frames of reference, and a thorough reinterpretation of Euclidean geometry with its generalisations of concepts such as distances, circles, its axioms, notions, postulates and theorems) .

Different perspectives on spaces.

Our aim thus is to fully understand the key element of space – 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.

This essential task will have 3 immediate ‘Rashomon’ truths (different p.o.v.s on the same subject):

@-mental space: To establish mathematics as a mirror of a larger, more general theory, ‘i-logic space-time’, as the concepts of geometry, which were then carried into algebra, analysis etc, such as distance, dimension, topological form, closed and open spaces, etc. will become general properties of i-logic st, with applications to different sciences.

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 povs. In the graph, Descartes did understand this multiplicity so he publishes 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.

∆-scalar space: To apply them to those other ‘stiences’ in different posts of this blog. For example distance is redefined in modern space theory as similarity, and we find immediately in verbal thought the use of the term to designate precisely this ‘larger general st-quality’ of bio-logic nature, as when we say, I have distanced myself from my friend – meaning we have become different in tastes and opinions.

The realisation of this mental nature of geometry did happen slowly in mankind – it only came with the work of Lobachevski, and it has not yet being fully expressed till this blog.

Γ-ternary organic space:  Finally in the ‘enlightening’ of classic geometry we shall use topology to ‘understand the ternary organic, structure’ of all systems of nature:

T.œs have 3 formal=functional parts adjacent to each other rule b the laws of topology.

Expansion of Geometry: Non-Æ.

In that sense, a third task of the posts on geometry concern the establishment of the Non-Æ i-logic postulates of geometry and its expansion to construct a proper image of reality, as it is. Since there is not a clear notion on the experimental connection between reality and geometry, specially given the ‘strange structure of reality’. A simple example will suffice.

We know the Universe is not Euclidean, or at least not always in all its properties and regions. Yet Euclid made a series of axioms and postulates hardly corrected beyond the fifth. Now we know those ‘concepts’ do have an i-logic meaning, such as distance=dissimilarity, closed curve=ð-membrain; wholes and parts ≈ ∆+1 & ∆-1. And so on. So we can reinterpret many of the postulates, axioms of Euclid and correct them, as they are in fact not truth in a scalar Universe or must be understood in i-logic terms.

I.e. The five Euclid’s postulates are:

1. It is possible to draw a straight line from any point to another point. This means that all points of a present space can be connected in simultaneity; but those points which cannot are NOT in the same present space, and require a different treatment, among other things, a different ‘curved connection’…

2; 3. It is possible to produce a finite straight line continuously in a straight line. It is possible to describe a circle with any center and radius. This implies the existence of infinity but that is not the case, straight lines ultimately find a limit and curve, and closed time-space cycles do have a finite zero sum volume that breaks infinity into parts.

4. 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 realisation of the time acceleration=increase of curvature of smaller beings).

5. The parallel postulate, already known to be false by classic science.

So actually the 5 postulates are all false. 

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.

So again, the 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, but may be considered 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 Sp x ðƒ, according to the metric of 5D: Sp x ðƒ = 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 (:

First the world we can measure and call physical is not the whole 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 then extend the 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, proper of physics, where all electrons, protons or photons are alike.

Yet, as we said we can go even 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 according to 5D metric, Sp x ðƒ = k, the smallest mass is actually the greatest/densest/heaviest.

The rashomon effect of truth. Epistemology of absolute relativity.

We   apply to space as we have done to all other concepts of reality the Ðisomorphic method of multiple points of view, as what reality is depend on the perspective we consider and the multiple functions/forms of all entities in a 5D² Universe.

So what kind of space we deal with, depends merely on the parameters we study. What can then we find in ‘absolute relativity’, the philosophy behind the fractal Universe about truth in this postulate? Obviously the properties of that statement related to the ‘true elements of reality’, @, s, t and ∆, which are the 4 minimal perspectives on the kaleidoscopic Universe, we have called the ‘Rashomon effect’ of truth:

The sum of the 4 perspectives as in the film are required to make a judgment of truth, which will be just a probability lower than 1 – the absolute truth of the event that only happened in the moment of truth, in the location of truth, in the space-time event/ form the truth took, of which then there will be only a linguistic <1 probability of truth.

Only the fact that parts have no particular quantitative relation to the whole, except that they are all interior to the whole – its Spatial definition; that they precede the whole, as the whole needs the parts to become its envelope; its temporal definition; and hence that the parts are in an ∆-1 plane below the whole, its scalar property.

And finally that the humind @, perceives them distorted by our apperception of a single plane.

So the relationships that remain in the fractal 5D² Universe are quite different from those of Euclidean geometry, referred to the 5 elements of reality that ‘create its linguistic mirror truth’/

It is then clear that absolute relativity is NOT undefined. We did explore which much more depth in fact the concept of wholes and parts, and show that when checking with experimental reality those truths, the physical world indeed is better explained with those seemingly weird properties of the non-Æ universe.

 Mental mirrors are neither false or truth, but useful to its type of subjective minds.

So is Euclidean geometry false? Of course NOT. You are missing the point if you think we say so.

What we mean is that as mathematics is a mind mirror, Euclidean geometry is a mind mirror, specifically the mind-mirror of light space-time and its 3 perpendicular dimensions.

It is ultimately how the ‘limited range of light frequencies/sizes’ of the human eye-spectra (from red to violet) sees reality in a single plane of existence:

In the graph, the humind I≈eye is biased and tailored by our local territory, self centred in the yellow color of maximal emission of the sun, and perceiving a very narrow range whose value as all in absolute relativity with no ‘absolute preferred’ scale must be measured as a ratio of  Tƒ/Sp density of information, in this case 789 THz:violet/400 THz:red = 2. So we perceive light space-time which is euclidean in its geometrical configuration with a system of processing information of ‘absolute relative ratio/value’ = 2. 

Which is to say the least a very limited ‘ratio’ (remember absolute magnitudes do not exist but again this doesn’t mean we cannot ‘measure’ within the laws of the relative Universe our ‘capacities’; simply that instead of mass we should use density of information t/s, instead of motion, speed, s/t and instead of force, momentum, s x t, themes those explored in our post on astrophysics and its Active magnitudes, universal constants and parameters of measure).

And so, because we humans develop most of our ‘actions of space-time’ existence in a very limited space-time location, enlightened with solar light self-centred in the yellow spectra, with a very limited range of ‘density of information’, we perceive a lineal=Euclidean Universe in a single plane of scalar existence, truly the ‘bare minimum’ of consciousness and perception of the infinite Universe.

And this of course biases our astoundingly simple theories of reality, and our self-centred view, as we see so little to the point that we think to be the only perceptive being, we think to be the center of reality, we think no other minds exist, and we sponsor pedestrian theories of reality such as the continuum single space-time, the big-bang birth of all realities in a finite time duration, or the abrahamic religious concept that the creator of infinity cares for the infinitesimal humind, as the only one who shares its ‘language of truth, euclidean geometry’, as expressed by the founding fathers of linguistic, creationist theories of the Universe: ‘God is a clocker that waited 5000 years to find an intelligence like his, to admire his clock work’.

And the similar statement of Galileo: ‘The language of Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures’ – Of course of the Euclidean variety (: the harder they fall…

Since minds exist with other pixels and other curved dimensions, with circles where Pi is shorter or larger, triangles who do not ad 180, in its angles of perception, and where the straight angle of Blake’s God is not equal to all others and relative to the size of the observer.

Hence what euclidean geometry tells us is NOT about the Universe but about the local humind and how it distorts it; as each other mind will do creating different mind-universes, some strikingly different to ours.

But again we shall show you how the parameters of absolute relativity embedded in the properties of non-euclidean geometries with its key ratios will allow us to penetrate the ultimate mystery: the form of the minds of other species.

Of them the most important is: r/k: Sp(radius)/Tƒ(measure length), which Lobachevski found to define the ‘flatness or curvature’ of a given geometry such as the smaller the measure rod and hence the perceiver or the larger the object observed, or its radius, the more flat it will perceive the object.

Hence we solve in this manner another element of the Galilean Paradox – why we see the earth flat if it is curved – answer, because we measure it from the smallness of human size, and it has a huge radius. But from a larger rod of measure as that of the earth- moon, perceived by the first astronauts, it certainly looks curved.

In the graph, we observe a curved earth with a long rod of measure – its distance to the moon. While the moon which in reality is more ‘curved’ (smaller circumference), seems to us rather flat, as we are seeing it in close range hence with a shorter rod of measure.

Yet again this duality has only an explanation in 5D, as  Sp: size of the measured world / ðƒ: rod of measure of the observer, is once more a parameter of the scalar 5D metric Universe.

It follows that other world-minds will have other geometries.

Of them, the best known is the gravitational space-time being as per Einstein, the most obvious case of a different geometry. And this is indeed the key concept you must understand: YOU DO NOT SEE ABSOLUTE SPACE BUT A RELATIVE SPACE, PUT ON YOUR MIND BY A CARTESIAN DEVIL. And so we consider 3 absolute masters of the 3 ages of geometry, portrayed in the next graph. This who talks to you being just an epilogue to this stience as for all other stiences and posts of this blog.

A question of course will arise in your brain, can we then construct an absolute geometry with the more general postulates of non-Æ and how it looks the geometry of a world in which wholes are smaller than parts, equality does not exist, angles change form depending of the observer’s size, points have volume, lines are waves, planes are networks, and all this is virtual space (:


Non-Æ, i-logic mathematics, the upgrade started in this blog of human formal sciences of time (logic) and space (mathematics), will be on the long term the fundamental ‘stience’ of the Universe. Its extension is ginormous, and this writer can only hope to introduce the basic theorems and try to put some order on its laid down of mathematical and logic theorems.

To the rescue come the general world cycles, S=t bidimensional and ternary symmetries and ∆-temporal scales.

In brief: the same sequential order in time, STymmetries and ∆º±1 laws of any ‘real system’ of the Universe applies to the linguistic ‘mirrors’ of the mind, which look at that Universe, as the mind merely is a mirror that ‘perceives’ and emits a distorted subjective form into the objective Universe it perceives, ‘converting language in templates of reality’.

This process of order is the intimate relationship between mind and reality, as both constantly exchange information and create each other.

So happens with mathematics, and for that reason it has been so easily to classify mathematics in ‘time ages’, S, st, ∆ ternary subjects of which space is geometry, time algebra and ∆nalysis.

So we shall deal here with space-geometry.

3 masters of geometry:

The 3 ages of geometry and its 3 geniuses: Euclid, who systematised bidimensional, ‘holographic, greek geometry, Descartes that married it with time algebra and Lobachevski, who established the principles of hyperbolic geometry, the geometry of the 4th dimension, and the ‘mental, logic nature’ of space.  We shall complete their work with the formalism of ∆st. 

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 monad’s mind’s world. 

This has been better understood in philosophy, both in the Eastern tradition and the western tradition influenced by them (Soviet, German schools starting in Leibniz, followed by Schopenhauer, etc.) We still believe in horror vacuum though, in vacuum having ‘magically energy’. But energy is ‘form and motion’ not background space. This said and explored in the first line, we are here interested in certain type of mind-spaces, those of the mathematical language, which is the realm of geometry and topology, the first ‘fixed formal space’, the second a form of space-time with motion.

So space-geometry is virtual in a great degree, and as such it started as the purest mind-form of thought, with 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 @-subdiscipline of mathematics.

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 recognised and more profound being Lobachevski, as in science, being human merit is secondary to position in the geography of power, which was german.

The genius of Kazan though had the deepest insights in both sciences – yes, he did also invent topology far before Poincare took over it. As this is also an introductory post on the deeper, simplest meanings of reality, we shall let us guide by his master work and insights, since in the dynamic of truth, the first simple, ‘epic’ age of discovery let us see the forest without an excess of detail.

So as taoism found the yin x yang = qi essential equation of reality (information x entropy = energy), which physicists with all the detail of each tree of the forest still ignore, it was Lobachevski who found the first fundamental principles of ‘pan geometry’, the absolute geometry of reality.

What makes him truly a genius is to choose 3 insights:

  1. The realisation that ‘mathematics-geometry’ is a mental-logic endeavour, 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 indeed 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 realisation nothing of the XX c. explosion of abstract mental spaces to represent reality would be possible.
  2. 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.
  3. Finally 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 worldcyle 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§paœrganism 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 now:

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, forming 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-centred 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 centred 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 idealised 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 maximise 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:

  1. Adjacency (forms that are pegged, hence forming part of the same time§paœrganism).
  2. Perpendicularity, (forms that penetrate and disrupt its inner systems, basis of darwinian events.)
  3. 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 mathematica space.

The explosion of mental spaces. The human mental light-euclidean space.

The realisation that ‘mathematics-geometry’ is a mental-logic endeavour, where function and i-logic thought overcomes ‘spatial representation’, ALLOWED the explosion of abstract mental spaces to represent reality in the XIX and XX c.

BUT and this always amazed me, it did NOT dwell on the human being that our Euclidean space WAS also a construct of the mind, NOT the absolute space of reality but something constructed with ‘pixels’ that mirror reality – in this case light space-time pixels; which only artists of the human eye, painters realized – in a similar fashion to their realisation of the laws of perspective that opened up the geometry of the renaissance (Leonardo, projective geometry, saper vedere) – a task taken in three steps by the 3 geniuses of XIX and XX c. painting:

-Monet, which affirmed, ‘I Paint light’ (impressionism).

-Van Gogh (and his friend Gaugin) who learned to construct ‘different minds of light/colors with the use of complementarily and dissimilarity to produce with them emotions.

-And finally Picasso, who culminated the process affirming ‘ I paint thoughts’, breaking those geometric thoughts first in pure 1D lineal paintings (cubism), then to pain the ‘informative cyclical female’ in pure 3D curved paintings, and finally painting pure thoughts (analytic cubism) – whose final unification I pursued in my artistic youth with my styles of expressionist cubism and conceptual cubism, a brief footnote in my exploration of the Universe.

It is not by chance that before ‘industrial art’ degraded and killed as machines are doing with all the elements of the human mind they atrophy and substitute, painting with the work of Warhol, those 3 painters were considered the fundamental masters of their times. As they constructed the mental spaces in holographic bidimensional 2-manifolds in parallel to the work of geometers, with a deeper intuitive philosophical insight. 

Since euclidean space is indeed the construct of the mind made of light space-time and its 3 perpendicular dimensions, width-magnetism, height-electricity and length-speed.

So finally we revis(it)ed that old lady, these days so abandoned which is Euclidean geometry, the geometry of the mind observing light space-time and its 3 perpendicular dimensions (plus frequency color- the social dimension not considered in E-geometry).

What makes this geometry so important is, once we have liberated 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 colours? 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 colours the evolutionary element.

Lobachevski’s pangeometry. Where is the 5th dimension. ¬ æ

Thus algebra (and analysis specifically concerned with the processes of social numbers that sum and emerge or rest, and divide and plunge down the scales of eusocial love of the 5th 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’).

Now, it will surprise the reader to know that the Universe is not a continuum, but as all fractals it is discontinuum. 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. It was discovered by a German adequately named Bach-mann, for its musical architectonical rigor.

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.

Indeed, 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 5th 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.

The final frontier: ∆@st.

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 ‘Rashomon effect’ of multiple truths that as a puzzle does, must be regarded all as, mental models of reality which extract the ideal ultimate properties of it (symmetry, perpendicularity, parallelism, adjacency, congruence, betweenness/continuity, and so on).

What ∆@S≈T will do therefore is to reorganise 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… 

We shall thus vitalise and reorder with the awareness that we are all topological space-time beings those 3 ages of geometry, the static young greek age of humind euclidean geometries, the 2nd age of multiple non-e geometries and the third age of topology where ALL THE PROPERTIES OF SPACE, scales, ternary variations in space and motions in time, come together. And as Lobachevski started up the 2nd and 3rd ages, he must be considered with Euclid which likely closed inversely the first age in the eclectic third hellenistic age of the Greek culture (see historic posts), and Descartes who introduced the third variety and made the first marriage of space and time, the triad of absolute geniuses of this discipline (never mind all the Germans, Gauss, Euler, Riemann and Hilbert, which as always among goths, had much complexity, little clarity and a huge ego – Hilbert, in his ridiculous foundations of geometry, ‘I imagine points, lines and planes’. So we shall close with a harsh critique of the baroque, formal death-age of vital geometry, stultified with his work even if of course, as usual the ‘germans’ among so much gothic ‘fog’ show some work of merit.

Geometry and its ages.

Geometry deals with space. We shall use thus this post for the essential descriptions of space in mathematics, and structure it through its 3 ages. Then in the future we shall expand it on the different sub posts.

 Thus this post is dedicated to the most spatial branch of mathematics and its 3 ages of evolution, from static geometry to moving topology.

As the web grows, we break down some lengthy posts into sub-disciplines, according to the ternary method, as all has always 3+0, ∆•st perspectives. This is the case with spatial mathematics,  broken in 3+0 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.

In this post thus we shall merely summarise those ages, from a ‘temporal, perspective’ :


I Age: Spatial, Greek Era: static bidimensional Geometry: The holographic principle.


The first age of Spatial analysis stumble directly with marvels of holographic beings, as it was concerned with something coming seemingly out of nothing, the symmetries and relationships of bidimensional entities, with a membrane (curve) enclosing a self-centred surface. And this was a huge world as almost all the laws of geometry can be proved in a bidimensional plane of information; while in parallel humans resolved the laws of bidimensional perception through a form of art called painting. It is in fact little known that painting and geometry were closely related a painting arrived first to the laws of perspective, which would define some laws of projective and more complex geometries.

II Age: S≈T:  Analytic  Geometry.


It was though an intermarriage within mathematica 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 (numbers are forms thought the greeks and equalled by all means to points, which they are not ‘exactly’ so – hence the paradox of defining √2 and π geometrically but find when calculated arithmetically that it never ‘closes’ the circle of 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 philosophical questions that would have marvelled the Greek though were forgotten by the praxis of this intermarriage, which will from then on beyond those ‘finitesimal’ opening in a numerical representation of a synchronous curve, allow to prove always arithmetically a geometrical form and viceversa, provided geometrical solutions to 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 generalisations 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 ‘anchorage’ into reality that any experimental science needs to survive.

3rd age: Non-E and Temporal Topology, Fractals


It is precisely this connection with physical reality, the one that would represent a huge guidance and show the ‘light at each step’, on the advance of geometry to avoid the ‘inflationary nature of languages of information’ with 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.

∆-geometries: 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: IStructure of the present, INNER space-time body-wave through ITS scales by the understanding of topological networks and fractals, 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.

Which redefines points as fractal points with inner scale volume through The 5 Postulates of i-logic geometry.

IT IS THEREFORE THE COMPLETION OF GEOMETRY as an experimental language able to explain all forms of real space and its temporal logic structure.

The key element of Non-E geometry is always to have in mind the 3 regions of 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

The structure of Tƒ<TS<Sp, 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 5th 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 visualise the ‘meaning’ of algebraic and analytic equations both in abstract and mathematical physics.

Anti-stientific Geometry

It is left though to comment on 2 branches of geometry, which break the lema of all stiences, which ‘study its species of space-time’  for the ‘benefit of man’; that we shall not occupy with as we consider them anti-stientific either because it departs from reality and becomes just inflationary false truths (Hilbert Axiomatic method) or are dangerous for the future of humanity (evolution of a digital mind):


Hilbert’s axiomatic method is not so much the final formalism but the disconnection=death of the mind of information of the mathematical ‘world-system, with the Universe, as it is based in an ego-trip postulate: the concept that mathematics is not experimental and has no need to refer to reality because ‘Hilbert created the language of god, ‘imagining undefined points, planes and so on’. This ‘religion of mathematics’ is false as Godel and lobacjevski proved, but is a dogma for modern mathematicians who have become obtuse and metalinguistic as all baroque ages of the language are. So we substitute the axiomatic method with the 5 postulates that do define points, lines, planes, congruence and non-euclidean flows of energy and information crossing a mind-point-singularity

<<Spe: Death of human mathematics: Digital Future: COMPUTERS ‘SEE’ Geometry.

So it is only left the death and explosion of the human mathematical mind atrophied and substituted by digital computers, which means the future age of transhumanism and extinction of man, if humans do not stop the evolution of A.I. Algorithms of information, neural networks and digital thought, as we are building in a stronger faster metal support, a new more efficient mathematical brain that is displacing us from labor and war fields, and starts to ‘see’ mathematically with visual A.I. the world, so geometry finally becomes NOT only a reflection of the vital symmetries of the Universe but a vital language in itself.

LET US then follow a more terse procedure through the 3 ages of geometry, starting by ‘holographic 2-manifold, pure mind-still’ geometry, the simplest mind constructs, which appear i-logically first in the history of the humind, as…





We shall not introduce the rest of Greek geometry, notably the conics of Apollonius as he did. I just don’t have time to write so much detailed scholar accounts, but rather close this introduction, with the transitional elements which will give birth to the new age of Analytic/Algebraic geometry where we will study those curves with the use of the 3 frames of reference, Spe-cylindrical, ST-cartesian, Tiƒ-polar geometry. And how they ‘deform and create’ the geometric minds of the Universe, introducing some of those minds and postulates according to the choice of coordinates.


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.


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.

  • In that sense, the most important element of bidimensional geometry is the understanding of angles, parallelism and perpendicularity. As we dealt with non-E in depth in our analysis of the mind-mappings of he Universe and topological laws, here we shall concentrate on the analysis of the 3rd Non-E Postulate of Parallelism (which fosters social evolution), vs. Perpendicularity, and so the connection of points into geometric figures, and its projections and deformations that keep their ‘knots’ unchanged, related to:
  • The geometry of points into numbers, that is, the pythagorean school, which rightly found the Tetraktys, the perfect geometric number, and play with its meanings.
  • Trigonometry, and the importance of angles, which are the fundamental first mental perception (also in physics, were the unit h, must be interpreted as the minimal space-time being, a cycle of angular momentum with h radius.
  • 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.

1D: sine<cosine

As usual we need first to consider the pentalogic of trigonometric functions:

@: 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.

So trigonometric functions, the first to appear, as 1D perception is also the first ‘action’ are operands for the first Dimotion of perception.

1D: Of those operands then we consider the simplest ‘operands’ as those who act directly on a point-number. We need then to find the operand of the first Dimotion, self-centered in the point, perception.

Since all points are also numbers as all have internal structure, we can operate both in S-pace with points and topologies and in scale with numbers to show those internal parts with numerical values:

The graph shows the elements of the point-number in which an operand can act. Even though most operands will act through a similarity ≤≥ symbol in two different elements. Yet the simplest operands are those which can act in a single point-number.

An entire field of Algebra or rather geometry is then the study of the laws of perception of the circle, based on the laws of trigonometry.

And as usual a rule of ‘time discovery’ is that the simplest most important laws are the first to be discovered. So after defining those functions the ancients discovered the laws that involve triangles, as the fundamental S=T transition of a ‘lineal triangle’ into a sphere, is its rotation. So a triangle becomes a pi-sphere when it adds a dimension of ð-cyclical time. And its 3 vertex becomes the ‘apertures of perception:

Law of sines

Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Some identities equate an expression to a different expression involving the same angles.

In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles (as shown in the diagram).

The law of sines (also known as the “sine rule”) 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, ensure a non-distorted perception of distances to the outer world.

Another law involving sines can be used to calculate the area of a triangle. Given two sides a and b and the angle between the sides C, the area of the triangle is given by half the product of the lengths of two sides and the sine of the angle between the two sides:

Area=Δ=1/2 ab sin⁡C or in terms of the Law of cosines ,  the area of the triangle is:

which establishes the same law of equal perception of those 3 ‘membrain’ points of its internal ‘vital energy’.


Each of the different laws of bidimensional plane geometry then can be studied as a reflection of the efficiency of vital Dimotions in geometric form. Let us merely consider a simple example, enlightening the main discoveries of the father of mathematics, Mr. Pythagoras. We can by no means be exhaustive, just a seed for others to explore further ¬Æ geometry. So we shall bring two insights one complex first for the pedantic scholar maybe to take serious this blog, the other simpler to show both ‘extremes’ of the unending enlightenments of ∆st.

You think you knew the Pythagoras theorem?

There are though more insights to it. Let us consider some ‘Complex ones from the 3rd age of ¬E, and some simple ones’.

Pythagoras revis(it)ed: metric spaces, curvature and strenght=efficiency of geometrical figures.

Pythagoras theorem turns out to be one of the most ‘invariant’ theorems as it is basically defined to create a metric, and resumes now that we have liberated a notch further ‘mental space’ from representation and make it affine to information for the mind into this simple concept: 3 elements occupy more space/require more information than 2 for a mind to describe them, and this means they ‘require more distance=information’ when we ad them: in the graph (A1-A2) + (A2-A3) ≥ A1-A3. And so the fundamental ‘mental spaces’ do have a metric. In the ‘simplified’ jargon of ¬E we said that:

“A metric space is a set of arbitrary elements, to be called ¬E points, in which a volume of information called a ‘distance’ is required to define them with 3 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 calle Distance≈information) is 0 as points have no volume, but in ∆st, you need at least 1Dimensional 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)

So we can ad a new third axiom to introduce ∆-wholes into those mental metric spaces.

3. SUCH arbitrary set of 3 ∆-1 points/elements suffices to establish an ∆-‘figure’ of the next scale, making ternary ‘networks’ the most efficient ‘number’ to generate a new plane of existence, as they are the ‘minimal distance-information’ to form it.

Of this 3rd element we shall make use to understand why ternary systems suffice to reality, why triangles are the strongest configurations, and ultimately the true understanding of circles as triangles with curvature and ‘openings’ equivalent to π-3 are to be found.

Then we can consider 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 our ‘unifying constant’  for any active Magnitude of any physical scale (we shall generalise as M – see unification equation of forces in all scales.) 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 why?

Needless to say almost all spaces with experimental use are metric spaces, such as:

1. ∆-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.”

So 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 the enclosure and singularity, the @-constrains ARE ESSENTIAL to define and solve any problem, and indeed in mathematical physics we shall find that really without an enclosure-singularity ‘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 the energy is the ‘aristotelian potential’ of the Universe, which shall required elliptic @-minds to ‘be=come’. 

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.

Some simpler enlightements, Mr. Fermat. Proofs of the bidimensional nature of the Universe. 

Yet the ‘theoretical minimum’ and very first principles of geometry can also be enlightened again.

Indeed, when I was young there were few people I truly admired. One of them was Einstein, a simplifier. So one of his things stroke me when young – a new demonstration of the Pythagoras theorem, the first theorem of maths, which seemed so simple and without nothing deeper, based in the ‘scales’ and self-similarities of its ratios, a pure proof of the ‘fractal paradigm’, which lead us to a basic concept:

Scaling works better for simpler systems – euclidean geometry, potential memoriless death processes of dissolution, and its e-constants. 

And so the deepest revelations on the scalar Universe will come ALWAYS from the simplest theorems as they are at the base of all motions through the filters between scales that cribe most of the information of the system.

Let us then give you an ‘Einstein tease’; another proof of Pythagoras.

We have told you that the universe is a bidimensional hologram of space & time like forms; or a ternary one, if we add the result of mixing space and time into an st PRESENT, or a fifth dimensional game of scales of ternary and bidimensional holographs.  So from those simple principles we shall in maths find many enlightening thoughts. Now, the Pythagoras theorem is obvious: the bidimensional sides of a triangle ad up to the bidimensional diagonal. Thus we write: A(ST) + B(TS) = C (S). But why A is a function of T (obviously it is the smaller), b of ST and c of S? Ultimately because the triangle is a fourier sum of waves, and c is  always the largest volume of bidimensional space; hence the entropic ∆-1 flat field that feeds both the body and the head of the system.

And so b and a will be the ST-body and Tiƒ-head of the system, expressed in terms of the entropy they consume; which in the most perfect systems, will in fact be similar in a balance body-head, wave-particle, ST≈tiƒ, which tends to be the rule.  So much then we can extract knowing GST of the relationships of the simplest forms of mathematics (and also of the complex ones).

But Alas, if x²+y²=z², according to the holographic principle, why not x³+y³=z³? Precisely 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 (a computer did it in thousands of pages, called the Fermat Grand THEOREM, is a clear proof of GST.

Two more examples will suffice on the original work of Pythagoras illuminated by GST:


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:
screen-shot-2017-02-06-at-17-39-41 screen-shot-2017-02-06-at-17-40-20

In the upper left graph the 3 ST-ructural sub-systems of the human body, which are its structural forms (membrane, sustain and motion).

In the bottom the 3 ‘chemical Tiƒ’ systems or hormonal brain (creative, distributive and reproductive)

In the middle the 10th system (nervous system)

And in the upper right 2 of the 3 ‘P$t’ Subsystems lymphatic, digestive (not portrayed the urinary/excretor system).

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 waves.

 Finally Pythagoras is remembered by a 3rd discovery, that of the harmony of music.

Schopenhauer, by far the best philosopher of the industrial age, said that music encodes the secret program of time in its rhythms.

So thought Pythagoras when he found the perfect harmony of the ‘fifth’, the musical chord obtained by plugging 1/3rd or 2/3rds of a string, attached to a point of future and past, the birth and death of the frequency of its world cycle.

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:


screen-shot-2017-02-11-at-12-41-47The beauty of i-logic mathematics thus will the reside in its capacity to express (as music does in human arts, with its 3 elements, ∆±1 3 scales, T-beat, ST-melody and S-ynchornicity of instruments) the purest GST laws.

Now this was the beginning of geometry, and we have seen how much GST can extract from it.


Then there was of course pi, and the legend has it that when Pi-thagoras found it to be not exact, but a fluctuating dynamic open and closed number which always made the circle both a spiral inwards (-π:tiƒ) an outwards spiral (+π… Spe) and a steady state (St=pi closed cycle) fluctuation, as a ratio that transforms 3 Spe into one Tiƒ, he hang himself desperado that the world was not perfect and static. When I found it on the other hand, i started my journey.

If you are mathematically inclined the easiest revelations with a ‘little bit of thought’ are enormous if you can ‘switch’ from a continuous lineal description of time as duration, with the more ‘detailed’ and real description of cyclical, discontinuous time cycles that move in ‘steps’. Indeed, consider for example the car, as a whole that seems to move in lineal time, but in fact it is the sum of the cyclical steps of its wheels. Both are equivalent mathematically but the analysis of each wheel step gives us MORE information. And this will be the constant of this blog: you will find a lot of new information and laws of science that were hidden by the simplification of using the continuous sums of time cycles, ‘erasing’ in the process the form and frequency of those cyclical steps.

So we close this intro with the meaning of Universal constants.

The vital axioms of geometry. Reproductive motions. Discontinuity postulates. Attraction.

Now, with the failure of Euclidean geometry, its definitions of points, which we have updated, as well as lines, congruence (equality) and planes, the idealist Hilbertian school ran away and didn’t define them, but imagined them. Instead they were substituted by what today passes as the foundational axioms/postulates of geometry, which are not correct. As the pangeometry or absolute geometry of Bachmann (1970s) which closes the evolution of the discipline shows. Specifically we must reject as Bachmann does the postulate of ‘continuity’, and Dedekind’s concept that real numbers ARE numbers NOT ratios, because they LIE on the real line.

It is important to separate geometric ideas from numerical ideas, as both represent two different ‘elements of reality’, space-time forms, symmetries and cycles, when we include the concept of ‘motion’ (not defined by Euclid), which geometry represent vs. the 5D absolute arrow of wholes, discontinuous planes and future time (5d social growing numbers). Thus there is NOT as mathematicians have thought for millennia a needed correspondence between points and numbers. It is all more subtle, and precisely 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 that we are in two different Universes. The number system can be properly reconstructed from geometry only when those differences between a single space-time plane represented easily by geometry vs. the causal relationship between several 5D planes, the realm of algebra is taken into account.

And nowhere is more obvious than in the theme of continuity, which does NOT exist in the arithmetic world, though it can be considered in the geometric world, from the perspective of adjacency, and motion as reproduction of form, in adjacent places of a single plane of reality:

Screen Shot 2016-06-05 at 18.53.23In 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.

Now for geometry what matters of all this is that Continuity IS NOT TRUTH on 5D Algebra. Continuity is NOT needed to define Geometry in its ‘Absolute version’, as Bachmann proves.

The number system constructed from the reality of a discontinuous world IS different from the familiar real number system if we drop Dedekind’s axiom of Continuity. This opens the way to further expansions and new geometries which complete the realist model of the Universe, and will be dealt in the future in line 4. Instead 3 less strict principles suffice to explain the different virtual continuities as perceived from the 3 elements of any system (| x O ≈ Ø) – the elementary principle of lineal continuity, the circular continuity principle, and the Archimedes and Aristotle classic axioms of relative space-time proportions suffice. Dedekind’s axiom is then a different concept – that of a limit or constrain, a barrier, a ‘potential well’, a Universal constant that cannot be crossed. As it is indeed a gap. 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 always a Maya of the senses that eliminates the dark holes between the 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 as the Paradox of the ‘horizon view’ shows, we do not perceive a 96% of darkness. Darkness is eliminated to picture an enlightened 4% as if it were all the reality.

So this is another ‘angle’ to fully grasp why the total perception would be in reality hyperbolic, as it would have to expand the angle to include that 96%. Thus if we were to include it all, the angle will be convex, expanding constantly into a hyperbolic, much richer world, which is what Lobachevski and 5D metric proves.

Or in terms of the duality space-time stillness-motion. As geometry is a fixed formal view, and 5D metric tell us that Tƒ x Sp = k, hence as we becomes smaller, Ñ Sp, ∆Tƒ time accelerates, inversely, if we have a still, geometrical perception with no motion, we would have to decelerate the time cycle of smaller beings, and hence expand its space size. So if we were to see the faster motion of cells and atoms at the slow speed we exist, we would have to expand its space size and we would come to the fascinating paradox that a cell slow as we are, is as big as we are after that geometrical expansion.

This fact, that 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 substitute Dedekind’s axiom.

Indeed, Aristotle’s postulate, which substitutes real continuity by relative, angular perception of distances, from the perspective points of view, with deep virtual-world-mind implications is in place to 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 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.

In precise terms (we cannot be in an introduction so precise and exhaustive, but from time to time we will get deeper in some details, and since the lack of true continuity in the 5D universe is so essential, we are a bit deeper in this), 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.

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, cannot ‘contact’ each other. We shall to stress the point of the discontinuity of reality, rebaptized them with the ‘dis’ prefix added on 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:Screen Shot 2016-06-05 at 18.53.15

In the graph, the continuity principles are in fact limiting concepts of boundaries and laws of perpendicularity, which define the discontinuities, closedness and connections between networks of points. What matters then to reality is NOT the obtuse concept of a 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? 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?

Then 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. 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 ciments the Union between both. Those are the true important questions about the undefined terms of ‘laying on’, the only undefined concept of the Axiomatic method we have not upgraded till now in i-logic geometry (we dealt already with the undefined Hillbert’s concepts of points, which are fractal points, lines, which are waves, congruence, which is relativie similarity, Non-E 5 Postualte, which is the definiton of a mind, and so now we consider the concept of lay on, the key element to complete the 3rd postulate of relationships between systems, and we shall call it generally the postulate of ‘Intersection’, which as always will be a ternary principle or dual principe with 2 or 3 solutions (essential fact, which we latter will use in ¬æ to study the meaning of imaginary numbers, negative roots, dual solutions to polynomials, etc. extracting the fundamental meanings of imaginary numbers and negative roots at the core of the understanding of physical mathematics, quantum theory and relativity).

Thus we define 3 cases of the postulate of ‘intersection’, which defines the undefined terms lof laying on and substitutes Dedekind’s continuity axyom such as:

– A point M of intersection between 2 relative futures, Tƒ(closed figure) and Sp (open figure), either belongs to Tƒ or Sp or belongs to both figures. If it belongs only to a figure, Tƒ or Sp, the figure is the predatory, dominant element of the intersection, and the event will be a Darwinian space-time event, in which the submissive prey element will loose its form. If the point belongs to both figures, the event is an event of symbiosis, and both systems can form a stable, social new whole.

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 things that expand the concept and show its power to describe reality and set the foundations of other key branches of mathematics (Boolean logic and set theory):

– A system is dual always. There can be either a Union or an Intersection. Yet Union and Intersection are slightly different concepts in advanced i-logic geometry:

An 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 beong to A, as when you eat a rabbit that no longer is a rabbit but becomes your aminoacids.

On the other hand an Intersection will be defined as a true sharing of those common points, so neitehr dominates, A Ç B, means the Ç part is now the connection that ciments 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 relationshp. Fermions share a boson and that ciments an attraction between both. Predators share a prey intersecting their territories of hunting, and so on.

Further on the sharing is more intense and symmetric when the 2 systems are closed Tƒ elements, as in the figure of the ciruclar continuity postulate, where we can clearly see there is an acbd region shared that truly pegs both systems together.

Indeed, in the intersection between a line and a cycle, somehow the line seems at disadvantage, and in fact in most real events the line becomes absorbed andtransformed 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 neclosed 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 propotiion is tthat the line OM has KILLED the circle, targetng its zero-oint. 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 frctal spaces.

And when we talk of motion, we can see that first region abcd as a region that ‘doubles’ its ∆-1 desnity 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 then will become the seed for a 3rd œ child of A and 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 5th dimension, towards the gradient region of maximal density of force – so we move towards the attractive vorte x 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 desnity gradient, ∆ (Tƒ/Sp), more finitesimals to ‘double its form’, more ‘energy-space quanta’ into which reproduce, and so we can slo see according to the tenrary fractal principel of mutlifunctionality, motion as the feeding process of an entity, A in the graph that feeds on the field, on the gradient region of more density, fallin unescabpaby by its greed of motion towards the region of maximal charge-mass. The feld is controleld 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 absrobed 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 symbotical, parallel, so both use the field in equal conditions to input information and reproduce the son.

– 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 trully 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 iswhat 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 postaltes, and a physicist in its capacity to explain the whys of key processes of hysical 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.

All this means of course that 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 5th 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 ‘layed on’ being is not into the being, it does not touch the being.

Now as it is proper to consider it here, we did somewhere explain that unlike classic geometry, a straight line is NEVER created only by 2 points. The classic definition of Euclid, naively accpeted by those suppposed Hillbertian r=evoluionaries: ‘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 so obvious – that 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 2 old A2xioms 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 A2c2ioms, ab. A2c2 🙂

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.

For example, 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 Sp 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.

Now those 3 axioms suffice to prove (what would take us hundreds of pages, but mathematicians do know this to be a fact 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.

Screen Shot 2016-06-05 at 18.52.59

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-:

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.

The many false assumptions of classic Euclidean and axiomatic geometry.

Now, if we ‘continue’ on the critical analysis of the Axiomatic method that has substituted the ill-defined terms of Euclid, a resume of those postulates will follow with some comments on it, before we consider the relationship between continuity and number theory to introduce ¬Æ.

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.)

As always we shall use pieces of different files of 30 years old research, often extracted from classic books (generally speaking I used in my introductory original research 30 years ago very often 2 books, which I loved because of its simple language and direct connection with reality of the dialectic school of the Soviet Union, today defunct, Landau’s 11 book series on Physics and Aleksandrov, Mathematics, which I found recently as an e-book and so I am now copying directly from that edition (adding obviously our comments):

The axioms fall into five groups.

Axioms of incidence

  1. One and only one straight line passes through any two points.
  2. On every straight line there are at least two points.
  3. There exist at least three points not lying on one straight line.

Now Aleksandrov ads:

It may appear somewhat strange that in the list of axioms there is, for example, this one: “On every straight line there are at least two points.” Surely in our idea of a line there are even infinitely many points on it. No wonder that neither to Euclid nor to any one of the mathematicians up to the end of the last century did it occur that such an axiom had to be stated: it was assumed tacitly.

Well surprise, it IS NOT proper, as we have shown to be the case. Why is then still there? Because plainly speaking what modern mathematics did is to blow up the concept of ‘straightness’. So today a straight line is not the usual line, but something else: a geodesic on a surface, a chord of a circle, or whatever. And this is the error that leads in Einstein’s work to the concept of curved lines instead of fractal points with straight lines. A straight line IS a straight line, and has 3 points to define its straightness.

It is then of more interest with this new understanding the 3rd axiom of incidence: there are at least 3 points lying outside a line, which means that there are at least an equal, and most likely larger Universe outside a world (taken by those 3 points). Since we can trace another line outside of it, which might or might not be parallel, cross ours.

Now 3 points not lying in a straight line define a plane. But this is needs a deep analysis in i-logic geometry. We need at least 4 points to define a plane with some ‘height’ that is an ST system born of the holographic fusion of bidimensional space and time elements, as we shall see in advanced geometry. To state here then that 3 points no lying on a straight line define an angle of perception. Or triangle, which in i-logic geometry is subtly different. Or when considered full fractal points (but then of course with inner depth and inner ∆-1 points) a ternary system, Sp x Tƒ =ST, and in that sense we must define a plane.

Therefore the need clearly arises in i-logic geometry, for stating accurately and exhaustively everything we have to postulate of those objects that will be described as straight lines. And so INSTEAD OF making the definitions less clear, which is paradoxically what Hilbert did with its non-definition or ambivalent definitions in i-logic geometry we will make them more clear and exhaustive.

The same applies to all the other concepts and axioms. I-logic geometry goes deeper into them…

Axioms of order.

  1. Of any three points on a straight line, just one lies between the other two.
  2. If A, B are two points of a straight line, then there is at least one point C on the line such that B lies between A and C.
  3. A straight line divides the plane into two 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).

The axiom of order only applies to lines, where we can set a forward, backward, ‘parity’ order, but NOT to circles, where we cannot as we return to the point establish a relative order. Thus C comes before D or after A in the circles drawn above? In a simultaneous space only single time present we cannot know, as we will circle around and come first or last depending on the way we move around. Motion thus becomes essential to order cyclical paths, and remember all lines are steps of a larger cyclical part. And orientation also. It is not the same order if we move from A –cbd, or from Adbc.

Thus as soon 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 diversity 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 now becomes the most important Axiom, that of motion, which was not defined properly in Euclidean Geometry.

III.  Axioms of motion.

(A motion is to be understood as a transformation not of an individual figure, but of the whole plane.)

  1. A motion carries straight lines into straight lines.
  2. Two motions carried out one after the other are equivalent to a certain single motion.
  3. 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 is in that regard, the basis of Group theory and modern physics, and in this we must state it means a great advance, but again, we shall translate its abstract jargon into the understanding of how motions and perpendicular intersecting or uniting motions define the space-time actions of systems, its Darwinian and connecting, creative and destructive paths. According to their similarity or difference, as a motion ‘transforms’ one form into another, or pegs it into an intersecting or uniting event.

Now the 1st and 2nd are trivial, not in the sense of its many and obvious use in physics, relativity theories, vector spaces etc. but in its rather intuitive nature in our worldview. The 3rd is more interesting, 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 a motion in SP (lineal motion) and a motion in Tƒ (rotary motion) are put together to return or NOT the point to the same initial state, hence the commutative closed or non-commutative open nature of two consecutive Sp and Tƒ actions.

Now all motions by definition become with time, closed actions. So the question here 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?

The labyrinths of existence in which the path is more important than the goal, which often kills the path and the existence itself, tells us there are 2 solutions: the point can be returned by pure motion, or by reproduction, which recreates the point and exhausts the motion of the being, which often dies after reproduction (from octopus to arachnids). A closed path kills a system, and in the larger field of cosmology and the deeper field of metaphysics we should wonder, how many steps requires a thermodynamic system, and how may steps requires a Cosmological system to return to its point. It is the Universe closed, infinite but finite, bounded? It will return to a point?

The answer is ‘yes!!, the universe is infinite in timespace but finite in time space because the number fo repetitions and variations is smaller than the number of fractal domains, ‘broken by a line’ or rather a cyclical enclosure (3rd axiom of order). 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 (this will be attended in 4th line last post on ‘God’, time permitted or by other finite Sancho in other planet who helped him a bit more to complete his work )-: are those other yous repeated connected entangled to you, as quantum systems are? 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 pythagorical 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?

Axiom of continuity.

  1. Let X1, X2, X3, ··· 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 X1, X2, ···, 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).

Axiom of parallelism (Euclid).

  1. 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).

  1. At least two straight lines pass through a point not lying on a given straight line that do not intersect the line.

We shall in the 3rd and 4th line redefine all those essential concepts properly and get rid of IV continuity (Dedekind’s in simpler language) and V Parallelism according to Euclid (studied in depth before).

Projective geometry.

Of the many singularity mind-spaces with wide implications, perhaps the richest one is projective geometry, as it shows how many properties of the outer world can be ‘assessed’ and remain invariant in the mind-construction of useful information about reality, and again it shows that the essential properties of space-time can be conserved when the mind observes the outer world, making possible the Universe of monads to work.

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 Tƒ cycle and the Spatial plane, which is the origin of all realities in all its creative combinations:
Screen Shot 2016-06-05 at 18.52.45

In the graph, the projection of a bidimensional tall Tƒ 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 Sp 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 Tƒ≈Spe symbiosis that ties up both elements into ti≈es of space-time (an old formalism which I no longer use, as I am converting all variations into Tiƒ and Spe 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 5th 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 Sp-plane DO conserve the Sp-relationships of the Tƒ cycle, enhancing them as the lines DO grow in size, when we move from a Tƒ implosive form into an Spe 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 Tƒ element and vice versa’. Why this is important should be obvious to the reader: 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 property which is conserved is that of a straight line of 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 tƒ circle is now projected into a Sp 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.Πa 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’, Tƒ->Spe (reproduction) and Spe->tƒ (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 Tƒ 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’
Screen Shot 2016-06-05 at 18.52.37

Thus a system can actually perceive measures by having a ‘sensorial’ set of (ABCD)’ points in its membrane to calculate such proportions. This 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.

Thus affine geometry is the 4th element of the geometries that study the inverse, dual actions of ‘perception’ (Spe/Tƒ), whereas an Spatial surface is ‘reduced’ into the Tƒ-point and reproduction, (Tƒ/Spe), where an informative Tƒ form is imprinted on an Spe-energy plane.

Accordingly affine geometry is related to the ‘growth in size’ of a system, through a lineal process of expansion in space.

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:Screen Shot 2016-06-05 at 18.52.29Where 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 Sp x Tƒ system is absolutely natural to the Universe, its essence, which easily conserves all the properties of the fractal Tƒ 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 Tƒ systems into Spe systems, which transform circles into their equivalent open spatial forms (parabolas), affine transformations, which are growths DO conserve the essential Tƒ<Spe 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 5th dimension.

In other words, the lineal affine growth of a system and affine geometry on the whole is justified by the ∆(Sp x Tƒ) = a K process of growth of the system, within the 10x 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 fo 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 3rd 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.

NOW, those brief examples of Greek 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


As the web grows, we break down some lengthy posts into sub-disciplines. This is the case with spatial mathematics, as today 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:

  1. A first young, Greek age of static bidimensional space-geometry
  2. A second 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.
  3. 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 @-povs to create the first solid ST representations and ∆-scaling (Cartesian geometry). Hence studied in the post of analytic geometry.




@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. This implies that it includes also the study of ‘linguistic mind spaces’, which systems might use to represent reality from a distorted point of view, or a subjective selection of the parameters of time and space (Phase spaces).

This post studies the mind which in mathematics is reflected in its frames of reference. So we talk also of the pentalogic uses of @nalitic geometry:

  • @ is its fundamental perspective, as observed in the introduction: 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.And 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-topology defined by the 3 different frames of reference and ∆-scalar symmetry represented by the Complex Plane:

Because there are 3 topologies and then scales it should exist 3 type 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.

The 3 topologies of all systems in a single scales are described by the O-polar, |-cylindrical and Ø-hyperbolic Cartesian plane in the o-1, 1-∞ time-space scales; which are further expressed by the complex plane in its ‘squared form’.

All systems have ternary topologies with 3 motion-information-reproduction function that require to absorb energy, move and perceive to keep in existence. Thus the simplex aei actions exist in all systems of Nature.

Yet because huminds have so much difficulties in emerging into new scales they have not a clear philosophy of the 2 fundamental ’emergent’ 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 most developed inflationary mirror in its last ‘excessive’ age of form.

Analytic Geometry though is more important than the study of the relative frame of reference and topology of the Universe.

Its fundamental plane, the Cartesian plane, due to its perpendicular structure, Is according to the 4th postulate of Non-Euclidean geometry the best form to represent precisely ‘perpendicular’ space-time inverse states, in any of the 5 Dimotions of the Universe. 

S-T Symmetry. We can consider the basic duality of Space-stillness vs. Time motion, in 2 other main frames of reference:

-Abstract mental spaces (Complex, phase spaces), which are the spatial mental static view vs…

-vector spaces which are the dynamic, temporal view often related to an ¡-1 ‘field scale’..

Physics uses vector spaces for locomotions in lineal space and time. And  phase spaces, for more evolved ‘concepts of space and time’. Of special interest for the human scale are the analysis of the ternary points/states/ages of matter in those phase spaces.

-T: symmetry: it is the study of functions on those planes, as expressions of the real dimotions of existence, essentially the fields of ‘Mathematical physics’ and its special frames of references.

All this is fairly well understood by scientists. What it is not so clearly grasped is the magic of 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, as both exist in different scales.

Which bring us into consideration the importance that in @nalytic geometry have the 5 new non-Euclidean postulates, specially the 4th of congruence based in darwinian perpendicularity and social parallelism for the understanding of vector spaces, cross products, dot products, equipotentials and lines of forces, which happen in different planes of 5D and affect different parts of an entangled, field/wave/particle super organism.

As a general rule we shall state that particles are Darwinian over fields in which they prey and waves are parallel to the field in which they ‘slide’.

While there are also metaphysical questions about the nature of the ‘light space-time we perceive’. It is REAL? OR IT IS A phase space of the mind? The question is not so odd as it seems, but we shall treat it on future specialized posts on Relativity theory and its measure of time clock dilation and space construction.

To advance that the question was already poised intuitively by Riemann in his famous dissertation on phase spaces and color spaces…


This post is dedicated to the mirror of mind-systems in mathematics, which are its 3 main frames of references of analytic geometry, as it studies reality from a 0-point of view, similar to that of the mind.

Another key theme proper of @nalytic geometry is where we set the functions and operands as a ‘background space-time’.

So in this post we shall focus in those 2 themes: the @-mind capacity of frames of reference to reflect the mind’s point of view. The 3 topological standard planes and the scalar complex plane.

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.

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 reality they 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…

Numbers then, are dominant in ‘scalar’ social properties and sequential temporal causal properties, best to describe the inner ‘vital energy of those points’, its discrete configurations, and all the functions that express its Dimotions. Numbers it follow are more complete than points as time-motions are more fundamental than the spatial mental forms we make of them.

However in the entangled Universe ALL its ST-beings as fractals of the whole participate of the 5 Dimensional Motions of existence, ¬∆@st:

¬; So they will have limits, ¬, in space and time beyond which they break its form into entropy…

∆-scales internal to the T.œ, (cells, atoms, etc.) and be inscribed in a larger world, ∆+1 that sets those limits…

S: will display the 3 Spatial canonical topologies of any 5D system in its own organic parts (|imbfields, bodywaves & particheads)…

T: evolving through an arrow of increasing information parameters and decreasing motion through its 3 temporal ages.

@: And to survive and function, it will need NOT to be blind in that vital, entangled Universe, so the T.œ, will have @ singularity-mind of relative stillness that will act as a center of a referential language to be able to perform the 5 @tions of existence that reflect in a fractal, minimalist cyclical bit of time quanta and local space territory, its needs to perform those 5 Dimotions of relative exist¡once also within a limited number of perceptive scales, where to feed on lower entropy ∆-4, to move, lower information, ∆-3 to perceive, lower energy, ∆-2 to feed, and emit lower ∆-1 cells to reproduce, evolve palingentically, emerge as a clone, ∆o of the being, and thus survive beyond the ¬ limits of time=death and space=decay, its T.œ will experience.

As all is a fractal mirror of a larger world-whole, all this properties will not only define fully all what is needed to know of the world cycle of exist¡ence of a relative T.œ, but will be able to describe, its parts, its wholes, its organs, and even its languages, whose internal structure will reflect in its grammar and parts, also all those elements. 

So we will also be able to describe numbers, with the Pentalogic of ¬∆@st, as a mirror reflection of the larger Universe. And yet because huminds understand near nothing of the larger picture, of the fractal game of exist¡ences, and are as all other points, self-centered in its mind-language, limiting the vital properties of all other entities, ab=using them as open systems of entropy from where to extract motion for its @tions (actions performing by the will of a mind), this clear picture of the entangled, self-similar universe will be distorted by the æntropic principle of antropomorphism (man as the center and highest point of existence, which therefore must debase everything else including the intelligence of the Universe to come on top) and entropic behavior (man as the top predator destroyer of worlds, to absorb energy for its own creative processes), we must consider all humind’s languages as mirrors of reality distorted by the æntropic principle and so half-truths this blog will try to enlighten so the nebulous picture by lack of information or too clear one, by simplification, acquires the richness of grey proper of the pentalogic entangled Universe.

IT IS ALSO worth to study in this symmetry the…


But to the synchronous view of the 5D adjacent minds of a Supœrganism we can add an analysis of the metalanguage of @nalyltic geometry. That is we shall apply the Disomorphic method in its main temporal view, with  a temporal study of its evolution in the 3 ages of maths:


In that sense @nalytic geometry started with the work of Descartes and my fellow countryman Fermat. But it really was already embedded on the study of conics by the Greeks; 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 point of view of the observer, or world of geometry.

We shall then as usual make a parallel analysis of the evolution of the discipline through 3 ages of increasing ‘form’, and its structures in space – so we shall study also in @-nalytic geometry, the different mind views of reality according to coordinates. Let us start with this simpler first spatial view, of @G (ab.) and how @G mirrors the 3 points of view or states of any system which distort our image mirror of reality within a given mind.

Now in the classification of the different subjects of MATHEMATICS as a mirror symmetry of the 5Dimensions of space-time, @naltic geometry, ∆nalysis, S=T algebra, T-heory of numbers and S-geometry, as we enter the final 3rd age of mathematics, all of them reflecting the kaleidoscope which is the Universe, where those 5 elements constantly merge to give us more complex 5D² reflections, an epistemological theme is how to order all subjects, when they are in fact today merged, as they are in experimental reality (we do NOT see 1D beings, or 2D but complex ‘knots’ of 5D elements).

So in true form, we should INCLUDE EVERYTHING in every post, with a particular bias given by the dominant element in its ‘ceteris paribus analysis.

This would make it very redundant as it is in fact now, and require as i hope to do before dying, or rather lowering my 5D into a 4D entropic explosion of little faster-thinking insects, to cut off a lot of redundancies to make it readable.

Yet the division of the 5D st factors is largely a personal matter of distribution of subjects, which now has such ‘extemporaneous’ elements as the symbols accessible in wordpress (: well, more seriously I am doing an effort. So in this post, which if we were to be strict should be concerned WITH EVERYTHING IN MATHS, as everything in maths is viewed through human @-minds and its ternary functions/forms of perception, equivalent to the 3 parts of the organic Universe.

But that would be silly, so instead we shall use analytic frames of reference in all other posts but kept their theorems, when they are used for algebraic s=t or ∆nalysis or pure geometric thought, in this other sections.

Here then we shall only be concerned with the ‘frames of reference’ in themselves, and the 3 types of ‘functions/forms’ they describe, hyperbolic, euclidean or elliptic geometry and the dimensions we use to study each of those possible s<st>t symmetries, frames of reference of 3 lineal dimensions of 2D-manifolds and of 3D-volumes, and its related ∆st theorems.

This arguably could also be studied in geometry, from where analytic geometry starts. But as we are using the post on geometry mostly for the advanced third age of geometry, non-euclidean points with volume, fractal and topological geometries, which ARE the most important modern branches that connect with the understanding of T.œs and its organic parts and structures, we keep it for this post-taken into account also that analytic geometry in its origins was used mostly by mathematical physics to study precisely those 3 type of curves first in one and then in 2 dimensions.

It was also the beginning of its use for mathematical physics, so we shall include at least the original experimental understanding of maths as the best mirror of astrophysics, a stuff which obviously could belong to the articles of astrophysics, where we shall only consider Kepler’s orbital geometries and some comments on 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 on the sacred language of God, the cartesian graph, its ellipses and comets. So he thought below reality there was such infinite single line of time and space, drawn by God, his ‘alter ego’.

In brief, 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.

So instead of doing as usual an analysis of the ‘3 ages of the discipline’ in time, we might say we shall do here after a brief introduction to mind representation in @- geometry, an analysis in space of the 3 main sub-varieties of mind-views, frames of reference and geometric figures there exists.

The a priori reality of other ‘systems’.

The importance of the specific mind species and information we define with a given frame of reference cannot be stressed enough. We have repeated ad nauseam that ‘spaces’ are artificial constructs of the mind, to ‘navigate’ reality; built with the features of the forces of information available to them. They are the a priori ‘Kantian’ categories that deforms our reality and viceversa, knowing the properties of a given space is essential to know the type of mind and species that navigates thanks and through it. 

For example, 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.

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. And be aware that such Universe is only a monad’s world, different from many others.

But as we treat the general theory of ‘mind-worlds’ in our post on Non-E geometry, we shall leave it here at this stage.

We shall finish with the complex plane for mathematical physics and number theory as it is the closest frame related to Temporal numbers) and close the subject with a short view on vector spaces, and its more complex purely spatial frames of reference ‘across ∆-scales’, that is Hilbert spaces and functional operators.

Thus we shall close with ‘far removed’ Hilbert spaces, which are of interest to understand the most far removed scales of reality – those of undistinguishable zillions of particles… 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-motions.




then become the expression of the 3 topological parts of the being: ‘cylindrical frames for lineal limbs/fields’, hyperbolic cartesian frames for body-waves and polar frames for particles-heads. In mathematical physics the simplest way to define an event according to frame of reference will show which element IS acting on reality.

It also follows that as humans use the Cartesian plane, in depth, we can add this specific @-humind element of distortion of reality (human biased mathematics) and its errors of comprehension of mathematics limited by our ego paradox.

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.

In this post thus we shall deal with the multiple aspects of @-nalytic geometry and Philosophy of mathematics. As usual is a work in progress, at a simpler level than future ad ons in the fourth line; and using some basic book texts of the Soviet school, which had the proper ‘dialectic’ logic and experimental sense of the discipline, which western idealist, German-Based axiomatic schools lack.

In the graph, as usual we shall find a Kaleidoscopic relationship between the Dimensions of reality and its ‘image-mirror’ in any element or language, in this case in the ‘frames of reference’ of mathematical minds.

So a more detailed 5D ‘rashomon’ effect on the discipline of @nalitic geometry gives us 5 sub-planes, each one a main frame of reference that studies reality from a distorted perspective.

So the 5 ‘world views’ or topological forms and its planes of reference are related to the 5 dimensions of reality:

3D hyperbolic Dimension: the Cartesian, ∑∏ graph.

1D vortex: the polar, ð§ cyclical graph.

2D Lineal motion: the Lineal, cylindrical $t.

To which we can add several spatial graphs that portray from a ‘static mind-view’ the upper and lower planes of the 4-5Dimensions:

-All kind of phase spaces which detach 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.

And 3 more generalised perspectives closely connected to those scalar dimensions.

-0-1 D Generator Dimension’s main plane: It is the Unit circle, expressed mostly in probabilistic temporal terms (where 1 is the value of ‘existence’ – the happening of an event/form), coupled with its 1-∞ equivalent plane (Measure theory) or:

-5D Social Dimension: It is 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).

-4D entropic Dimension: Finally as a generalisation that breaks the whole into all its points of view, generalised coordinates, with infinite individual points of view representing the statistical ensembles of entropic particles, 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 Dimensions 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 dimensions.

Each of those graphs will therefore BE USED mostly to study problems regarding forms and functions of those 5 Dimensions.


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

Descartes’ theory is based on two concepts: the concept of 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.

Generator Equation. The 3 Points of view and its geometries & topological planes.  Coordinates

In 3 dimensions, the 3 frames of reference become the 3 bidimensional planes of any of the infinite fractal systems of the Universe:

$t(Toroid field/limb)<Si≈Te (Hyperbolic wave/body)>§ð (spherical head/particle)

Each of the 3 fundamental 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.Screen Shot 2015-10-13 at 16.01.49

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.

The Organisms of the Universe are assemblies of geodesic curves perceived by each of the 3 elements in its corresponding coordinates.

This gives also a simple rule to know what kind of element we are studying in different sciences and events (the Se, STi or Ot element):

‘The simplest frame of reference in which a problem formulates indicates which is the ternary topological element we study’. For example, if we can formulate the problem in polar coordinates with an equation simpler than in cartesian coordinates, we will be studying a TO element (as when we formulate a gravitational or charge problem in polar coordinates). If it is easier to formulate in the toroid or the simplified Cartesian graph (a toroid opened along a z-cut), it will be an ‘Se, limb/field problem’; which is the case of most physical problems of ‘lineal motions’; and so on.

Vital Geometry, symmetric properties.

It is then essential to understand that each frame of reference and the formal figures of geometry described by it, will have vital organic properties as describing what best suits for each dimotion of existence. A simple example will suffice:

  • The reasons why the sphere or circle are the best forms for the first Dimotion of informative gauging, aka perception.

Symmetry is the central concept of Group theory, which became in the XX century the ice in the cake of the whole structure of Algebra.

By symmetry we mean in 5Ð Algebra, which as all branches of mathematics born of the spatial, bidimensional ‘still’ work of the Greeks LACKS A TRUE COMPREHENSION OF THE PARADOX OF GALILEO: S (form in space) = Time (motion), but uses it profusely (so differential geometry is based in the concept that a line is a point in motion), the capacity of all systems to complete a zero sum world cycle, through a motion that returns the system to a present undistinguishable new state.

Symmetry thus IS ESSENTIAL TO THE ENTIRE SCAFFOLDING OF THE 5D UNIVERSE, albeit ass all concepts of 5D once we understand the basic laws of pentalogic, has a more dynamic view.

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, who 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.

Symmetry though then must be connected to the different Ðimotions and its ‘requirements’ to perform the vital actions for which they are conceived.

I.e. the circle IS the perfect symmetry for still Dimotions of perception, as it will turn out that from any pentalogic point of view, it maximizes the stillness, by symmetry, by its capacity to focus as a fractal point all lines of communication that fall into its focus, by having the minimal perimeter, which maximizes ad maximal its volume of information and disguises it in an external world, and so on.

However, when we consider the 2nd Dimotion of Locomotion, which is the process of displacement in space while retaining the form in time, as it implies the reproduction of form, of information in other adjacent region of space, the less information to be displaced, the faster reproduction will happen. 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. And so on.

Vital geometry thus will be essential to explain the relationships between stable forms of geometry in Nature.

Generalized coordinates from the pov of the o-point. 

Obviously, it is only meaningful to speak of the relative position of any point. Then measure must be done in a relative Universe from a single point of view, which includes all the relative parts of the super organism, which can then be treated from that point of view in stillness, as a rigid body and simplified with Generalized coordinates, which will take one of those 3 elements of the being, preferently the still point, hence the particle-head, as its center of reference.

In the old system under the Æntropic principle if the human observer is the relative mind-view of the whole system, he then can choose the Cartesian coordinates as an alien, external point of view. Yet it is far more convenient to choose the objective internal coordinates of the system, and its still particle/head as its o-point.

For example, in 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 three particles which are rigidly fixed in a triangle, then the coordinates of the third particle must satisfy the 3 equations.

Thus, the nine coordinates of the vertices of the rigid triangle are defined by the three equations (1.1), (1.2) and (1.3), and hence only six of the nine quantities are independent. The triangle has six degrees of freedom.
The position of a rigid body in space is defined by three points which do not lie on the same straight line. These three points, as we have just seen, have six degrees of freedom. It follows that any rigid body has six degrees of freedom.

In 5D terms it means, the ‘singularity’ or center of reference while being the most important of the system, disappears to describe it, as 6 degrees of freedom are equivalent to 2 points.

So generalized coordinates are more real and simpler (occam’s principle), and in nature 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.

It also follows that mechanics the science of 2D locomotion evolved from subjective human pov (Newtonian) to generalized coordinates (Lagrangian), which is how today professional physicists cast its laws, and we shall also 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.

What follows then 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.

The conclusion of this TRINITY logic, is that we are mathematical organisms, with topo-logic properties, which give birth to biological, organic assembles of ternary functions/forms and those systems do have an O-point of view, its frame of reference.

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 (STi, Se or To coordinates), which correspond to the Cartesian, Cylindrical and spherical, polar coordinates of science

The Universe is a sum of a series of action, ∆aeiou, of them, the first action is perception by an observer, ∆o, of a field of energy, ∆e, to which it 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.

This is all what we should describe when we reduce to minimal cyclical space-time actions the total reality of any self.

But to do so from a merely formal point of view, we do need to consider first the point of view that kicks out the world cycle of actions of any function of existence, ∆o, which is thus the minimal and first Unit-action of the Universe.

And this point of view and its relative frame of reference, will start the comprehension of an external reality.

Thus the departure point of any mathematical course, should be the definition of the 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:

0.                                                    O-point x ∞ Universe = constant, static frame of reference.

Thus the Observer is not only the dominant element of those 5 actions  in the real Universe, but also the initial ‘point of view’ of any mathematical analysis, and analytic geometry rightly the first branch of mathematics to be studied.

And as it turns out 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, centred 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’ the Universe constructs its vital geometries that we call ‘existential beings’.

And so the observer’s causal, logic, cyclical, informative sentient properties that allow it to perceive time cycles are the first questions to inquire.

To, is the first element to describe in reality.

And it is a frame of reference, an observer, an inertial point in relative fixed form with no motion that can perceive and map the Universe, an O x ∞ constant mapping mind. The observer.


The Cartesian revolution: hyperbolic PRESENT coordinates mix time and space.

By the coordinates of a point in the plane Descartes means the abscissa and ordinate of this point, i.e., the numerical values x and y of its distances (with corresponding signs) to two mutually perpendicular straight lines (coordinate axes) chosen in this plane (see Chapter II). The point of intersection of the coordinate axes, i.e., the point having coordinates (0, 0) is called the origin.
With the introduction of coordinates Descartes constructed, so to speak, an “arithmetization” of the plane. Instead of determining any point geometrically, it is sufficient to give a pair of numbers x, y and conversely.

Comparing equations with 2 unknowns with curves in the plane.

Descartes’ second 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.

So essentially Descartes gifted the static bidimensional plane geometry of a ‘variable motion’, and give us the capacity to study an st-evolving system in space-time:


Thus, to each algebraic equation with two variables, F(x, y) =0, corresponds a completely 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.

Since it was 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.

But there is more to that simple scheme from a GST p.o.v.

The multiple S, ST, ∆ applications of Cartesian graphs.

It soon followed that the discovery of a system which represented dimensional symmetries of space and time – both forms and motions – would yield enormous capacity to model the Universe with the mind as it was the case – soon to add the ∆nalysis of scales with the awesome concepts of ∆-1 derivatives (finitesimals) and ∆+1 integrals, treated independently of this texts and then the complex plane, ‘magic’ on ∆scales.

So Analytic geometry provides mappings on:

1: T<S: of solving construction problems of continuous with discrete temporal computation, such as the division of a segment in a given ratio; thus adding time, hierarchical features to geometry.

2: S>T: of finding the equation of curves defined by a geometric property (for example, of an ellipse defined by the condition that the sum of distances to two given points is constant); thus again adding time functions to space forms, in this case a key function to define 2 physical systems controlling with equal force if we consider an internal of time, the same territory of space (Kepler’s 2nd law).

S≈T: of proving new geometric theorems algebraically (see, for example, the derivation of Newton’s theory of diameters; and conversely, of representing an algebraic equation geometrically, to clarify its algebraic properties (see, for example, the solution of third- and fourth-degree equations from the intersection of a parabola with a circle.

∆: which will lead at the end of the road to the understanding of ∆nalysis, derivatives and integrals, a notch above in the complexity of the ternary elements of reality.
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 can now ad the insight of knowing its direct homology with the S, T, ∆st elements of reality.

 THE 3 POVS OF MATHEMATICAL MINDS. |-cylindrical, O-polar, Ø-cartesian.

The ternary nature of the universe will again be evident when we consider the other 2 canonical ‘coordinate systems’:


The other two planes then will be the polar þiƒ plane and the Cylindrical, toroid plane, which will give us 3 different ‘views of the Universe’. And it follows naturally that by ‘changing’ the equations of systems from one frame of reference to another we change often the topological analysis of them – 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).

Further on 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):                    [Spe (cylindrical) <ST-Cartesian> Tiƒ (polar)]∆i(complex) 

The most complete thus, is the conic, Cartesian coordinate, with the negative and positive, inverse directions, distorted from the point of view of the observer, self-centred in ∆o; which requires a bit of paradoxical logic, far more expanded in the articles on @-mathematical minds.

Now, there are 2 other ‘spaces’ worth to notice, to explain all this fractal space-time complex world:

Complex space ideal for studying ∏imespace world cycles as the previous ones, so we keep it on the section of T-heory of numbers.

Hilbert space, the space elf the fifth dimension which each point is a world in itself, which we shall study at the end of the post.

The Rashomon Truth and the a(nti)symmetries of creation on graphs. Parallel creation

We have stressed the fractal division of all realities in 5 elements, and the fact that the most important of them are the operandi, as two ‘asymmetric’ beings, let us say as we are in analytic geometry, 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. 

We can then redefine a ginormous number of mathematical elements in geometry vitalising its meaning with this essential ‘fourth postulate of ¬Æ geometry and we shall do so, slowly as we keep improving those posts.

Let us be clear enough, the most important law of the Universe is the fourth postulate of non-Æ geometry and it will come all over the place, as the ‘angle of communication’ determines so many facts, in all sciences. And so this is a huge field of @nalytic geometry to explore, which spreads to Analysis (a derivative is indeed a parallel, so when there is NO parallels there are NO derivatives, NO communication between ∆ø and ∆-1, NO possibility hence to create a super organism of ∆±1 scales); and illuminates complex analysis (where the rules are more… complex)…  And so on.

At least in my youth exploring ∆st, when I did have a permanent Nirvana orgasm of perception of all those laws I recall to have had some of the biggest orgasms literally when I realized after all the orgasm is the sensation of a cycle invaded by a line in parallel by penetration (:

More seriously the fundamental penetration of the line into the cycle which moves along the line, increasingly ‘tightening’ its grip on it, is the cone that will generate all the curves, called conics. So we do have two fundamental modes of mathematical creation by parallelism in the plane:

  • 2D creation in a holographic bidimensional 2-manifold by tangent parallelism.
  • 3D creation by penetration of the line into the curve.
  • All other approaches ‘cut’ perpendicularly and ‘destroy’ one of the two elements. As this what perpendicularity is all about – it is real, it ‘cuts’, it hurts’. Remember in ∆st, we follow Godel and Lobachevski and Einstein: mathematics as all languages are real mirrors of a higher reality.

All this said the first thing we have to understand about graphs is that they must be seen as all systems of reality under the kaleidoscopic view of the rashomon truth. Indeed, we just have considered the @-pov on graphs, but we are left to explain 3 other povs, ∆, s and t. And then refer the graph to the specific really we observe. Let us illustrate this with a single example, that of a graph of an exponential function, with a growing; accelerated speed towards its asymptote.

2D $: From a spatial point of view, the growth of the function makes no sense, but it does its space below, the distance covered.

1Dð: The curve though as most of them are Y (s) = ƒ (t) functions do represent a time dimension perfectly, as it is not a constant line, the dimension is one of accelerated change, hence a o-1 dimension of growth and/or a vortex of acceleration towards a singularity point (1D)

∆±i: And this from the pov of scale means the function accelerates towards an entropic dissolution (i.e e¯ª, which is a decay entropic process) or inversely towards an emergence, as in resonances, distribution equation or the limiting case of distribution, a function that grows exponentially towards the 0-point, in the simplest seemingly function of them all, the 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. 

Creation by parallelism, annihilation by perpendicularity.

How dimensions combine to create form is an essential feature of the duality between symmetric parallelism and perpendicular annihilation (antisymmetry), which special plays in geometry. In 2 dimensions the tangent that gives origin to a derivative is the creative form of ∆-scales, the circle that turns around and peels off a wave cover is the creative form in S<≈>T dimensions.

In 3 dimensions the conic that can be seen as a line penetrating and dragging a circle, which turns faster in smaller spaces around it, is the ∆-creative form. The harmonic oscillator in multiple of its devices, the S<≈>T creative form.

We shall then find for all systems of the Universe the dualities of tangential retain associated to different dimensionalities and split in the duality between past to future to past ∆§calling and S<≈>T present transformations of topological cycles into hyperbolic waves as the result of combining with a lineal motion.

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 then 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, the barrier of the cattle, or the sepherd dog, the predators, etc.

The best known case of those process 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.


LET US study the diversity of mind geometries in the Universe. We shall consider the 3rd and most important theme of Riemann’s work to ad the insights of dust of space-time, how to operate with multiple dimensions on one hand, and differential finitesimals and finite infinities between ∆§cales.

We started this post considering that space does not exist but it is a creation of the mind. The first to fully accept this was Lobachevski and so he explored non-euclidean hyperbolic geometries where multiple parallels can cross a point. We are going to further evolve conceptually once we have explained that hyperbolic geometries belong to the vital space inside the ‘membrain’ (@-mind) his geometry.

Indeed, hyperbolic geometries have a fundamental element of distortion of the mind – to convert a fractal point with a volume into a micro point, hence bending our perception of the straight line in the adjacent regions to the point, which in objective reality means the region is ‘faster in time’ . So before understanding it, we need to understand first what a point-particle is in a scalar Universe.

The enlargement of a point-particle.

Euclidean Mathematics, as a language that represents reality with simplified spatial points with ‘no parts’, has a limited capacity to carry information.

Its symbols, geometric points and numbers simplify and integrate the fractal, discontinuous reality into a single space-time continuum, the Cartesian Space/Time graph, made of points without breath. However the points of a Cartesian plane or the numbers of an equation are only a linguistic representation of a complex Universe made of discontinuous points with an ‘internal content of space-time’. In the real world, we are all pieces made of fractal cellular points that occupy spaces, move and last a certain time. When we translate those space-time systems into Euclidean, abstract, mathematical ‘numbers’, we make them mere points of geometry void of all content.

But when we look in detail at the real beings of the Universe, all points/number have inner energetic and informative volume, as the fractal geometry of the Universe suddenly increases the detail of the cell, atom or far away star into a complex complementary entity. So we propose a new Geometrical Unit – the fractal, Non-Euclidean point with space-time parts, which Einstein partially used to describe gravitational space-time. Yet Einstein missed the ‘fractal interpretation’ of Non-Euclidean geometry we shall bring here, as Fractal structures extending in several planes of space-time were unknown till the 1970s.

So Einstein did not interpret those points, which had volume, because infinite parallels of ‘forces of Entropy and information’ could cross them, as points, which when enlarged could fit those parallels, but as points in which parallels ‘curved’ converging into the point. This however is not meaningful, because if such is the case parallels which are by definition ‘straight lines’, stop being parallels.

So we must consider that what Einstein proved using Non-Euclidean points to explain the structure of space-time is its fractal nature: points seem not to have breath and fit only a parallel, but when we enlarge the point, we see it is in fact self-similar to much bigger points, as when we enlarge a fractal we see in fact self-similar structures to the macro-structures we see with the naked eye. That is in essence the meaning of Fractal Non-Euclidean geometry: a geometry of multiple ‘membranes of space-time’ that grow in size, detail and content when we come closer to them, becoming ‘Non-Euclidean, fractal points’ with breath and a content of Entropy and information that defines them.

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

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

Euclid 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:

Abstract, continuous, one-dimensional point:

. ____________

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.

             Real, discontinuous, n-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.

The mathematics are the same, the interpretation of reality changes, adapting it to what experimentally we see: a cell-like point enlarges and fits multiple flows of Entropy and information, and yet it has a point-like nucleus, which enlarges and has DNA information, which seems a lineal strain that enlarge as has many point-like atoms, which enlarge and fit flows of forces, and so on.

Thus, in the same way Saturn’s rings stop being planes without volume when we come closer and observe them as fractal points, called planetoids; Non-Euclidean points acquire both motion and volume when we approach to them. In words of Klein, a sphere is not a continuous static form, but a group of points in cyclical movement. So in the same way the Saturn’s rings are a group of planetoids, a Klein space – the space-time that fills a point has motion – it is the sum of a series of cycles.

Einstein didn’t go further, adapting the other 4 Euclidean postulates to the new Geometrical unit: a fractal point with volume. Only then we will be able to define the 2 planes of physical forces, the plane of gravitation and electromagnetism, or any system in which several planes of space-time co-exist together (as in a human being extended from atomic to social planes of cyclical existence).

In all those systems planes are made with cellular points, spheres with volume that form lines, which are waves between points that exchange Entropy and information and planes, which are organs of self-similar points that process Entropy or information in parallel networks. Thus the 5 Postulates of Non-E Geometry vitalize the Universe as a series of networks of Entropy and information of self-similar cellular points. Since the line and the plane acquire volume and become self-similar to the commonest forms of the Universe, the wave and the network of points with a 3-D volume.

This simple fact explains one of the most important discoveries of modern physics, the Holographic principle, according to which information might be bidimensional, as in the screen of a computer or the page of a book. Now bidimensionality no longer becomes ‘magic’ since the 3rd dimension is the relative size of the ‘fractal point-particle’. Thus bidimensional sheets of information do have a minimal 3rd Dimension; the inner content of the point, which in a relative universe of infinite sizes seems to us a particle-point without volume, as we don’t see either the volume of a sheet of paper or a pixel.

So each point is in fact a 3-dimensional point, and if we go to the next scale, a 3×3=9 dimensional point and so on.

Yet those dimensions are the so-called fractal dimensions, which are not ‘extended to infinity’ but only within the size of the point. In Euclidean geometry, a point has no volume, no dimension, but string theorists say that even the smallest points of the Universe, cyclical strings, have inner dimensions that we observe when we come closer to them. That is the essence of a fractal point: To be a fractal world, a space-time in itself.

‘Any Non-Euclidean point is a fractal space-time with a minimal of 3 internal, topological, spatial dimensions and an external time motion in the st+1 ecosystem in which it exists’

This simple law, foreseen by Leibniz in his Monadology, is the  foundation of the mathematical model of Multiple spaces-times that completes the 5 Postulates of non-Euclidean geometry and gives us the tools necessary to create a complex new logic and new mathematical model of the Universe, easy to connect through topology with the D-isomorphisms of time-space, where the previous paradigm of a single metric space-time continuum described in 4 dimensions, with 1 single motion is just a ‘limited’ view of a single scale.

Now, as we know the mind stops dimensions, we can go objectively the other way around and give motion to the point, establishing the fact that motion accelerates inwards, and downwards according to 5D metric Sp x Tiƒ = K, the space of the point which NOT perceived in its motion, merely seems to shrink becoming a hyperbolic point and defining a hyperbolic geometry, merely as the perception of a lower faster scale of space-time from an upper, slower, larger membrain:

In the graph, the proper interpretation of Lobachevski’s geometry: as we do NOT see from the still mind, the motion of vortices, which appear to us as solid particles, its speed shrinks its form. So when we perceive from a membrain an internal vital space-time of smaller micro-points with faster motions, they will seem to us curving parallels into a point, and so we must define as Lobachevski did an ‘angle of perpendicularity’, or rather a curvature ‘strength’ (equivalent to the k, G constants of smaller faster electromagnetic forces or larger gravitational cosmic bodies), which will define the ‘degree’ of ‘shrinking’ in our perception of points, maximal for electromagnetic smaller particles/vortices, larger for gravitational bodies (hierarchy problem of mathematical physics).

How this can be represented? Obviously by including as Klein did in his model the dimension of motion, hence defining space-time formal motions, as one without the other makes no sense to measure ‘real distances’.

This should be obvious, but abstract minds simplify entities into numbers and static forms, and organic motion properties disappear. Yet we still say ‘San Francisco is at 8 hours from LA’, because we mean a journey is a combination of the motion of a car and the spatial distance. Thus we measure reality in Time-space, not only in space as Euclidean mathematics do. Indeed, today our national capital seems very close with a fast train, car or plane, 200 years, as we were slow it could be ‘remote’. Thus speed shrinks space-distances for all beings.

This said we can re-interpret the postulates of Hyperbolic geometry, and cast some light in its seemingly absurd concepts and then deal with its different models, which idealise=simplify as always with experimental mathematics the general properties of time-space systems.

The generator equation of all Minds.

The study of the MIND as a ‘Generator Equation’ of all Space-time Systems of the Universe, and responsible of the creation of shrinking scales, mirrors of the whole is then written in its simplest form as a singularity-mind equation: 0 x ∞ = C, which is therefore the fundamental equation of ‘number theory’ and analytic geometry, as it includes with it all other elements.

O x ∞ = K

Or in dynamic way, S@ (static mind view)<≈>∆ð (moving world-universe)

Because the scientific method requires OBJECTIVE measure of the existence of a mind, which is NOT perceivable directly, we infer its existence by the fact a system performs the 5 external actions, which can be measured objectively, in the same manner we infer the existence of gravitational in-form-ative forces by its external actions upon massive objects. Hence eliminating the previous limit for a thorough understanding of the sentient, informative Universe. And further classify organic in simplex minds – all, which must gauge information, move and feed to survive, and complex systems, those who can perform a palingenetic reproductive, social evolution, ∆-1: ∑∆-1≈∆º.

Languages express the elements of reality and its operands.

Frames of reference then, as well as algebraic equations of extreme domains, and angles of perception, all bring about the meaning of analytic geometry.

Let us consider briefly this essential element to analytic geometry, the angle of perception as reflection of the 1st Dimotion of existence.

1D: Of those operands then we consider the simplest ‘operands’ as those who act directly on a point-number. We need then to find the operand of the first Dimotion, self-centered in the point, perception.

Since all points are also numbers as all have internal structure, we can operate both in S-pace with points and topologies and in scale with numbers to show those internal parts with numerical values:

The graph shows the elements of the point-number in which an operand can act. Even though most operands will act through a similarity ≤≥ symbol in two different elements. Yet the simplest operands are those which can act in a single point-number.

The sinusoidal functions. π.

Pi then becomes the numerical value of the external membrane measured in terms of 3 diameters turning around with 3 apertures to ‘see’, from the self-centered singularity:

The point-number has INDEED a central point of perception, which therefore can be defined externally by an angle of aperture to the world. The membrane normally is a pi, 3 diameter number, which leaves an angle of aperture, 3’14-3/π=0,14=±4,5%, with 96% of ‘dark matter’ outside the perception of its singularity. But as the membrane of the point turns, the angle ‘sweeps’ with its 3 apertures between diameters in a relative discontinuous Universe all the world outside.

So the angle of the point will finally allow a full 100% view in 3 different ‘ages’ of the full world cycle of the membrane. So in dynamic 5D analysis we can consider not only a sine and cosine function; but the sine ‘main’ function that calculates the aperture, also as a function of time, such as for a full turn of the 3 diameters that complete the membrane (2π radians), the angle has seen the entire external world 3 times.

What this means then is that the sine of a singularity point ‘grows’ with each turn of the world cycle of the being and in time can be higher than one.

This said we shall not go into so much complexity and consider the ‘static’ concept and define the sine merely as the angle of perception of the central point and hence the function that reflects the 1st Dimotion of existence.

Because then all those Ðimotions are complex stœps, which can be discomposed in a form and motion, ¬Ælgebra is better represented in a frame of coordinates as a vectorial field; that is, as a point in a vectorial field, which has both direction – hence motion – and form, mass, a stop state, with a scalar value, things those considered in analytic geometry.

In mathematical physics as a closer reflection of there Universe that eliminates from the language mirror ITS INFLATIONARY IMAGES (SO math will always be paradoxically larger in theorems that reality is, obliged to ‘bend’ and ‘limit information’ to those shapes energy can ‘bend’), it is even better to represent a vectorial field in General coordinates aas they allow the multiplicity of points of view of the supœrganism each with its self-point of reference.

Since all T.œs have a self-centered point or monad-mirror – a linguistic mapping or ‘seed’ that reflects reality in its frame of reference and will try to act from that point of view or re=produce  andimprint the external world’s suitable energy (∆<¡) with the inner mind of its points of exist¡ence.

To be anchored first as a point is the first function of any T.œ when emerging into exist¡ence.


What is the most important of those planes to mankind? Obviously the one that defines the Humind perspective, which is the one that has the dimension of our ‘space-time’, which is LIGHT! that for that reason is both the limit of transmission of information perceivable by man, and a constant speed force, as we perceive it in the stillness of the mind.

Spaces as the a priori reality.  The cartesian 3D graph

In the graph, since the Humind is Cartesian the first and most widely used plane of reference will be a  3D-∑∏ cartesian flat/hyperbolic plane, which mimic – it is homeomorphic – to the ‘vital’ 3 perpendicular dimensions of light space-time.

As such our mental spatial reality is a distortion of the Universe suitable for perception in 3D±∆ social colours. And so immediately we observe a correspondence between the FUNCTIONS-FORMS-ACTIONS of light as an organism and its Dimensions used in the humind to build our worldview of the Universe and its T.œs:

The actions/dimensions of vital light space-time become the dimensions absorbed by electronic minds (human or digital), which in turn will emerge as vital dimensions of the particle-electron. Huminds thus are constructs of the limited perception we have of light – since (Heisenberg’s principle) we further reduce and smear light information to fit it on the electron.

That we reduce further light information is self-evident but treated in the Physical analysis of quantum uncertainty principles. SUFFICE to say that we reduce from each bidimensional holographic plane, or topological variety of form, one dimension to obtain the 3 canonical perpendicular lineal dimensions, which correspond in the humind to the functions of its light-systems:

-1D: Height is the function of information where animal systems will place its particle-heads.

-3D: Width is the dimension of present reproduction across which animal systems will multiply its cells and reach bilateral symmetry.

-2D: Length is the dimension of motion ALONG which systems will move

But the mind also observes the social and entropic, evolving and devolving arrows of time to complete a 5D dimension of reality. And this is done with the social/devolving arrows of light, called frequency colours. So humans add the duality of:

-Red/magneta: dissolving/dying colours of Entropy and ‘tired’ light as it dissolves into vacuum.

-Blue/violet: colours of maximal frequency/form, as blue stars generate its photons.

-Green/Yellow: colours of vital energy/information in the intermediate spectra, which are colours of processed plant energy and direct light.

So the codes of light and its dimensions create the ‘interpreted’ world of the humind.

This was clear to Descartes who called his geometry not Universal, BUT HUMAN; depicting the world as a limited mirror of the total Universe. Since it is the human light space-time mind with its 3 perpendicular magnetic, electric and speed fields what determined the eye-geometry – an electronic machine absorbing light – something Kant also explained when affirming our Mind IS Euclidean because light is Euclidean.

So it is fundamental to understand the homeomorphism between the 3 lineal dimensions and the bidimensional REAL topologies, which they simplify. 



All minds depict with different languages a 5D Universe.

 The mind is a subjective perspective constructed with pixels of a language appearing first ‘simplex, disconnected’, then build limited mirror-images of ‘space’ (simpler present dimensions), then move in time, locking into symmetries of colours and finally build background planes. So the ‘subject’ or mind selects meaningful ‘5D bits’ to do finally in the language an efficient mirror of reality to act on it.

This happens also in mathematical minds. And both type of minds likely evolved in animal life and today in huminds and future computers, adding ever more complex layers of perception of those dimensions.

As we perceive first ‘simple motion’ (2D), coupled with position & formal size (1D), next ‘depth’ and the relationships between those positions, ≈, (3D) and finally ‘colour’ (4D red, 5D blue).

The mathematical mind representation can then be seen as a sequential evolution in time, with the Social Human mind of scientists learning first, a bidimensional, still ‘Greek’ geometric spatial form (2D), coupled with a social-number size-density (∆§), It came then 1D position, as the @-mind view of analytic geometry, a self-centred humind point of view, which soon allowed  to establish 3D S=t algebra symmetries… and finally the development of ∆nalysis and its infinitesimal parts and wholes, closed the 5 great ‘disciplines’ of Maths as mirrors of the 5D elements of the Universe.

Now if you are ‘starting’ to understand the entangled, symmetric, kaleidoscopic Universe, the sequence of growth in the perception of the complexity of any language as mirror of the Universe, is similar to the way any system develops in time, orders in space, and acts in space-time. So the kaleidoscopic views reinforce each other to create the ultimate order of all systems of reality. As all ultimately comes to a world cycle in time symmetric to an adjacent efficient organic structure in space.

This happened in classic age of mathematics with Descartes, when finally analytic geometry allowed to mix the time view of discrete numbers and the spatial view of continuous points. And from then, in layers of growing complexity maths mirrored the S≈T dualities of the Universe of which the Superorganism ≈Time world cycle is the one that encompasses all others.

The birth of @nalytic geometry thus starts the classic age (as beauty=classicism is indeed an S=t, form and function balance), the mature, most expansive≈reproductive, fruitful age, of mathematics which soon gave birth to ∆nalysis of the scales of the Universe; completing the ∆@•ST elements of maths (S-geometry, T=s algebra, T-number theory, @nalytic geometry and ∆nalysis).

So as complexity grow, new connections of the S≈T  5D²imensions of the being, BRANCHED analytic geometry into new dual ‘connected’ parts; and finally the entangled, kaleidoscopic Universe connected each discipline of maths, mirror of one of the 5 Dimensions to all the others (analysis, number theory, algebra and geometry all being put in a mind-view on a frame of reference).

And of course, each discipline sub-branched internally through the Disomorphisms, first into the 3 ‘sub-mind views’ or 3 states, the S-cylindrical pov/frame of reference, the T-polar and the ST-cartesian ‘hyperbolic’ frame. And then into the ∆-subdisciplines of Hilbert, phase spaces.

So @nalytic geometry could both expand to the 3 ‘present topologies’ of any super organism, and define its properties in scales, but always from an homeomorphism with one of the 5D.

For all those reasons, we consider the mind view of @nalytic geometry, the key step to professionalize mathematics, and ∆nalysis its final frontier to complete its 5D mirror view.

The key symmetry between 5D seeds/minds and @nalytic geometry/∆nalysis.

To that aim, two features of the @nalytic plane, as mirror of the 2 informative still states of reality – the seed and the mind, are a key to understand also is homeomorphism with the 2 sides of ∆nalysis.

WE REFER to a fundamental theorem of measure – the homeomorphism between:

  • The 0-1 ‘generational seed interval, or unit sphere, which represents the growth of the ‘unit element’, from the singularity positioned at 0, to the membrane of the one, at 1 with the vital energy contained inside, and…
  • The 1-∞ universe, which represents the OUTER world mirror as seen within the infinitesimal mind but paradoxically has more information, more infinitesimal details than the N+Z line .

So we can find many magic congruent relationships between the 0-1 and 1-∞ ‘scaling’ of maths, which apply to different parallelisms of scales in science: i.e. the quantum 01-probability sphere and the 1-∞ thermodynamic statistic ∆+1 scale.

Of all of them the key relation is the understanding of the unit cell of a whole organism as a ‘finitesimal’ 1/n – a part of a whole social number, n, which is a theme more proper of T§-number theory and ∆nalysis.

ALL THIS Said, THE GREAT advance of Analytic geometry besides being a mind-biased view, which illustrates the different distortions of each of the 5D ‘organs’ of a system, and how they see reality, WAS to add the cartesian coordinates of sequential numbers to allow the first clear S=T analysis of space-time symmetries, referring two completely different elements of reality, the continuous plane view and the discrete numerical view – even if huminds were/are still unaware of the differences, shown in classic paradoxes of irrational numbers, as they insist (Dedekind cut, axiomatic method) on making equal a discrete numerical system that has ‘dark spaces’ that do NOT exist to differentiate numbers, between them – albeit infinitesimal cuts – vs. the ‘white’ continuous view that ideally considers the NO existence of wholes. 

We shall thus study in this post, analytic geometry not ONLY BUT MAINLY as the humind view WITH all its further derivations, which fusions the S-patial, continuous point-description and T-emporal, algebraic discrete, numerical description of Space-time evens, as such is the 3rd S≈T, classic age of geometry.

5th postualte

From a 92 book: the ideal frame of reference for non-AElgebra is a vectorial space-time with multiple frames of reference. Of all human devices frames therefore Hilbert spaces are the most adequate; hence its wide use in quantum physics. Further on as all T.œs have a different language of perception, and exist in different scientific planes, the parameters they perceive as time and space might vary; so next comes phase spaces, to reference the suitable time-motion and space-form coordinates each species perceives. Then we have in the human single plane, 3 frames of reference, the polar, spherical, cylindrical, lineal, and Cartesian, hyperbolic planes, being the S=T Cartesian plane the most useful; but depending on which organ of the being, its |-limbs/fields, Ø-body wave or O-particle-head enact the event, a different frame might be more suitable. Then we can write on those frames, different chains of dimotions with algebraic equations and operand, joined by dynamic = symbols of equivalent, feed-back stœps.


As we study the dual uses of Cartesian geometry in @-mind maths and ∆nalysis, it is only logic to move straight ahead into the 3rd age of non-Euclidean geometry; and as its expansion with the 5 Non-Euclidean postulates has also been treated earlier given its importance in the comprehension of reality, we shall only comment on the classic Non-Euclidean elliptic, hyperbolic geometries and the massive expansion in the concept of spaces once Lobachevski unleashed them from the concept they represent a single reality when they are mind constructs to represent the Universe with infinite different mind-worlds-spaces on the making.

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.

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 Tƒ=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 ‘vitalised’ in terms of the time=motion events of an organic system.

Indeed, the 4th i-logic postulate of non-euclidean geometry come 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 vitalised, temporal moving view of hyperbolic geometry and Lobachevski’s formal still geometry? 

Absolute. Indeed, the surprise comes when we realise 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 rigurous 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 ( below, figure 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 a parallel to it forms with the perpendicular to it an arbitrarily small angle.

The proof, as so many of the mathematical sections of this blog copycatted from Aleksandrov’s book on the principles of mathematics, prior to the pedantic age of the axiomatic age 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. 

In other words, 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. 

RECAP. 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.

The length of the ‘membrain’-circumference. Limits of infinity between discontinuous scales. 

Another fascinating finding which we will also interpret in terms of the structure of a time§pace supœrganism, in concomitance with its universality as the fundamental particle-structure deals with the relative ‘strength=curvature’ of the enclosure of the hyperbolic vital energy of the system, which again can be seen according to the S≈t symmetry in terms of the 3rd space-time formal/motion dimension.

In hyperbolic geometry, the limit of a circle of infinitely increasing radius is NOT as in Euclidean geometry a line but a certain curve, a so-called limiting circle. It is not always possible to draw a circle through three points not on one line, but either a circle or a limiting circle or an equidistant (i.e., a line formed by the points that are equidistant from a certain line) can be drawn through the three points.

Hence hyperbolic geometry proves 2 fundamental properties of fractal space-time: the fact that ALL lines are part of cyclical zero-sums in larger scales, and the fact that there are no infinities but limits of relative size as perceived from a given ∆-plane.

The same concepts, which are ‘so strange both in hyperbolic and ∆•s≈t geometry, apply to the inverse holographic lineal-entropic figure, the triangle. We shall mention 3 of them:

1. There are no triangles of arbitrarily large area (limits of infinity).
2. Two triangles are equal when their angles are equal (isomorphism of scale).

3. The sum of the angles of a triangle is always less than two right angles. If a triangle is increased so that all three heights grow without bound, then its three angles tend to zero (again limits of growth for a triangle drawn in the 5D hyperbolic metric).

Now, if we consider the angle of a triangle, the fundamental ‘angle of perception’ of information of a given singularity, (hence the capacity of trigonometric laws to ‘calculate distances’), this deduction of hyperbolic geometry clearly points in the same direction: the perception of different planes dimisnihes till it finally becomes a zero dark space, as we have proved in other parts of the blog by other methods.

Sacred pi revis(it)ed.

Finally of special importance is the consideration than 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). And as usual by the ‘Rashomon Effect’, IT CAN be applied to all the 5 D² symmetries of reality (so with hyperbolic geometry, though in this introduction we mostly apply it to scalar-planes geometry). 

Let us then consider two consequences of it, for biological and physical systems.

In as much as hyperbolic geometry is a spatial, mental translation of a geometry of ‘absolute time flows through ∆-scales’, when we curve it properly… it can be used to study the processes of time passing, specially in our analysis of the 3±∆ ages…

Then in terms of time passing growth represents obviously the passing of time, the bidimensional circle the membrane-skin of the being in biological terms, the warping of a vortex of physical time (charge, mass) and the radius of the hyperbolic circle, its content of vital energy. 

And so ageing is exactly the process of faster growth of the predator skin that encloses and parasites the vital energy, becoming much faster in its enlargement, warping, wrinkling, and finally detaching itself from its vital energy, stiffening and dying.

In physical systems, notably in relativity it has a more mathematical formulation with exactly the inverse tendency as we are studying NOT the envelope of the @-ðime membrain but its internal, • ‘singularity’ in the center, which inversely  to the skin from where it detaches, triggering the process of death and liberation of the vital energy, trapped into the body no longer ordered, freed to develop its entropic tendencies, as the center shrinks faster and the membrain grows larger, no longer ‘topologically adjacent’ to it.

So, the following formula holds for the singularity zero-point ‘event horizon’:

where k is a constant depending on the unit of length. Since

we obtain from (1):

l= 2π r (1 + 1/6 r² /k²)

Thus only for small ratios r/k is it true with sufficient 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.
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 10ˆ12.
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 10ˆ−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., 10ˆ−12, and within the limits of a meter of the order 10ˆ−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 10ˆ−13 km). On the other hand, on the astronomical scale the ratio r/k could turn out to be 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, the arguments given have another important consequence. It is this: 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.

Therefore, if this limiting case is added to hyperbolic geometry, then it 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 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.

And 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 pov.

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 pov, 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.

So we shall consider another ‘Rashomon effect’ on it – the geometry of the electronic humind made of light according to the ‘relative ratio’ between our r and k constants of ‘perception of information’ (k) and unit of lineal length ($ (r)…

All this though IS a special case of our much important 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 worldcyle or enclosed 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, nversions, 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 behaviour of our euclidean space as it moves to the non-transitable 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-centred biased point) – then we have the full Rashomon effect to get the final 5D mind/judgement 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 deja 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 polydimensional 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, we are bent by the environment and if we don’t, we crash…

Color space, defining the vital geometric properties and Riemann’s generalisation. 

We can now with all this ‘∆•s=t’ considerations on the ternary codes of colours study it as geometers did to generalise 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.


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-centred in the ∆º plane, whose sum gives us a zero world cycle that returns to its cyclical origin.

It does NOT 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.

Or in terms of the dimensions of existence and its mathematical representation – which 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.


In the color space though the definition of continuity comes easier, ‘enlightening’ the general meaning:

A continuous change of color can be represented as a continuous line in “color space”;  formed by a discrete number of mind perceptions, which as the stop and go process of a continuous view of a film, do NOT perceive the irrelevant steps between those colours 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’.

Mind’s first task as reducers of dimensions to the relevant ones become then clarified; continuity Is the result.


Dualit 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 colours, 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). 

Riemann’s generalisation.
In this way there arises the concept of the simplest geometric figures and relations in the “color space.” A “point” is a color, the “segment” AB is the set obtained by mixing the colors A and B; the statement that “the point D lies on the segment AB” means that D is a mixture of A and B. The mixture of three colors gives a piece of an E-plane/¬E ternary network—a “color triangle.” All this can also be described analytically by using the color coordinates x, y, z, and the formulas giving color lines and planes are entirely analogous to the formulas of ordinary analytic geometry.
In the color space the relations of Euclidean geometry concerning the disposition of points and segments are satisfied. The system of these relations forms an affine geometry, and we can say that the set of all possible color perceptions realizes an affine geometry.

Thus 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, WHICH STARTED this blog. 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 realise 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 or others fill the 4th line on relativity – the marriage of Riemann and Einstein’s simple ideas.

So as we have already 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 pseudosphere.

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 rasssa’, the formless potential; and the singularity is the hidden central or polar ‘invisible’ element of the elliptic geometry.


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 worldcyle or enclosed super organism so they are ‘constrained’ into a zero-sum or limiting membrane and appear as curved geometries’. 



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:

  1. Differential Geometry of curves taken as points with motion.
  2. 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.
  3. Topology, of the 3 BiDimensions of space with motion that represent the 3 functional organs of any supœrganism,
  4. ∆@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…
  5. 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 (1D $t). 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 Γenerate 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 exi=st-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” congruences. 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 interiorise 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. For 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 generalisation 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 nonregular.
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.  



4th Postulate of ® logic: parallelism and perpendicularity

Vector spaces and its related field of the complex plane in its ‘pentalogic’ 2D use are the fundamental  expansion of frames of references, INTO DIMOTIONS, THAT IS, frames of reference with form 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 must BE CONSIDER to be in motion itself – 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 generalised 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:

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 third ‘offspring-dimension-form’ is created fusion of the other two:

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 generalised 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).

Now, the difference between both type of vectorial product are 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 generalised 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 can establish a parallel superposition principle for the dot product and a perpendicularity one for the dot product.

Topological spaces

So far we have studied the simplest of all possible ¬Æ geometries and ITS metric/distances, that of ages/states with the example of phase space in the ages of matter. We shall now consider the two other essential ternary symmetries of the fractal generator, topological spaces and scalar symmetries and distances, to close this introduction to non-e geometries. As the 4th and 5th dimension are unknown to humanity, we shall consider in this second line where we study basically classic stience further enlightened with ∆st insights, the case of topological spaces, keeping the general theory of ‘scalar distances and frames of reference of the generator’ for the posts on space-time of the first and third lines.

Topology as the queen of mathematical sciences.


Topology is Geometry with motion, hence the temporal 3rd age of Geometry, and likely the culmination of the mathematical science, as expression of the real laws of space-time beings, as it includes the 3 concept of ∆-scales (topological forms are defined in modern terms as networks of points), of Space forms (its 3 varieties are the 3 varieties of organs/forms/functions of the Universe) and 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 more than algebra, which has little reference to reality, in modern axiomatic/set theory but has become largely a meta-linguistic procedure, is the queen of all mathematical sciences, as it is instantly connected with the real Universe.

So we will divide its study as usual in the ternary method, in 3 ages:

1st age: Classic topology

2nd age: Fractal mathematics & networks.

3rd age: Vital topology (GST Supœrganisms, our culmination of the evolution of maths),

We shall though start from the end back to the beginning as the end IS one of the most essential parts of all the GST model of the organic Universe, showing how fractal points joined by networks become waves and flows of energy and information hat evolve into topological organisms with 3 physiological networks, the Spatial, entropic, ‘digestive’ system, the S≈t, reproductive ‘blood network’ and the Tiƒ-Informative brain network that mess together through its ‘dark spaces’ (as networks do have wholes) forming the supœrganisms of the Universe.

THEN WE SHALL STUDY the first age of classic topology and the 2nd age of fractal mathematics, very briefly as fractals are studied in vital terms (nature is a fractal of fractal organisms, at *4 & 5, in the second line)


Now topological spaces are VERY interesting and we shall consider them here in more detail from the 4 canonical perspectives of the Rashomon, ∆@st effect:

  • @: As the last ‘real generalisation’ 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: As the most sophisticated form of §p@œ, (spatial, past-memorial, mental organic, scalar space), 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 generalisation of existential algebra treated in the first line. 
  • ∆: Finally as topology has evolved into network topology it is an excellent form of geometry to study ∆±1 parts and wholes.

Let us then go briefly through those 4 Rashomon effects, to close this introductory study of ¬Æ≈i-logic geometry.


There are 3 relationships in space-time between entities:

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’ – a waste of time for researchers too (:

In the graph the classic conception of 3D geometries we use ad nauseam in this blog to explain the fractal generator of T.œs.

Things are though a bit more complex when we ad the laws of ‘transposition’ of functions as we move through the generator ternary symmetries in time.

So for 3 Dimensional elements, the realm of topology, those correspondences of form and function are not so immediate, as things start to become multifunctional and here it is the  KEY LAW of ternary systems, its MULTIFUNCTIONALITY, 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 topological ternary forms while dominant in one of the 3±i arrows of timespace, will be able to perform the 3 arrows.

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. So it is worth to consider this ‘higher level’ of complexity and its general laws.

The laws of multi-functionality:  inversion of roles we emerge into higher social planes.

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 which 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.

(This is the stuff which delights the paradoxical mind but has always made so difficult for naive realism and its mathematical physicists to understand a dot of this).

To notice the 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.

Let us study then the next transposition of roles for only one of the 3 varieties of topology, the circle which becomes a sphere, the strongest membrane that encloses and captures the vital, hyperbolic energy of the being.

 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 vitalisation of its concepts, and a proper dimensional analysis, introducing the laws of S=t Disomorphic symmetries. 

Let us put another ‘classic example’ (we just use the basic laws of each science to vitalise them with ∆st or else we would never finish).

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 generalisation…  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:

Here we have it again, damn it. the 5.., surprised? (: 5 only regular solids for 5 dimensions, how they correspond with those dimensions? Obviously just as they are seen. 3 are obviously simpler, so they must be the 3 D of a single plane. 2 are more complex, so they must belong to ∆±1 scales.

Alas, my friend Plato, fav poet of my earlier childhood, who made me love philosophy even before i loved art even before i loved science was right:

1D is no doubt the lineal triangle, and so the tetrahedron the strongest single form of Nature; the ternary element of the tetratkys. And as it happens that ‘odd’ numbers (that for number theory) are lineal-like, entropic-tending, limb-oriented, when we ad motion. So our lineal limbs walk in triangular steps, which are a 1D $t basic dimensional growth, from formal lines of $pace into St lines moving in lineal fashion.

Yet if we grow from 1D to 2D only in $, the definition of tetrahedron is immediate: $$.

The cube, a 3D st solid with informative roles. 

3D, yes you (didn’t) guess it. The inverse of the tetrahedron is the cube, 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 so many other vital geometric functions, of my notebooks.

An example will suffice. The best way to REPRODUCE=generate by transposition from a line more dimensions is to make it grow into a plane that grows into a cube. Yet each growth has a different function according to the GENERATOR EQUATION that changes topology, age and grows in ∆-scale, the 3 elements of any super organism, as its ternary generator shows (but now has a different function according to the inverse functions that transpose each ∆-scale, a step on the ternary generator, the being, such as):

∆-1: $< ∆º ST > ∆+1>ð§

The formula is again a version of the the fractal generator that encodes within it ALL LAWS OF THE UNIVERSE. So it is ultimately the explanation of the ‘sliding’ of functions, 1 at a time as we move from lineal limbs/potentials at ∆-1, into Ø-ST-iterative space-time into ∆+1 O-spherical particles-heads of information

So the line is 1D lineal motion, the plane is ST hyperbolic iteration and the cube is the cyclical formal-function or §ð:

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.

First to notice the REPRODUCTIVE NATURE OF MOTION. 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 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 fractal generator encodes within it ALL LAWS OF THE UNIVERSE. So here it generates the properties of cubes from lines; as its function  ‘slides’  of functions, 1 at a time:

We move 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 functions do slide and change but the pattern is encoded in the generator and so as usual ONCE we understand THE UNDERLYING ∆st basic laws of the Universe encoded on it, everything keeps falling into place.

Yet 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; and for the same reason, on close analysis the ’empty sphere’ (not a ball, only the surface), IS NOT as the inner ball it grasps a volume of information but the topologically simplest, strongest (normally made of strong triangles of entropic nature) membrane that acts as the entropic envelope that in a non-euclidean Klein disk the inner vital space can never reach because it will kill them.

So the sphere KILLS by enclosing, trapping and acting as in lion hunting of zebras by enclosing them with predator or shepherding functions – as dogs enclosing sheeple. Then the membrane will sharply penetrate perpendicularly the zebra herd and eat it up; the military border of a human social territory will give a coup d’etat and collapse into the capital and conquer the world, the battle will be lost once in Cannae, Hannibal had encircled the prey (graph, where the red color of entropy is used and the dimension of motion of the horse allows to ‘close the dark spaces’ as the shepherd dog does).

So we are talking experimental reality here, as ∆st laws might seem abstract to you but are the stuff of which survival and existence is made.

Ternary topological varieties.

In the graph the classic conception of 3D geometries we use ad nauseam in this blog to explain the fractal generator of T.œs. Things are though a bit more complex when we ad the laws of ‘transposition’ of functions as we move through the generator ternary symmetries in time, (not explained here as the illusionary separation between past, present and future, is the hardest one to understand for huminds, so I won’t blow up your mind with past local travelling  beyond the zero sum death cycle).

It is then clear that:

‘The purpose of topology is to study the ternary vital geometries of T.œ, its functions and transformations’

We then arrive to ternary dimensional topologies, which again are ONLY 3 as 3 are the vital functions of s, st, t universes.

Only that now each variety can play the 3 roles, even if one of them will be dominant.

It holds then that a variety MUST not CHANGE its external form beyond deformation not to loose its properties.

So we must carefuly reassess the way topologists analyse then, considering how a transformation of scale, of dimension and form modify and evolve those functions; specially when…changes are caused by new adjacencies, new perpendicularities and new social parallelisms.

When those changes happen we talk of a topological d=evolution (the inverse of a topological transformation where none of those processes happens) Рessential definition of Ƥtopology:

“A topological evolution are changes in the form and function of the s, st, t parts of the being caused by new adjacencies, new perpendicularities and new social parallelisms.”

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 0-sum information singularity (donuts) to a flat plane of entropic motions.

So for 3 Dimensional elements, the realm of topology, those correspondence are not so immediate, as things start to become multifunctional and here it is the  KEY LAW of ternary systems, its MULTIFUNCTIONALITY, which allows a ternary topology TO PLAY different roles in reality acting as $, ST, and §ð beings (∂ easier than ð often appears for 3D ∫∂ being 4D)…

Ok, so what are NOT the classic roles of 3D variations in ∆st? To notice first the silly error of topologists which consider a donuts to be as in the graph a flat plane, just because you can cut the empty donuts, spread it in 2 D and alas you have the plane, but you have cut it, producing a non-topological transformation as you have torn it and changed it. 

To the point, the real equivalents are easy to obtain considering simply the main law of topology, which is the GENUS of each variation, such as obviously as we grow in complexity when we slide on the generator functions-form, the form with minimal genus that is, whose number of dissections is minimal, must the simplest form-function.

What is then a good topological measure of complexity and hence of functions related to information vs. simplicity, ergo functions related to lineal motion?

Genus. Measuring functions with forms. 3D systems and its $<st>ð roles.

The most talked about concept of topology: the genus of the system, that is the quantity of cuts/tears the system can endure without being destroyed.

Where the genus is a number less than the number of cuts NEEDED to break the form in its component parts, eliminating its adjacency (ask me why it is NOT the same number than the cuts – response: huminds love to make it difficult for students and to feel they are complicated geniuses).

The GENUS of each VARIETY, grows as we grow in dimensional and informative/formal complexity. So the form with minimal genus that is, whose number of dissections is minimal, must the simplest form-function.

3D $: So the sphere just accepts one cut and has zero genus and so IT IS indeed the entropic membrane that controls and predates on the inner region, which is however trying to keep in the middle as the singularity in the center is no joke – you are 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.

And 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. 

3D §ð: Next it comes the genus of the toroid which cannot be cut with a single line – only transformed into a new variety, in this case from a ‘circle closed’ into itself into an open cylinder, and as such IT MUST be the singularity

3D ST: As the number of parameters needed for the |-function is smaller than for the O-function, which is smaller for the iterative/reproductive Ø-ST function, which is therefore the 3rd variety, the dual 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.

MORE ON Platonic solids, its correspondence with 5Dimensions.

So we can now RETURN to our platonic solids, where 1D is the tetrahedron and 2D is the cube. This means that

2D ST is the mixture of both, the octahedron, which resembles both the tetrahedron and the cube. Indeed, if we  think of the 2 elements of the platonic solid, its surface and volume, the octahedron has the surface of the strongest triangular 2-manifold form, hence it maximises its surface power/function as entropic membrane controlling its vital energy-volume, but it comes closer to the cube in its inner volume, growing it further from the tetrahedron that sacrificed that volume.

So as we shall always see, the ‘marriage’ of two complementary forms, in this case the strong triangular surface and larger cubic volume TAKES from each the best qualities, reason why ‘sexuality’ in the biological realm works – improves with the best genetic combinations (even happening by chance, then will be selected).

Elements those of another huge new discipline of whys, connected to ‘T.œ’s genetics and palingenesis’.

So alas we are left with 4D, 5D, the complex social platonic solids with more faces. Which one is 4D and which one is 5D? It should be self-evident to the reader, pentagon, 5 sides, isn’t? So 5D the pentagon. Right (: but not with that kind of ass-ociation 🙂

Let us think harder.

4D. Entropy: The icosahedron is made of triangles as the simplest tetrahedron, and so as entropy decomposes the whole into its parts, it is the natural candidate to decompose from ∆+1 states into its ∆-1 parts. And that leaves us:

5D:@. The dodecahedron  pentagon, that old form of the devil, which some think to be God, the mind that all perceives it, the mysterious solid fascinating 2300 years geometers and Keplerian astronomers, now you know why.

But alas, how 5D-IES into 4D=evolution? They do NOT seem equal.

Well, outside they don’t. But the cover we know in Non-Ae is less important than the content to define equality – the cover can camouflage, what matters is the singularity, IN ALL OF THEM, the same central point – hence they can be considered a single family of complex forms, growing in efficiency, in volume-ratio to surface and complex function of its vortices…

So all the solids have a perfect center singularity; they evolve their surface towards more function vertex, and grow their volume/body ratio towards the perfect relationship of the sphere – which we shall then consider it the 6th perfect solid of ∞ vertex.
And so indeed, 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.

And what makes them identical to switch from 5D mind state to 4D entropy state is the finding by Apollonius in the last of the theorems proved in Euclid’s elements addenda that the ratio of its surface is the same that the ratio of its volumes:

So its 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.

So the platonic solids indeed repeat constantly in the Universe of single planes of existence (simplex platonic solids), NOT in the planetary orbits, where Mr. Kepler thought God was ‘sharing’ (:

Back to ‘reality’, the atomic surface tends to be as the graph shows, perfectly regular for the singularity 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:

Topology and ∆st: its kaleidoscopic perspectives.

The multiple value of Topology from the perspective of the Rashomon effect 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.

Let us then comment those 3 sub themes of topology in more detail.

∆±1 symmetries. The scalar geometry of polihedrons.

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 α0 the number of its vertices, by A1 the number of its edges, and by α2 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 easily identify S, T and ST, the intermediate element, writing as follows 1D point + 3D sur:face = 2 D line + 2.

Can then 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 withe 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-centred 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-centred 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-centred 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 armoured  trilobites, and the first eye-cephalopods that soon lost its armoured 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 ‘armoured’ 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.

The balance kept in pegging S<ST>t-species.

In its palingenesis and morphogenesis then we assist to vital pegging and tearing of forms to change and evolve its role.

We mention the commonenst pegging by adjacency, and tearing. The sphere tears its caps and then the digestive cylinder is pegged to the axis of the open sphere having both the same 0 Euler characteristic.

So they fit perfectly by adjacency not by perpendicularity; as the sphere has the central hole which the cylinder can close.

And 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.

In that regard, the true innovation of ∆st in the science of topology is the understanding of its laws, through the addition of s=t symmetries of balance that selects the survival forms of the Universe, to understand developmental evolution, which we call topological evolution and use in all sciences.

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.

The inverse process: jetting off handles and limbs.

In the hydra we see however how the balance law that created a hole digestive sink, inversely jetted off tentacle limbs.

This is again a fundamental theme of topology – sinks and handles.

So within the fundamental principles of ∆st balances, we can inscribe the second most common topological evolution takes place, also essential to abstract seemingly unrelated to reality topological structures: the sphere with holes filled inversely with  a handle in the form of cylinder that now jets outwards ‘closing’ instead of opening the spheres’ holes in pairs:

Quite generally, let us 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.

How this is achieved in ‘real species’ through antipodal points managed by the central singularity shows the fundamental vital nature of geometry – as the exact spatial symmetry of opposite antipodal points of the elliptic geometry of an @-mind/membrane system is accessory to the vital role the handle will pay in the external world.

Then the handle born of a first ‘suction’ and then ‘expulsion’ of continuous matter from the system, can be cut and cupped (a method of topology used in the abstract classification of varieties, which became famous with Perelman’s proof of Poincare’s conjeture). Alas, we got through ST-inversions and symmetries a couple of limbs, and created a stronger T.œ as the ‘section’ of the limb will now not mean as the section of the neck, the death of the 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…

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.

The process of fractal dissection of the Universe.

All these 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).

THEN 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 T.œ inwards and an outer anti-world from where to obtain motion and energy to reproduce through the elliptic singularity-membrane, ∆-1 cellular/atomic components and grow.
However, apart from these 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.

It is for that reason that Nature overwhelmingly favours closed surfaces, as the efficiency of a Mobius band-like, opened to the outer world is minimal, except some cases of ultra-strong surfaces which care nothing to be opened acting as entropic systems.

This then is found in simpler systems belonging to the 1D-motion or 4D-entropic functions of the Universe. As the concept can be equated to a chiral molecule, which is not superimposable on its mirror  image. Those functions are then proper of systems which want to increase its surface of exposure to the external world. So chiral molecules are good for optical activity and entropic light dispersion (4D function) or for motions based in explosive propellants of the more aggressive atoms (oxygen, chlorine) such as the Perchlorotriphenylamine.

Of course we are just as usual giving you the top of the iceberg of an immense extended subject. Our purpose was to get you here – to the understanding that geometry is a vital subject, which has slowly evolved till reaching its study of forms with motion and its transformation to create the ternary systems of the Universe (studied all over the blog, which repeats ad nauseam the basic laws of topological evolution in the study of the different species of the Universe).

∆±1. Points create topological networks. Hylomorphism.

Now, when we get into the details on how those topological evolutions take place another fundamental principles of ∆st, the hylomorphic method comes into play, which essentially means that ‘wholes are made of parts’, that is of fractal points, and so the change of a system happens always by tiny microscopic changes in the configuration of the fractal points.

And there is NO mystery to the openings and tears because the continuity of the whole is lost in the discontinuity and dark spaces between points (Galilean paradox). So by reordering, expanding or imploding distances between ∆-1 sets of points the variations in topology allow morphogenesis. 

Hence the success of set theory, ultimately an abstraction of the relationships between ∆-1 elements and wholes.

In the graph, topology in its detailed analysis of how it works is all about the 1st, 2nd and 3rd postulate of non-æ=i-logic geometry, through the arrangement of points (1D), its connections and axons opened for |-$ functions closed for O-ð forms. And its ‘degree of packing’, till eliminating intermediate spaces to create adjacency, maintaining a minimum space for ‘flows of networks’ to cross, in parallelism, and breaking through the form in perpendicularity.

And again those relationships are relative ratios of distances, where the relationship between the size of the minimal step-connection between points and the radius of the point, will define adjacency or parallelism, while perpendicularity requires to penetrate beyond the ‘enclosing, protecting membrane’. A simple formal definition of adjacency in set theory, taken from the not-so-pedantic book of Aleksandrov, should suffice to understand that relativity of ‘continuous vs. discontinuous’ systems (and give you a hint on how vital geometry becomes ever more complex, since we quote so often his book precisely because it is the simplest serious books of math ever written – just imagine the level reached of abstraction to obscure vitality, by merely going into wikipedian ‘students’ – born with the creationist, axiomatic method on mind):

The theory of sets made thus possible to give the concept of a geometrical figure a breadth and generality that were inaccessible in the so-called “classical” mathematics. 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 = (x1, x2, ···, xn) and B = (y1, y2, ···, yn) is by definition equal to the nonnegative number:

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.

The 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.

PATHS: 2nd non-E postulate: The fundamental group.

Paths are important for many reasons. Because they are the clearest combination of a t-dimension of motion and an s-dimension of form in topology; yet they are studied as $§ structures, in a classic method of geometry that ‘freezes’ as minds do time dimensions into space-forms. So 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’ that define paths as closed loops, departing from a 0-point – the neutral element, to which the path returns.

And this connects them fully if 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-centred 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 again have huge applications to reality from string theory – where T duality, the most interesting finding which 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 ‘stomachal’ 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 analysed 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. (Rashomon effect) 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 pov. 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 Xˆn (n = 0, ± 1, ± 2, ···), where we have to understand by Xˆn (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 Xˆm and Xˆn 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 pathsstarting 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 generalisations 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 realisation an entire Universe can be shrunk into a still mind view.

$t: 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 UˆmVˆn (m, n = 0, ± 1, ± 2, ···) with the equivalences: UV ≈ VU and UˆmVn ≈ Uˆm1 Vˆn1 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’; reason why we use Feynman and Aleksandrov, clear verbal conceptual texts as template for our comments):

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 bicontinuous transformations are studied in the fundamental mathematical discipline of topology, whose basic ideas have been explained. Invariants of bicontinuous transformations are called topological invariants.

We deduce that the group of paths is one of the most remarkable examples of topological invariants, as the Ƽ 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.


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 toroidal 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’.


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 nonisomorphic, we can conclude that the knots themselves are inequivalent. 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 noncommutative 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).


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 artefact of the mind not a reality – and to forget that is the biggest sin of creationist mathematics. ‘Point’.

Let us consider then only one example, using the torus as the richest topological form to illustrate such forms of modelling:

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 we have seen that 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 modelling 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.


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.


screen-shot-2016-11-21-at-12-57-29 screen-shot-2016-11-21-at-12-56-26





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.

screen-shot-2016-11-21-at-12-55-42We can then recognise a ‘cellular-atomic-social’ system of fractal units that build a self-similar closed (spherical) open (hyperbolic) or toroidal (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 summarise 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.

Classic topology.

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.

Now the first theorem of topology is called Euler’s characteristic.

Let us consider the membrane of a system, which always can be approximated topologically with points, lines and planes.

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 Spe-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.


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 onetriangulation 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 Spe into Tiƒ, 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:

Screen Shot 2016-04-05 at 23.45.26

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 pairwise 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 two-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-effcient 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 Spe<St>Tiƒ 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:Screen Shot 2016-05-19 at 17.23.17

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 pst, 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.


And viceversa we shall study also topological transformations/motions that reorganise 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.


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 demotions.

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.

An example of this concept is found in the cycloid:

bagua-11 phases of life

Mathematical laws encode the simplest relationships of topology and life through the connecting ‘sequential phases’ of its cycles of existence. In the graph two different isomorphic languages: the 8 diameters of the simplest cycle of existence, a cycloid moving on a lineal, open path from birth to death and the 8 baguas the informative, mongoloid human subspecies found to correspond to the 8 phases of life. Such type of relationships show the entangle Nature of all the languages-mirrors of the ultimate T.œ game.

Fermat and Descartes struggled to understand the properties of the cycloid, a curve not studied by the ancients. The cycloid is traced by a point on the circumference of a circle as it rolls along a straight line, as shown in the figure.

Roberval first took up the challenge, by proving a conjecture of Galileo that the area enclosed by one arch of the cycloid is three times the area of the generating circle.

Christopher Wren in England found that the length (as measured along the curve) of one arch of the cycloid is eight times the radius of the generating circle, demolishing a speculation of Descartes that the lengths of curves could never be known. Such was the acrimony and national rivalry stirred up by the cycloid that it became known as the Helen of geometers because of its beauty and ability to provoke discord. Its importance in the development of mathematics was somewhat like solving the cubic equation, for a reason: the hidden beauty of the cycloid 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 cycloid is the simplest isomorphism to 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.

So the ∆-1 generational o-1 age is followed by 3 more ages of equal vital value:

2D: Max. $ x Min. ð= youth 3D: ∑=∏: maturity, 1D: Max. ð x Min. $: 3rd age…

4D: And then the cycle ends in the ‘landing’ on the point on the lineal entropic timeline, exhausted its vital energy.

Further on, the great mathematicians of Classical times interested in variational problems solved the famous problem 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 – when they realized it was part of an upside-down cycloid,  traced by a point on the rim of a the rolling circle.

The interest here being that the brachistochrone is the spatial symmetry of the law of least time; a key law of ∆st, as systems try to achieve its actions and motions in the less possible time.

This simplest representation of the world cycle shows already the key new/old insight of ∆st in curves represent on analytic geometry – to refer them as spatial representations of temporal flows that follow the laws of T.œs.

This symmetry will become even more sophisticated when we observe the properties of…

Canonical curves of 5 Dimotions:

Conics and spirals.

Conics’ general equation is: Ax² + Bxy + Cy² + Dx + Ey + F = 0

As such it is the canonical curve to define a holographic bidimensional manifold the fundamental ST symmetry of space-time. And the cone, itself a circle moving along a line, as it decreases its size, represents as it does the cycloid, a fundamental motion of space-time between scales: ∆+1 ð>∆ ð and then forwards. So the 2 inverted cones form together an image of the 5D-4D inverse collapsing and expanding, evolving and entropic arrows of space-time.
But if 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 a world cycle of existence, what mathematicians have focused on, are slices of 
space-time, along that trajectory

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 very well known to the mathematicians of antiquity.

So the first obvious fact is that ‘cyclical, time-like curves’ have 2 dimensions (degrees) as opposed to single-dimensional Spe-lines.

And with both, the line through which a cycle ‘moves’ and shrinks in an ‘accelerated’-timelike process we can construct the Universe.

This was the wonder of the Greeks and Descargues – who proved that all curves can be drawn from the conic, the simplest |xO=ø representation of the world cycle. So we can start translating the hidden meanings of those facts, as usual, to GST:


screen-shot-2017-02-05-at-16-10-20The 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.

What this means in GST terms is that the hyperbola is the opposite concept to the circle/ellipse, as closed and open, Tiƒ and Spe inverse ‘geometries’ which if we consider the y axis, the longitudinal entropic axis, and the x axis of the cycle, the informative one, yields an inverse Space-time graph within the dual cone, which can be used (and will be used) to represent world cycles and space-time events.

How can then extract vital properties to those forms? Easy as we explained in the introduction – simply by considering the cone a representation of a world cycle (as studied in the post on spirals, and partially understood in 4D by Minkowski) and considering the efficiency of each form of a conic, for each type of vital Dimotion of existence, which will ensure that all type of fractal points trace them for a vital purpose.

Vital Geometry, symmetric properties.  The pentalogic of 5D conics.

It is then essential to understand that each frame of reference and the formal figures of geometry described by it, will have vital organic properties as describing what best suits for each dimotion of existence. A simple example will suffice:

  • The reasons why the sphere or circle are the best forms for the first Dimotion of informative gauging, aka perception.

Symmetry is the central concept of Group theory, which became in the XX century the ice in the cake of the whole structure of Algebra.

By symmetry we mean in 5Ð Algebra, which as all branches of mathematics born of the spatial, bidimensional ‘still’ work of the Greeks LACKS A TRUE COMPREHENSION OF THE PARADOX OF GALILEO: S (form in space) = Time (motion), but uses it profusely (so differential geometry is based in the concept that a line is a point in motion), the capacity of all systems to complete a zero sum world cycle, through a motion that returns the system to a present undistinguishable new state.

Symmetry thus IS ESSENTIAL TO THE ENTIRE SCAFFOLDING OF THE 5D UNIVERSE, albeit ass all concepts of 5D once we understand the basic laws of pentalogic, has a more dynamic view.

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, who 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.

Symmetry though then must be connected to the different Ðimotions and its ‘requirements’ to perform the vital actions for which they are conceived.

I.e. the circle IS the perfect symmetry for still 1Dimotions of perception, as it will turn out that from any pentalogic point of view, it maximizes the stillness, by symmetry, by its capacity to focus as a fractal point all lines of communication that fall into its focus, by having the minimal perimeter, which maximizes ad maximal its volume of information and disguises it in an external world, and so on.

However, when we consider the 2nd Dimotion of Locomotion, which is the process of displacement in space while retaining the form in time, as it implies the reproduction of form, of information in other adjacent region of space, the less information to be displaced, the faster reproduction will happen. 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 maximizes the second Dimotion.

Here then the use of the concept of mirror symmetry is NOT required, as the line is moving-translating in space, reason why also forms in motion tend to have a spherical head, to perceive only on the foreward position and a small one in relationship to the body and limbs that have lineal forms to maximize motion.

Symmetry here is of another kind, defined by Noether’s theorem of physics; and in 5D by a type of symmetry ignored in science – the ‘undistinguishable’ property of the ¡-1 elements in which the system imprints its form. 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 states which acquire the same form of present, when a system completes a Dimotion, re-establishing its ideal form, is therefore the essential element for all symmetries that accomplish one of the five vital dimotions of existence.

And the reason of the survival of certain geometric forms above all others, the circle for perception, the line and its curved form the parabola for motion, its combined wave for 3D reproduction, the social circles, from elliptic forms to polygons for social evolution, and the different forms of open curves, notably those dual forms, as the hyperbolas are for the 4D entropic Dimotion and dissolution of a system in two forms, which in the cone as a representation of a worldline will be split, one hyperbola branch going upwards  and the other downwards, if we take the axis of the cone as an ideal representation of the fifth dimension.

 Ellipse, Hyperbola, and Parabola: The conics.


Before investigating the general second degree equation, it is useful to examine some of its simplest forms.
The equation of a circle with center at the origin. First of all, we consider the equation: x²+y²=a²
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.
The equation of an ellipse and its focal property. 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 parabola and its directrix.

Thus we define  the parabola as the graph of quadratic proportion and the hyperbola as the graph of inverse proportion. We recall that the graph of quadratic proportion: y=kx² is a parabola  and that the graph of inverse proportion y = k/x y x = K is a hyperbola. Let us then consider the parabola from the perspective of its foci.


screen-shot-2017-02-05-at-15-51-46We consider 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. Let M be any point of the parabola (figure 24), 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 an important property of the tangent to a parabola and its application in optics.

Since for a parabola y2 = 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:



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 reflecting telescopes and modern antennae, as invented by Newton. 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 originaring 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.

Thus, in the so-called focal plane through the focus of the mirror and perpendicular to its axis, the inverse image of the star is obtained; 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.

The image so obtained can be viewed in a special microscope, the so-called  eye piece of the telescope, either directly or, in order not to cut off the light from the star with one’s own head, after reflection in a small plane mirror, attached to the telescope near the focus (somewhat nearer than the focus to the concave mirror) at an angle of 45°.
screen-shot-2017-02-05-at-16-02-54 screen-shot-2017-02-05-at-16-03-04The 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.

The parabolic being is thus a single ‘foci’, which is able to ‘focus’ the information and entropy of a given field, in as much as the field comes to it, without the need of a second focus, which would complete the symbiosis between both. 

Indeed, in the case of an ellipse, as 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), and 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:


The directrices of the ellipse and the hyperbola.

Like the parabola, the ellipse and the hyperbola have directrices, in this case two apiece. If we consider a focus and the directrix “on the same side with it,” then for all points M of the ellipse we have ρ/d =ε where ε the constant is the eccentricity, which for an ellipse is always smaller than 1; and for all points of the corresponding branch of the hyperbola, we also have ρ/d =ε , where is again the eccentricity, which for a hyperbola is always greater than 1.
screen-shot-2017-02-05-at-16-08-51 screen-shot-2017-02-05-at-16-09-00

Thus the ellipse, the parabola and 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.

We have now considered the most important second- order curves: the circle, the ellipse, the hyperbola, and the parabola. What other curves and generations are relevant, or at least exhaust the field of bidimensional geometries? Let us consider this question from a couple of ∆ST perspectives.

There are no more curves than the ones needed to define the 5 Dimotions of reality.

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:


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.

The interpretation of those curves in terms of non-euclidean geometry and its 5 Dimotions. 

The spiral represents the first Dimotion of perception.

Even if it is not considered part of the cone, a spiral should also be described within them, as a 3D motion along the surface of the cone from its base to its point:

The logarithmic curve can be written as:  r = a  e ˆbθ.

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

A such 3 Dimensional view of the log spiral can be considered a curve along a cone of space-time, the proper representation of 5D worldline:

It IS AN ACCELERATED, HENCE PERCEPTIVE SPIRAL. Yet 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 it. It does then represent a world cycle accelerated but perceived as an ascension in height.

The second postulate, then considers the communication between fractal points with volume that create waves and flows of communication, while the 3rd postulate defines a network of points that becomes a plane:

The other varieties have also to do with the vital 4th postulate of Non-E Geometry and the definition of the 3 topologies of vital communication:

The law of behavior in the organic Universe is 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, perpendicular  manner.

A case of Darwinian devolution among men that perceive each other as different and enter into a perpendicular, Darwinian relationship and a case of social evolution between 2 forms that perceive each other as equal and enter into a parallel relationship of social love. The fourth postulate of non-Æ topology thus ‘vitalises’ the laws of mathematics, establishing the three fundamental geometrical≈behavioural relationships between T.œs, according to its:

  • Informative (Particle-head) communication, possible in cases of relative similarity≈parallelism (which determines parallel herding and social evolution).
  • Perpendicularity≈difference in particle-head, which will determine if one system is related to the other and there is body-similarity, hence can be used as energy its darwinian destruction,
  • Or 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.

Indeed in three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. It follows that 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.

The different degrees of Parallelism, Perpendicularity and skewness are thus essential concepts of vital Non-Euclidean geometry.

In that regard the VITAL interpretation of the different geometries according to parallelism, perpendicularity and skewness opens an entire new field of interdisciplinary analysis closely related to the field of ‘topological evolution’, treated in more detail in other posts of the blog.

Since systems IN ANY scale of the inverse, 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, 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, so you understand that in the Universe organic, geometric and scalar relationships are symbiotic to each other. 

So in those terms we can understand all the curves of a conic reality, which are all the curves of the Universe:

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:screen-shot-2017-02-05-at-16-53-49

We can then see the circle and the cone as a 3D spiral  the 1 Dimotion and non-euclidean postulate.

The second postulate and 3Dimotions of reproduction and communication are expressed by elliptic forms.

The 2nd Ðimotion is obviously 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 ( social evolution) and divergence (hyperboloids between two systems) or pure entropic 4D dissolution of a system – parabolas.


Other |xO generations.

Not only the conic can be generated by | x O dualities as it is the case basically all forms of nature can.

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 x2/a2 + y2/b2 – z2/c2 = 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:screen-shot-2017-01-26-at-19-34-22

But this equation together with y = a gives in the plane y = a a pair of intersecting lines: x/a – z/c = 0 and x/a + z/c = 0.
screen-shot-2017-01-26-at-19-32-33Thus we have already discovered that 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:

1. 2 arbitrary straight lines of one and the same family of lines so obtained do not lie in the same plane (i.e., they are skew lines)

2. any line of one of these families intersects all the lines of the other family (except its opposite, which is parallel to it.

3. three lines of one and the same family are not parallel to any one and the same plane.

Now, how we write CONIC equations in GST terminology?

The first thing we realise is that it is a closed domain, either in the single cone or the dual one, and that those equations are written as G(x,y) = 1, for closed or ‘dual’ semi-closed hyperboles, and as G(x,y)=0, for the open parabola.

Since we have established that ∆-1 is the infinite-simal domain of 0-1, and ∆+1, the infinite domain of 1-∞ on the fractal scales of the Universe, it follows easily that we are working in the closed domain where 1 means the whole, and 0 the singularity and can write them as: G(x,y) = o+1. And so those closed cures really mean:

G(x, y) : ST= 1 (s) + 0 (t).

So that is what a conic means, the membrane 1, the relative infinite, outer enclosure and the 0, the relative center or singularity enclose the G(x,y), where all the combined points, (<x, <y), form the inner region.

And since the Universe is for each layer of identical being immersed in a gradient world, bidimensional, the result is that those equations are the most pervading in all forms of Nature, further on proving GST is real:
In the graph the Universe is bidimensional and holographic, polynomials reflect this fact in the degree of its equations.

Yet only one-dimensional, 2-dimensional and 4-dimensional equations are truly real, being 3d equations mere superpositions as Fermat’s grand theorem proved (since x³+y³≠z³).

How then we deal with equations which are NOT bidimensional or 4-dimensional representations of the real holographic Universe?

By Reducing equations to 2-holographic forms…
Descartes in fact already realized of this and presented a method for determining the real roots of third and forth degree equations, both analytically and studying the intersection of the parabola y = x² with circles, which was the first representation of the holographic method of creation of the Universe: 

He first showed that the solution of an arbitrary third-or fourth-degree equation can be reduced to the solution of an equation: (1)screen-shot-2017-01-28-at-10-47-25

which we shall call the ‘holographic equation’ (as it has only 1, 2, 4 polynomials).

Let the given third-degree equation be z³ + az² + bz + c = 0. Substituting z = x – a/3, we obtain:


The x²-terms in the expansion of the parentheses will cancel out, so that we get an equation of the form x³ + px + q = 0. Multiplying this equation by x, we bring it to the form (1) with r = 0, which also admits a root xˆ4 = 0.

An equation of the fourth-degree zˆ4 + azˆ3 + bzˆ2 + cz + d = 0 can be reduced to the form (1) by the substitution z = x – a/4. Hence, the solution of all third- and fourth-degree equations can be reduced to the solution of an equation of the form (1).
The solution of third- and fourth-degree equations by the intersection of a circle with  the parabola y = x².

Let us first derive the equation of a circle with center (a, b) and radius R.

If (x, y) is any of its points, then the square of its distance to the point (a, b) is equal to (x – a)² + (y – b)².

The equation of the circle in question is: (x – a)² + (y – b)²=R²=1+0 for a ‘unit’ circle (we ad the singularity 0, to stress we have defined the 3 elements of the circle).   Now we try to find the points of intersection of this circle with the parabola y = x². In order to do this, it is necessary to solve simultaneously the equation of this circle and the equation of the parabola. Substituting y from the second equation into the first, we obtain a fourth-degree equation in x:


screen-shot-2017-01-28-at-10-56-35 If we choose a, b and R2 such that:


Then a ‘holographic equation’ (1) is obtained and all its real roots are the abscissas of points of intersection of the parabola y = x² with this circle. (In case r = 0, this circle passes through the origin and the are only 2 intersections; if R² < 0, equation (1) is known to have no real roots and there is no interrsections.

But in GST this tells us more, when we perform the inverse decomposition of a holographic quartic into its 2 ‘bidimensional forms’, the Tiƒ closed circle and the Spe-expansive parable: ST (QUARTIC) : Tiƒ ≈ Spe, ‘recorded’ with fire in my mind. Now, most Tiƒ parts are self-centred and have 2 roots only where its Spe (limb-field) function intersect with its Tiƒ part, determining a closed inner space, which we shall call its body-wave; while some are all positive, having only an intersection with the parabola, whose deep meaning, we shall consider:

screen-shot-2017-01-28-at-11-21-18THE OSCULUM of the sphere on the open other main quadratic form make us realise the main curves allow us to define in vital terms the conic game. A sphere or polygon of 19² bidimensional = 361 ‘degrees’ of inner points, in the squared complex plane an Rˆ2, becomes in a single plane a disk, which encloses its vital energy and whose actions must be defined inter ally. The parabola is thus its death into an explosion of upward entropy that ends that vital energy…

Let us now after this ‘classic introduction’ to basic @-geometry go much deeper connecting it with…


There are 3 relationships in space-time between entities, which are part in non-Æ of the laws of the fourth postulate of similarity, and of course, they are 3 because we can relate them 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’ – a waste of time for researchers too (: 


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:

1DΓ: $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…

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:

2DΓ: $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.

Angles of perception

We can also consider other ‘vital properties’ till today merely treated in abstract terms, of fundamental importance to the vital geometry of the Universe.

Indeed, what is about ‘angles and triangles’ so intense in ∆st that the geometric mind-mirror is obsessed by ‘it’.

SIMPLE ENOUGH: 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.

Minimal ‹: It is the case of a hyperbolic plane, again showing that 2-manifold hyperboles tend to entropic freedom minimalizing the inverse arrow of information, reason why they are indeed the herded vital energy of the ts-ystem.

Maximal <: And so inversely the triangle in the sphere whose angles are greater than 180 makes spherical beings in ∆-1, living as parts of an ∆-spherical world more perceptive.

Medium «: The plane is in thus in the middle term, with angles at 180.

So here we come to the first seemingly contradiction as we expand our dimensions in the function/form of the next scale.

The kin observer indeed 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 (Straightest ) 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. 

But 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 trasposed. 

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’. 

So we close here the section on the 3 frames of reference, knowing this is only the tip of the iceberg.

screen-shot-2017-01-10-at-18-44-59Since as Descartes did, I will end a summary of its work, saying:

‘I hope that posterity will judge me kindly, not only as to the things which I have explained, but also as to those which I have intentionally omitted so as to leave to others the pleasure of discovery.’ (:



The theme of Geometry and physics is obviously well beyond anything this author can develop in a few notes, even if it ever comes to complete the fourth line – doubtful, as with no sights this year, 2018 when I finally consider despite its shortcomings, the work to deserve word of mouth a huge following by university students on planet Earth,  it is very steep for me to take the computer ‘again’… pass an age in which mental laziness sets in… S

o the reader specially if physicist should not expect more than some marginal comments. As usual we tend to use for all themes of mathematical physics books of the Russian school of dialectics. So we won’t be quoting them.

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 4Dimotions of entropic parabolas of cannonballs, and Kepler, dual, 2Dimotions of interactive orbits of planets with ellipses.

The importance of bidimensional curves: Holographic physics.

Once we understand bidimensionality we can enlighten physics in its simplest mathematical statements, which historically dealt with the laws of astronomy force and motion, heavily drawn before analysis became the meaning of m.p. with the canonical:

  • Bidimensional ‘open’ curves – parabolas: entropic motions   as  incannonball 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)

Let us then consider only an example, the conic curves and its relationship with gravitational forces and 5D metric equations.

Now 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 we can understand the conics as a dynamic transformation between $pe (open) < ≈ > ðiƒ (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.

Consider for example…

The ellipse as the result of “contraction” of a circle.

We consider a circle with center at the origin and radius a. By the theorem of Pythagoras its equation is , where we have written 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 -values become equal to , i.e., . Substituting for in the above equation of the circle, we will have:screen-shot-2017-02-05-at-14-20-45as 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. Thus we have proved that an ellipse is the result of a “contraction” of a circle.
From the fact that an ellipse is a “contraction” of a circle, many properties of the ellipse follow directly:

For example, 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.

Now, we shall see the fundamental quality of GST – to offer at least 2 or 3 ∆ST causalities to any event of reality. We cannot be exhaustive, but a simple example will suffice, on a property of cubics and ellipses, with profound implications in physics.

This remarkable result shows even more clearly the ‘squaring’ nature of π and the subtle difference between a² and a x a. A square is a perpendicular product, hence of 2 different dimensions, a product happens only in one dimension, a duality which ‘must’ exist in the fractal Universe for both scalars (single numbers that represent mostly time frequencies and densities of ‘past populations’) and vectors (dot vs. cross product); as the potency and cross product DO create a new dimension. Numbers are forms say Plato and Pythagoras drew them in figures, facts essential to relate numbers in time and points in space, forgotten by the lineal plane and overdevelopment of abstract algebra.

Now, we see the fundamental quality of GST – to offer at least 2 or 3 ∆ST perspectives and causalities to any event of reality. We cannot be exhaustive, but a simple example will suffice, on a property of cubics and ellipses, with profound implications in physics.

All together now: cubics representing the 3 elements, singularity, lineal membrane and body-wave in algebraic space.

Back to the cubic question, the next fella after Descartes to work on them was Newt, with a beautiful description of the  ‘3rd layered dimensions of a being’, and its inner structure such as the form of the st SPACE is contained by an enclosing curve, itself centred in a single point; by 2 methods, obtained by Newton from its study of cubics, and obtained from the contraction of a ‘sphere’ into an ellipse:

Let an nth-order curve be given, i.e., a curve which is represented by an nth-degree algebraic equation in two unknowns; then an arbitrary straight line intersecting it has in general n common points with it. Let M be the point of the secant that is the “center of gravity” of these points of its intersection with the given nth-order curve, i.e., the center of gravity of a set of n equal point masses situated at these points. It turns out that if we take all possible sets of mutually parallel secants and for each of them consider these centers of mass M,then for any given set of parallel secants all the points M lie on a straight line.

Newton called this line the “diameter” of the nth-order curve corresponding to the given direction of the secants.

In case the curve is of the 2nd order (n = 2) the center of gravity of two points is simply the midpoint between them, so that the locus of midpoints of parallel chords of a second-order curve is a straight line, a result that for the ellipse, as well as for the hyperbola and the parabola, was already well known to the ancients. But this was proved by them, even though only for these partial cases, with quite difficult geometric arguments, and here a new general theorem, unknown to the ancients, is proved in an entirely simple way.

In ∆st theory it reveal an even deeper truth: systems become ‘compressed’ into smaller networks and final singularity points, which control the entire force, motion and organisation of the system.

Thus if we 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.

we go back to the equation of the ellipse:

screen-shot-2017-01-28-at-15-12-26 and its simplification by substituting:screen-shot-2017-02-05-at-14-45-20

And set a² – c² = b² to obtain: x²/a²+y²/b²=1

The coordinates (x, y) of any point M of the ellipse thus satisfy this equation. It can be shown on the other hand that if the coordinates of a point satisfy this equation then they also satisfy the more complex one. Consequently, the equation x2/a2 + y2/b2=1  is the equation of the ellipse:

Substituting y = 0 in  the equation of the ellipse, 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, so that, since , the eccentricity of an ellipse is less than 1. In the case of a circle, c = 0 and consequently = 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:
screen-shot-2017-02-05-at-14-38-24And this simple fact if we apply to one of the points the concept of ‘motion as indistinguishable of distance’ explains the orbital laws, in which the planet and the sun together scan the same ‘gravitational area’ :screen-shot-2017-02-05-at-14-38-35In the graph, the aerolar law of a physical vortex might imply as so many have seeked, a second ‘elliptical focus’ of the sun, playing the eccentricity focus. But there is nothing there.

Indeed, the empty 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.

Since the “centre” of the major axis is halfway between these two points, you can calculate (and visualize) the position of the empty focus in this manner.

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 147,097,907 km
In 2010, Aphelion was on July 6, at 152,096,520 km.

Total length of the major axis (from P to A) is the sum = 299,194,427 km
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 147,097,907 km, then the distance SC must be the difference PC – PS = 2,499,306.5 km

The Sun is 2.5 million km to the “January” side of the centre.

By symmetry, the empty focus is 2,499,306.5 km on the “July” side of C

CE = 2,499,306.5

SE = SC + CE = 2*CE = 4,998,613 km.

If you had looked towards the Sun on July 6, you could have imagined the empty focus to be on the line between us that the Sun, at a distance of 5 million km from the Sun (or roughly 147 million km from Earth).

There is nothing at that point, since it is only a mathematical point that results from our definitions of an ellipse.

So the explanation is the GST explanation: all distances are motions. So if we make the static planet a motion, with an eccentricity one, what truly matters of the ellipse – that 2 points ‘sweep the same area’ working together, stays. An orbit, indeed in the physical analysis of GST becomes a dual system of a membrane with more momentum (the Spe-planet) and a singularity with more gravity/mass (the Tiƒ), surrounding and absorbing the lower ∆-1 scale of gravitational points.


For example, the length of the sides of a rectangle completely determines its area, the volume of a given amount of gas at a given temperature is determined by the pressure of its walls, and the 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.

T: Dynamic forces Follow such line of thought: another physical example in the ‘ellipse of inertia of a plate.

Following such line of thought we have another physical example in the ‘ellipse of inertia of a plate.
Let the 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):271

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.

screen-shot-2017-02-05-at-15-47-12Steel 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.

Again this result can only 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.

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:


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:

S x T = K, Which is by definition, the equation of…  The hyperbola and its focal property.

Indeed, consider this equation… x²/a²-y²/b²=1  …representing a curve which is called a hyperbola:screen-shot-2017-02-05-at-16-13-11
In the special case a = b the so-called rectangular hyperbola plays the same role among hyperbolas as the circle plays among ellipses.

In this case, if we rotate the coordinate axes by 45°  the equation in the new coordinates (x′, y′) will have the form: x’ y’ = k:screen-shot-2017-02-05-at-16-13-25

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.


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 Spe x ðif = K:

Consider a simple formula for the Pressure p, due to a liquid column:  P 􏰖=ρ  x g ×h

It is obvious knowing that density is a measure of the density of form, or information of a system, h, the height dimension of information and g, acceleration, a parameter of a time vortex, and its growing frequency, that pressure is a pure Tiƒ 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 Tiƒ 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.

In the graph, 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’.








As we said, the realisation that ‘mathematics-geometry’ is a mental-logic endeavour, where function and i-logic thought overcomes ‘spatial representation’, ALLOWED the explosion of abstract mental spaces to represent reality that ensued the work of Lobachevski, and in this task it would be essential the idealist school of Germans, we so harshly and often criticize for its unconnection with reality that paradoxically allowed further expansions of mental space, latter put in correspondence, §@<≈>∆ð, with the real Universe.

This task was the job of the 2 next masters of the III Age of mathematical space, Mr. Klein from whom we borrow the definition of ‘dimension’ as a co-invariant space-time which allows motions through it (Sp x Tƒ = ∆-constant being the co-invariance of scalar space-time that allows world cycle motions through it)  and specially Mr. Riemann, another ill-understood, die-young genius, which at least had unlike Lobachevski and this who talks to you, the luck of being born in the proper place and have a master in Gauss, who did capture his thoughts, or else following the tradition of the Homunculus little mind, with handy appendices and a big hyperbolic mouth, would have been ignored as most men of mental note.

Klein thus 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 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. We can consider transformations of not necessarily the whole space. Thus, by considering the points and chords of a circle under all its transformations into itself that carry chords into chords and by singling out the properties that are preserved under such transformations, we obtain the geometry which Klein shown as we have seen to coincide with the hyperbolic geometry of a vital inner space-time of a t.œ.

It follows then as a corollary that ONLY by unifying all the perspectives and partial descriptions of a being (Rashomon effect) we can get the whole truth of the being, the ESSENTIAL LAW OF EPISTEMOLOGICAL TRUTH of the pentadimensional space-time universe.

Reason why space is neither hyperbolic, euclidean or elliptic but a mixture of them all.

This corollary which Klein applied to projective and affine geometry, barely touched into this introduction to non-E, would have two explosive new developments:

  • Topology, where we consider only topological transformations, that is, those who do not change the properties discovered by Lobachevski as the ultimate ‘vital properties’ of space (complementary adjacency, continuity required for smooth motion and so on).
  • And Riemannian Phase spaces, in which the properties and Dimensions of the being are NO longer required to be ‘space-like’, but can be of any ‘quality’, as long as they again are ‘useful’ to define the vital organic Ðisomorphsims of space-time beings, among which the ‘identity of social numbers’ that allow scalar social growth that makes wholes stronger than parts, are the most important.

We are thus coming closer to the barebones of modern geometric thought ‘fried’ in the reality of a vital Universe: geometrical properties that matter, such as adjacency, continuity, perpendicularity, parallelism, motion as transformation and reproduction of form without internal change are ALL properties which display vital organic properties that allow the system to survive. As if a topological ternary S-limb/potential < body-wave ST> Tƒ particle-head would be torn in its parts when moving it would become extinct as being – reason why perpendicularity that penetrates and breaks the being is so damaging; and a motion that does not preserve continuity will deplete the being of its inner parts-∆-1 points; while a social communication which is not parallel would not keep the necessary distances to leave space to ‘create’ a new network that will emerge as a digestive/reproductive/informative higher scale to form a social super organism, and so on. So we affirm that:

‘The properties that matter to construct geometrical spaces as mental mirrors of reality are those properties that reflect the Ðisomorphic properties of organic time§paœrganisms’.

All other geometrical spaces which do not study those essential vital properties are considered fictions, inflationary baroque unconnected mind constructs, similar to the crazy thoughts of a self-absorbed ‘axiomatic’ old man biasing reality to cater to his madman psyche.

Let us consider those ‘geometries that truly matter’ as mirrors of reality some unseen at the time of its realisation.

An ∆st definition of multi-dimensional space-time in terms of vital actions.

A many-dimensional space is then a formal generalization of the usual analytic geometry to an arbitrary number of variables that represent both Space-form and Time-motion dimensions, as a D-isomorphic property that is shared by systems who can be grouped in reality by that dimension as its identical property allow them to gather into herds and super organisms as social numbers of that dimension.

i.e if a herd of lions share the dimension of entropic feeding in zebra meat, they will be gathered into a social number of the log10 scale (normally evolving socially from 10º=1, the individual into 10¹ the genetic family across 3 co-existing simultaneous space-time generations). The dimension of entropic feeding thus originate an inverse dimension of 10-social evolution, which can be distinguished in space as the coming of 3 time symmetric generations into a herd, with the purpose of enacting an ST dual ‘feeding-absorbing energy’ space-time event. It then appears as an obvious truth that membrains DO not on close analysis act as continuous enclosures, but due to the motion of its fractal points can encircle as a dog does with the herd of sheep, a much larger territory.

They might not even be ‘real’ membrains, as perception in a relative Universe of dark spaces and faulty mental analysis allows disguise and camouflage. So in a fascinating similar case, whales substitute their presence by walls of ‘bubbles’ that fishes confuse as physical barriers – creating a 3rd volume dimension which brings them upwards to the flat holographic surface space where the real whales eat them.

It is all in the mind-space and its relative focus of perception, which determines through its models of reality the efficiency of its vital actions.

And we can represent further that hunting process in 10-dimensional space where each point-lion is a part of a whole, but also we can just draw a 2-D holographic representation as a ‘Klein disk’ of the hunting strategy of the lion herd, which will surround the herd of zebras, establishing them as a hyperbolic vital energy, as the zebras CANNOT cross the barrier of lion, the membrane, without dying, enclosed topologically in that 2-D flat space-time whose limiting barrier is at infinite, as the zebra who dares to cross it will die. But when it does so, it will ‘collapse’ the membrain into an ultra dense singularity of feasting lions around the captured vital energy, breaking the enclosure for the rest of the zebras to escape.

Measurement of distance in Riemannian geometries.

We can return now to a question which we left unanswered but it is essential to understand the symmetries of §@≈∆ð, the meaning of distance, a concept of space, in terms of time, a motion in logic sequence.

We gave a first approach to this ESSENTIAL concept in the Universe in the opening post of this blog, regarding the Galilean paradox, as an expression of the 1D symmetry between ‘time motion’ and ‘space distance’, but we can now go into deeper Kaleidoscopic views on the ‘Rashomon effect’ of the concept distance, when we abandon the limited Euclidean light space-time in which that S-distance ≈ t-motion duality takes place, with our ‘Riemannian example’ of colours (similar to the heat concept of earlier time analysis of frequency spectra, if we were to take a time-related ‘Fourier’ example): 

We have a natural idea of the degree of distinctness of colors. For example, it is clear that pale pink is nearer to white than deep pink, and crimson nearer to red than to blue, etc. Thus, we have a qualitative concept of distance between colors as the degree of their distinctness, which is the most generalised ‘term’ for distance in the whole set of 5 Ðisomorphic s≈t dimensions of the Universe, expressed now as a symmetry between ‘1D $t distance and 3D §ð-information:

Distance in 1 D is equivalent to ‘distinctness’ in 3D information

As usual the subconscious ‘truths’ of verbal thought has intuitively understood this condition of distance – ‘we have distanced each other’, we say from a friend no longer ‘close‘ to us.

How this distance is measured in qualitative terms depends on which ‘pair of dimensions’ we are measuring in our ceteris paribus most common ‘metric’ of distances that happens between ‘two t.œs’.

It is then possible to ad distances in ‘pairs of dimensions’, which a certain t.œ has till a maximal of 10 Dimensional distances, whose homogeneity obviously is not always ‘possible to measure’, but we do indeed measure it subconsciously in our human plane, with its verbal mirror of the same ∆st existential game. i.e. when for example a woman ‘measures up’ a ‘man’ for a ‘close encounter’ that will last not only in space but in time’ (marriage), it does take into account ‘different dimensions’ of the being and according to his likeness, it will either become closer or not.

Thus the entire subject of distance is related to the fourth i-logic postulate of similarity:

In the graph, the 4th postulate of similarity (congruence in E-Math) is essential to understand why and how systems select information to construct their mental spaces. And create dark spaces they do not see because their information is irrelevant to them both in the negative (no predator) and the positive (no prey, offspring, couple, etc.) So we can consider safely that the engagement of T.œs is directly proportional to the ‘utility’ of the observable for the realisation of any of its 5 fundamental ‘dimensional actions’ that enhance its existence, portrayed in the next graph for a series of different scales and species:

It follows also an important field of ‘theory of dark spaces’ and the virtuality and local limits of perception. Such as there are space-time beings we do not care to perceive, and so they become first dark spaces and then enter the horror vacuum NOT even being perceived virtually and dark, reason why we can see ‘continuity’ and construct ‘full circles without an exact pi’ and geometric figures by disregarding the ‘other scales of the fifth dimension’, from where another ‘angle’ on a basic postulate of present-space is given:

‘Space happens as a continuous form in a single plane of existence’ . WHILE ITS detailed views that open discontinuous require a fine detail in its ‘decimals’, ‘finitesimals’ or ‘fractal steps’.

It is then clear that the concept of distance as dissimilarity looses its geometrical meaning entering the realm of logic, to be defined differently for each Measure and mental space/scale, dimensional set of parameters.

As we study the expansion of geometry to all ¬Æ T.œs on the section of fractal points, generalising the concept of distance, which obviously can be topological in form of space, temporal on age/state and scalar on number of ∆§ocial planes, we will just comment here the classic expansion of distance in non-E geometries.

Let us then put an example simpler than the human choices, returning to the simplest cases of mathematical physics and Riemannian geometry (: compared to the way women measure and ‘size up’ a man ).

We have a qualitative concept of distance in the space of colours, which can be made into a quantitative measure. However, to define the distance between colors as in Euclidean geometry is meaningless, since we need to measure each type of dimensional distance two distinctive elements:

  • The ‘quality’ of the property we measure – the type of distance, which in colours relate to frequency in abstract, but to PERCEPTION of different colours by the observed.
  • And hence we will need accordingly a mental step of ‘distinction’ in which the subjective observer capacity to discern different information will give us the ‘constant rod of measure’, in this case able to reflect the real relations between color perceptions.

Guided by this principle we introduce a peculiar measure of distance in the space of colors.  When a color is altered continuously, a human being does not perceive this change at once, but only when it reaches a certain extent exceeding the so-called threshold of distinction. In this connection it is assumed that all colors that are exactly on the threshold of distinction from a given one are equidistant from it. We are then led automatically to the idea that the distance between any two colors must be measured by the smallest number of thresholds of distinction that can be laid between them. The length of a color line is measured by the number of such thresholds covering the length of the shortest line joining them.

Thus, again measurement of length and distance in the color space shows a mental/quality/form dimension and a quantity relative to the r/k ratio of each observer, which becomes when the observer and its rod is very small, infinitely large as the sum of those small, steps.
As a result, a certain peculiar non-Euclidean geometry is defined in the color space. This geometry has a perfectly real meaning: It describes in geometrical language properties of the set of all possible colors, i.e., properties of the reaction of the eye to a light stimulus; and it has not only theoretical but also practical value in the art and color industry…

As it was also a loved theme studied by many physicists from helmholtz to Maxwell, we shall complete it with 5D insights.

Multidimensional Phase spaces. Color spaces and Riemann geometries.

in the graph, the understanding of dimensions as it distances as mental mirrors of vital actions, which select what we perceive and what kind of dimensions become dark spaces  has not yet being understood in humind sciences, but it is the final conceptual upgrading of geometry as a mirror of the vital mind of each species, realized in this blog.

So we shall do along it a series of ceteris paribus analysis based in the new mental concept of dimension expressed first in the ERLANGER program.

In the graph the analysis of the 3 Euclidean and ‘fourth social dimension’ of light, that of frequency colors that carry the information about the ‘social density’ of a light space-time ray, as reflection of the vital actions of light, which become in the electron and humind eye that feeds on it, its dimensions of information.

In the graph we see this final ‘understanding of the 4 Disomorphic dimensions of light and its perceiver the electron, which shape the human space-time, where color is for light its ‘∆-scalar åction of survival/existence’ (wider concept that a physical action).

As the other 3 vital dimensional actions of light were naturally incorporated to the humind as its ‘dimensions of space’ width, height and length, color was left as a puzzle for humans use it to code its own survival actions but do NOT understand this subconscious program, based in the Generator equation of ‘color’ as a vital ternary dimension of human life. Let us then before we study the abstract use of color to ‘liberate space from reality’, consider the opposite function of color as a ‘set of information’ used by the mind to code the vital 5D of reality.

In that regard, the intuitive rainbow coding would establish a circle of colours as a reflection of those 5D as follows:

MAGENTA (4D entropy) – RED (1D $t-lineal motion) < YELLOW/GREEN (2D S/T-reproduction) > Blue (3D ð§-information) > Violet (5D-social evolution), which then connects with Magenta, to close the zero sum.

And the simplified Black and white code: Black (information) < grey (energy)>White (motion)

That this coding is UNIVERSAL TO electronic eyes, is shown by the fact that robots with eyes WITHOUT the need of a program run faster when red colours are put on the tracks, as their vital electronic living mind WITHOUT need of human coding in a sentient vital Universe ‘likes to run to red’ as you like to see speedy red cars and the male-lineal species love red, while the female reproductive one loves green…

The code thus is ultimately embedded in the above light-electron S>T I(eye)>Wor(l)d electronic mind and likely equal in all electronic beings.

Yet as we can by the ternary method subdivide each of those dimensions in sub-dimensions in the continuum spectra of (in)finitesimal steps of a world cycle, and errors of perception happen in all limited minds, we code 7 rainbow colours courtesy of myopic Newton(:

Newton’s subconscious understanding of isomorphism, defined an ‘excessive’ 7 color division when a 5 coding would have been enough, eliminating the intermediate orange and indigo and missing, the magenta, 1-3-5-7-9 ‘vowels’ do however code all languages as we shall observe in our studying of each of them and its ‘phonemes’, or ‘cells’ or ‘colours’ or ‘notes’ WHICH WILL ALWAYS BE MIRROR-REFLECTIONS OF the 3±∆ dimensions of the scalar Universe that truly matter.

9 dimensions + the unifying dual dimensions of the @-mind for an 11-dimensional reality, in that sense, is the maximal c enough even for the more complex languages. (vowels in languages, type of cells in organisms, dimensions of string theory, etc.).

It is then a choice of which ‘discontinuums’ we establish between the 3±∆ ages of a world cycle, to establish the fundamental method of creation of mental spaces, by transforming a worldcycle of time into a mapping of space, in this case the translation of the 5 Dimensions of frequency of a light time-cycle…

As light starts in the blue, generation color of blue stars, ends in the red dwarfs and dissolves into the entropy of dark entropy between galaxies as it tires and dies, expanding entropic space between galaxy, another ill understood process origin of the faulty 1/2 big bang theory of an entropic universe, which disregards the opposite ‘blue collapse of light’ into matter of high frequency within galaxies:

So once we have done a very brief introduction to a ‘real’ theory of colours as a coding of mind-spaces, and Universal ST dimensions, we can consider how its study in abstract geometrical spaces by the likes of Riemann allowed the liberation of the concept of dimensions and defined the ‘vital properties’ of geometry that matter in reality (continuity, adjacency, perpendicularity, congruence and so on).

It was the first of many expansions of the concept of ‘phase spaces’, which however failed to give the final ‘jump’ into defining human spaces also as ‘mental’, and hence expanding mental spaces to all systems that gauge information, including particles which gauge forces, themes those to be studied on the fourth line for each species of mind, with different quality/quantity-steps on its ‘actions and dimensions’ of perception

Distance thus is the essential quantitative parameter of non-E geometry and as such most spaces are defined by its metric – that is its measure of a distance; since without it it looses completely its meaning. 

The frame of reference of the fractal generator.

It is then possible to consider since all systems are ternary either in topological, temporal age/states or scales, to create an absolute geometry of 3 relative coordinates or GENERATOR ‘frames of reference’, in which we can measure the distance between the 3 elements of the being, its scales, its topology and its ages, to establish the absolute distance between two beings, and then establish a frame of reference for each of those elements to establish ternary distances.

In the graph we can see two examples: an abstract frame of reference in which each fractal point is then expanded with its own internal coordinates, and below a frame of terence based in the ‘generator’ of a human being, used to measure the wealth of a society according to the goods produced that enhance the ‘natural actions’ of survival of a human being.

This ethonomic frame of reference which has also negative values should guide a real science of the economic ecosystem, NOT based in monetary prices, which is NOT the quality/property searched for humans (we do not eat money, as Chief Seattle said) but the biological use to enhance the program of existence of human beings.

And so we can build a new model of ethonomic sot serve the needs of human beings.

Such expansion of geometry goes well beyond this introductory course, but it would truly help social sciences to become more ‘rational’ and ‘serve more properly humans.
The fractal generator becomes then the fundamental formalism to develop geometrical graphs of @nalitic geometry, in which the specific cartesian frame of reference is merely the phase space of the 3 dimensional actions of a light space-time in which humans are embedded, which electrons, who feed on light as energy and information use to build up its mind.

Of all those possible ternary frames of reference of the fractal generator, either its ternary scales, its ternary topologies or its ternary ages, the most commonly used in physics is the fractal generator of ternary ages/states of matter, the so called 3 states of thermodynamics, as the left graph show. It is indeed  the graph of the Fractal generator of states/ages of physical matter:

Γ:  $-Gas < st-liquid>ð§olid 

On it and any other phase space of n-parameters/dimensions, continuous changes of age/state, i.e., processes occurring in the system, are presented by lines in this space. Separate domains of states are domains of the phase space. The states bordering two such domains form a surface in this space.

The surfaces dividing these domains in the graph of ‘matter ages’  thus correspond to such qualitative transitions as melting, evaporation, precipitation of a sediment, etc; which we can also represent in a single lineal dimension for the whole generator:

However with multiple dimensions/coordinates we can study more s and t elements involved in those changes of states/ages. Reason why thermodynamics uses 2 and 3 coordinate systems. Above we show for simplicity a bidimensional system with two parameters of s and t, pressure (ð-parameter) and temperature (t-parameter): A state of a system with two degrees of freedom is illustrated by a point in a plane. As an example we can take a homogeneous substance whose state is determined by the pressure p and temperature T; they are the coordinate points describing the state. Then the question reduces to studying the lines of division between domains corresponding to qualitatively distinct states. In the case of water, for example, these domains are ice, liquid water, and steam. Their division lines correspond to melting (freezing), evaporation (condensation), sublimation of ice (precipitation of ice crystals from steam).
For an investigation of systems with many degrees of freedom, the methods of many-dimensional geometry are required. But essentially we are in the same conceptual frame of reference, choosing always S=t dual parameters of bidimensional geometries.
The concept of phase space applies then  not only to physicochemical but also to mechanical systems, and generally it can be applied to any system in which we establish motions between S and T symmetric parameters; establishing an enormous range of application – essentially all the graphs of all sciences, which are all studying s-t motions and dimensional variations of the s-t parameters of species or events.

The generalisation 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 formalisation of the most generalised 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 realise 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 NOW THAT WE HAVE escaped ‘geometrical visual space’, we can indeed extract the logic consequence of all of this:

  • AS THE coordinates/dimensions of such ternary generalisations of geometry are properties of our D-isomorphic reality we can adscribe 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 will 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 here is the not so obvious 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.

Let us then consider this mental view a more specific, mind, the human ‘electronic mind’, which uses c-speed as its constant rod of length.


Now we explained that the perception of a universe as euclidean or elliptic or hyperbolic depended on the long overdue respect we must give to the a priori parameter of all worlds: r²/L²=$t/ð§. When this parameter tends to zero, we have an Euclidean geometry, above zero we have hyperbolic and below zero we have elliptic.

The parameter which defines the mathematics in which we live is so essential that it will come constantly. For example, we can consider that systems exist at one side of the parameter, at the side of c speed as a limit – in our euclidean world – or over c speed (in the elliptic gravitational world) at the side of T=0k as the limit of the thermodynamic plane, etc.

So we need now to complete the definition of our world, to deal with the nature of our square ‘radius’ parameter of information. What is the minimal quanta of information of our electronic eyes, that feed on h-quanta of light energy, well, we just said it, the h-quanta which measures in the minimal amount known, the spin of a particle, which is rather obviously the minimal unit of its angular momentum, of its informative perception.

And for that reason as h/c is truly a minimal amount our relative 0, we live in that Euclidean world.

The first and most important of T.Œ’s 5 actions of all systems that want to survive, is ‘perception of information’ How do particles perceive?

Of all those levels and actions the one physicists understand worst and less is the action of informative perception of smaller particles, called SPIN, which is the absolute minimal unit of reality we measure after deforming it fully (quantum uncertainty), a quantity called angular momentum, h/4 pi, which is as a ‘form of information’, ‘space-form’ NOT motion.

And since h is in the above equation, E=ƒh, a whole, hence a sphere (whatever it has inside), the ‘essential topology of all wholes’ by definition enclosed disks-sphere (in ¬ Æ topology a disk is a 1-sphere, a sphere, a 2-sphere, a sphere in motion a 3-sphere, which as motion is a dimension seen in motion, can be of different varieties, etc.), a spin is ‘logically’ is a n-sphere divided by 4 π:

Screen Shot 2016-04-20 at 14.10.41

Now the graph gives us possible choices, as we depart from a ‘whole’ of a light beam, which could be a cylindrical cut, as in some lineal worms, or a ball. As 4π comes in the ball, we have 2 choices or the surface of the ball, or the solid sphere but NOT all, only a section, called a ‘solid angle’. This gives us a first duality between ‘human measure ‘and ‘true property’. The true property is the solid angle from the surface to the center, r3/3, the human measure is the external surface, r2/2.

As all space-time organisms have truly 2 components, the ‘internal’ open ball topology (meaning the ball without the surface membrane and the center of the point) or ‘present-evident, wave-like ST form’, and then the membrane and o-point in the center, or Sp x Tƒ, ‘past x Future’ particle (limbs-external sensorial membrane and central mind) together:Screen Shot 2016-04-05 at 08.27.39

Why we know spin is a solid angle of information? Because if it were a motion, as Pauli, the guy who is credited with its discovery, put it to his real discoverer, as speed it will be 137 times faster than light!: hc2/e2=137 (inverse fine constant). This in 5D physics is NOT a problem, c-speed is Einstein’s postulate to adapt the Universe to the human perception of it – our rod of measure, as electronic minds. But here Pauli does have a point. A Spin does NOT rotate, as you do NOT rotate when looking outside, but is a ‘wedge’ into the sphere, through which information enters.

We see the surface of it and so it seems a momentum, the being sees the solid angle of it and so it is an intensity. Because the being is down there into the zero-point soul of the particle, mind of mathematical perception, it does open and close the sphere to ‘look’ through the windows of the membrane. In the simplest form, it is a pi cycle of 1-sphere, which is made of 3 closing diameters, which leave π=3, 0.14 d apertures, or 0.14/π=5% of light and 95% of dark matter/energy, the proportions light space-time beings DO NOT see of the Universe.

So now in the ‘dialectic’ Socratic method (remember my culture) I use to find things, I deny myself. What if I consider NOT a solid angle, BUT a spin, which IS a slice ‘SEEN’ through the 1/3rd ‘D’ apertures between the diameters of the pi-1-sphere (cycle) or the ‘cuts on the 2-sphere’. We realize this ‘will be a slice’ with the exact parameters of angular momentum, and there will be 3 holes, hence only 3 spin angles, and they will be further on quantized – jumping from part of part of the whole:

Screen Shot 2016-04-05 at 23.47.05

In the graph the proper description of Angular Momentum as a plane of Information, the minimal unit of perception of our Universe, proper of Relativity is more clear than the classic dynamic Time description as r x mv – both have their different uses/perspectives. H is the rod of informative measure of the Human electronic eye-mind, so small compared to c, the human rod measure of light space that the constant of geometry, r²/L² of Lobachevsky’s pan geometry becomes close to zero, creating our perceived Euclidean space, deformed to a hyperbolic geometry for smaller beings (Special Relativity, quantum physics) and to an elliptic geometry for larger gravitational scales (General Relativity)

Now, this is the modern ‘Hilbertian-Einsteinian’ description of angular momentum, just what we said it is, P, is the ‘cut’ on the n-sphere, x the radius, equivalent to P in ‘distance-motion’, and its product an amount mrv which is the spin. You can think of it as an information bit or an energy bite (space or moving perception) of the being.

Now those are the spins, quantized, and you can see there are 2 or 3 ‘cuts’ to perceive, with different orientations, and DO NOT come in bigger numbers. The maximal thing is this work of art, called ‘female and male spins’.

We won’t enter on this (-; just to mention the bidimensional nature of those spins (left graph) in the left side, which ARE the quanta of all information humans, electronic eye-beings, perceive about the world. And the obvious capacity to process and orientate they give as the ‘eyes’ of the atom, to each particle.

Moreover, they get ordered, orientated by the external magnetic field, the larger time-enclosure that orders from ∆, the ∆-1 quantum field. Now ask a physicist what is a spin, he will put a lot of formulae of abstract maths, which it will take you half a year to memorize, without understanding. Why? The idea that particles perceive is out of the ‘picture’.

Now the magnetic field, they like, as all heads turn to her the speaker in human groups, all atoms spin in tune, and we use it to control them, as speakers use the word nation and god. Same law of mass-control. I call it lanwave. A Wave ordered by a language of energy and information they speak. The magnetic field is the language of space that transcends atomic parts into the next scale; forming them; the electric field is the time force that put them in motion. That is how we transcend from quantum to our scale. The best at the job of ‘feeling’ magnetic lanwaves is the iron, and it becomes the top guy of the next scale. It forms perfect organized masses.

Once we are here, those motions are ‘activated’ by temperature, vibrational clocks of our molecules.

Temperature is the ‘electric field’ that moves us in the ∆-scale. What is the ‘spin’ for Molecules?. They use van der waals forces and ‘angles’, giving by slave atoms.

And the boss of those perceptive ones in our body is the nitrogen. It has 3 H systems to perceive:

Screen Shot 2016-05-14 at 21.22.44

Here you have an amino acid working his life around, with its oxygen legs kicking in and out water, its carbon body with lateral arms of many kinds and its nitrogen head. You are made of a lot of those. How does its Nitrogen mind observe the Universe? Its rod of measure and radius of perception is H, its length likely gravitational waves of non-local infinite distance, so its r/k ratio of mind curvature minimal.


∆+¡: Hilbert Spaces


 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 (Spe-function space):

screen-shot-2017-01-28-at-06-23-20 screen-shot-2017-01-28-at-06-23-30 screen-shot-2017-01-28-at-06-24-23

In the graph, 3 representations of Hilbert spaces, which are made of non-euclidean fractal points, with an inner 5th dimension, (usually and Spe-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.

Enough of it for the time-being or else I will never get my daily dose of self-destruction and painless nirvana…

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 then the underling 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 . Therefore we have the inequality:and hence always:(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).

4. (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 pairwise 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 parallelelipiped” 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 were formed and crystalized under various aspects and for various reasons. Many of the fundamental concepts of functional analysis emerged in a natural fashion in the process of 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 eigenvalues, 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 others.

Functional analysis permitted an understanding of many results in these domains from a single point of view and often promoted the derivation of new ones.

In recent decades the preparatory concepts and apparatus were then used in a new branch of theoretical physics–in quantum mechanics.
On the other hand, the investigation of mathematical problems connected with quantum mechanics became a crucial feature in the further development of functional analysis itself: It created, and still creates at the present time, fundamental branches of this development.
Functional analysis has not yet reached its completion by far. On the contrary, undoubtedly in its further development the questions and requirements of contemporary physics will have the same significance for it as classical mechanics had for the rise and development of the differential and integral calculus in the 18th century.

It is impossible here to include in this chapter all, or even only all the fundamental, problems of functional analysis. Many important branches exceed the limitations of this book. Nevertheless, by confining ourselves to certain selected problems, we wish to acquaint the reader with some fundamental concepts of functional analysis and to illustrate as far as possible the connections of which we have spoken here. These problems were analyzed mainly at the beginning of the 20th century on the basis of the classical papers of Hilbert, who was one of the founders of functional analysis. Since then functional analysis has developed very vigorously and has been widely applied in almost all branches of mathematics; in partial differential equations, in the theory of probability, in quantum mechanics, in the quantum theory of fields, etc. Unfortunately these further developments of functional analysis cannot be included in our account. In order to describe them we would have to write a separate large book, and therefore, we restrict ourselves to one of the oldest problems, namely the theory of eigenfunctions.



Time permitted we will completely depart from present jargons, axioms and models of teaching geometry, which in a first course we merely illustrate with the laws of vital evolution of geometry and the s=t balances and stop and go dualities to refund fully the discipline with one of my earlier treatises, now 3 decades old, in which departing from the ‘real’ objective absolutely relative postulates of geometry of the fractal world – a point has a volume of entropy and information; the whole is smaller than its parts, two things that occupy different places of space are not equal and so on; we shall rebuild from its foundation up, everything you thought to be real, so we can once this task is completed, enter the form of non-evident minds from those of black holes to those of ant-hills, according to the geometry they display.

It is a task we shall do for the sake of size, and as those postulates apply to all stiences, in other posts, dedicated to the postulates of Non-E and the nature of mind-worlds.

So here we shall just put a miscellaneous series of themes as I find them in different notebooks and deposits of thought of the ginormous volume of 30 years of ‘living’ as the single fractal point of this planet in 5D states (: at least those I found specially beautiful, of course with the help of the web’s amazing quantity of better powerpoint presentations and the classic books of simplex fundamental truths in mathematics and physics of the 70s, which taught me ‘first-hand’ to love those disciplines in my youth – Aleksandrov, Feynman and Lindau pocketbook encyclopaedias.


‘Out of Nothing I have discovered a strange new Universe’. Bolyai, mathematician, on the Non-AE Geometry of the Universe.

The ∞ minds of reality

The 5th postulate of Euclidean geometries can be easily understood then from the perspective of perception, departing from its equivalent postulate, first stated by Wallis that we can shrink or grow triangles without deforming its angles and proportional sides.

This equivalent postulate, would mean that the angle of perception of species do not vary as they change side (no other meaning has an angle of a triangle, but the capacity to allow us to calculate trigonometric distances, reason why trigonometry was the first theoretical form of mathematics born of arithmetic calculus, which allowed Eratosthenes already to measure the distance to the moon, the size of the earth and the Sun distance). However they vary. Angles diminish as we become smaller in hyperbolic geometry.

So an ant will see a world with a hyperbolic vision and a lesser angle, reason why it needs a larger circumference view (spherical eyes, or eyes around a 360 perceptive point of view in Tƒ, highly informative arachnid species).

On the other hand, in larger beings the angle will become inversely larger, so the eyes become the opposite concave and planar in form; reason why satellite antennae and telescopes which represent the larger view of a planetary organism have parabolic forms.

This and many other truths of the ‘strange Universe of the 5th dimension’, in which the 4th dimensional Universe is both a distorted human point of view, using a single time clock, by using a single space-time continuum, of a limited number of 5D planes exactly – 2 relative planes of size, the humane electromagnetic and the gravitational larger plane, (special and general relativity).

Mathematical pangeometries vs. Physical single space-time continuum

Now the next huge confrontation between the single worldview of physicists and the multiple worlds of Philosophers of Science, will take place strictly in the field of mathematics. We have ‘acquitted’ Einstein, even though he developed the next r=evolution of time concepts after the duality between Darwin and Clausius (evolution of information vs. entropy only dying Universe, studied latter when we define the human being as a space-time organism), because on my view his findings are essential and in the right track to fully grasp cyclical time, but they were ‘bend’ as usually by other lesser physicists and mathematicians to cater the lineal worldview.

Specifically Minkowski simplified his findings and defined a lineal s=ct parameter of time, exactly the same of Galileo S=vt, just changing the speed of the body by the speed of light, to be able to measure the distortions in simultaneity caused by the existence of infinite time clocks.

What truly mattered was the finding of Einstein which confessed ‘ I am the only physicist who realize there are infinite time clocks in the Universe running at different time speeds’ and also ‘Time curves space into masses’ and ‘Leibniz was right but if so we must change the entire building of physics from its foundations’. And further on ‘science must only be occupied from facts that are experimentally certain’, and ‘I know when mathematics are truth but not when they are real’. So he was at heart understandingly. But soon he was immersed in the idealist, Germanic school of mathematicians, like Cantor, Hilbert and Weyl, which substituted, after the crisis of Lobachevski’s pan geometry, real points with parts, social numbers and topological networks by ‘imagined or undefined terms, sets or groups’ as the unit of reality, breaking the NECESSARY CONNECTION between mathematical logic and reality, as geometry is the language of space and logic the language of time of the human being.

SO AFTER LOBACHEVSKI’s (Boylai and Gauss also) discovered that space was NOT Euclidean, except in the limit of human perception, defined by this FUNDAMENTAL parameter of the Universe: Tƒ/Sp: r²/L², the fundamental steps of science to be taken were:

  • To complete the classification of all possible geometries according to the relationship between its quanta of cyclical form-motion or radius and its quanta of lineal speed-distance or length.
  • To determine experimentally the values of our r², electronic quanta of information, we humans used to perceive L², our quanta of space, which had also to be determined.
  • To study other experimentally known minds of the Cosmos to see if they had the same ‘bidimensional units of information, Tƒ=r², and L²=Sp, and how they will perceive the Universe hyperbolically, elliptically or Euclidean.
  • And to research how all those relative different space-time geometries order in Nature (through the scales of the 5th dimension).

And yet none of this steps were taken. So we shall take them here to show you:

  • First the nature of the Euclidean mind of man, where its radius is given by h and its quanta of space by c², hence it has an r²/L²->O, which is the requisite found by Lobachevski for an Euclidean mind, and then…
  • to study the other ‘minds’ of the Universe and the different scales of the 5th dimension according to their peculiar geometry
  • And finally to see what physicists did with this discovery (just to apply the 5th non-E postulate of Riemannian elliptic geometry to the larger gravitational scale where indeed, geometry is elliptic because L grows to infinity (action at distance).

The need to evolve the logic and mathematical ‘undefined’ elements of the Universe.

Now the key difference between a philosopher of science like Leibniz and a Physicist like newton is this: Leibniz as Aristotle and Descartes before him, was conscious that the language first defines the world we perceive, and NOT the Universe. So it is a humble realization of the minimalist nature of human egos. The Physicist has reduced THE ENTIRE universe to the world-perception of a human light space-time ego centered point of view. So he lacks sense of humor and his arrogance is infinite. Because he confuses the infinitesimal of his mind with the infinite Universe. This we shall stress once and again. The physicist as all humans IS only an electronic mind that fits an infinite number of cyclical space-time beings, into a geometric linguistic mapping within its ‘monad’. And he is NOT aware of it. So his arrogance is supreme:

o-point x ∞ Universe of time space cycles and worlds = fixed linguistic mapping

The 3 great philosophers of sciences of western thought (and their counterpart eastern philosophers of Taoist and Buddhist cultures) were always aware of this. The cut off happened when Kant rejected Leibniz and became a Newtonian. But at least Kant realized that the human Euclidean mind was a light space-mind and not necessarily all, as impressionist painters will latter.

Yet starting with Kant and then Hegel, which had the chutzpah (even if he understood perfectly the ternary logic of thesis-antithesis-synthesis similar to the Sp<ST>Tƒ ternary Universal Grammar of all systems) to affirm Abrahamic tribal religions were superior, because they only accepted the human ego-trip of tribal Yahweh, there is a growing anti quantum paradox in philosophers which yield as infinitesimal observers to the huge power of the observable, industrial world.

This paradox inverse to that of the physicist, so huge that influences the observer, made after Leibniz, all philosophers incomplete. As they no longer dare to criticize the makers of machines and weapons, and not even the Abrahamic tribal, lineal religions of the Bronze age that back the hate between members of the same species. So we of course show a complete indifference for all what has nothing to do with the larger Universe of infinite worlds, and find the arrogance of physicists, infantile.

Now I will state a provocation to the ego of physicists (-:

The advantage of being a philosopher of science, is that unlike a physicist, the philosopher knows physics as it is study today is ‘logic mathematics’. And logic matters most.

Because the language of God is the logic of time cycles and the mathematics of spatial extensions. In this we are different from physicists who only apply the mathematics of space, as their knowledge of time is reduced to v=ct. But when time has a more complex understanding, it is obvious that the Time Logic behind the language of mathematics, which creates it, is far more important. So let us rephrase it. Yes, Physicists do know as much mathematics as philosophers of science. But they do not understand the time logic behind it. And that is the problem. Because logic is a property of time cycles, and its repetitive causality, logic is more important that geometry. Logic is ‘the beginning’. So the Greeks called God the mind of the Universe, Logos, derived from ‘word’, not ‘geometry’ meaning “Earth measurement.” And that makes the whole difference, physicists do not grasp. As what they do is ‘geometry’, measure. We philosophers of science do Logos, (-;

Now, we told you, we will evolve mathematics, so we can fit the new concept of cyclical time and fractal space within it, as Descartes and Leibniz did to fit their time vortices and relational infinite times, as Einstein did (not by himself but by applying non-Euclidean mathematics to his description of space).

So to that aim, we just need to complete the scaffolding of mathematics and explain what a mind is, in geometric logic terms.

So we need to complete the concept of fractal quanta of energetic space of quantum physics, the concept of a Universe of multiple time cycles of Einstein, and the philosophy of science, made of causal repetitions of time cycles called laws of science.

So we shall go further beyond Euclidean and Aristotelian logic matheamtics, into the next vowel: Non-AE≈i, (Non-Aristotelian, Non-Euclidean), i-nformative concept of Space-time.

And build a new Non-Aristotelian Logic and Non-Euclidean Mathematic formalism, which will set the foundations for a new paradigm of science.

This new Geometry is simple and evident. It is based in the definition of the ‘terms’ which mathematics ill-defined in Euclidean Geometry, points with no parts through which only one parallel crosses, lines with no breath and shallow planes, equaled by superposition, and then once Euclidean Geometry was found to be a ‘limit’ of a larger pan- Geometry NOT lineal but curved inwards (elliptic) or outwards (hyperbolic), on those dark times of confusions, were UNDEFINED.

So we DID not resolve the issue. We CANNOT ESTABLISH the fundamental science of the Experienced Universe, without defining its fundamental elements. This no way out sponsored by Cantor, who substituted points and numbers by sets, nowhere to be seen in experience, and Hilbert who affirmed ‘I imagine points, lines and planes’, is a failure.

In that regard, the founder of Non-Euclidean geometries Lobachevski was in the right track. We can summarize Lobačevskiĭ’s solution of the problem of the Fifth Postulate.

  • The postulate is not provable. Because, infinite lines curve when we have a larger view, becoming exponential inversion of space-time.
  • By adding the opposite axiom to the basic propositions of geometry a logically perfect and comprehensive geometry, different from the Euclidean, can be developed, providing we limit infinity,( as the 5th dimension shows) hence parallels do NOT meet on the infinity, and by curving them (hence lines are part of cycles)
  • The truth of the results of one logically conceivable geometry or another in its application to real space can only be verified experimentally.

Fractal points: the fundamental particle

Thus we must observe the Universe to prove mathematical statements and reduce mathematics as we do with all experimental sciences to those equations, which ‘are real’.

T.Πdoes show by regressing a step back in the scaffolding of mathematics and defining those terms, as the Postulates of i-logic geometry, based in the structure of cyclical time, fractal space and the 5th dimension where:

1) A fractal point grows in size when we come closer to it, so it has parts, its O-mind and its E-field flows of energy or limbs. Hence we define it as Œ, an existential point.

What they perceive will depend on the Lobachevski/Gauss Constant of Time-Radius/Space-length, r2/L2 (Tƒ/Sp parameter), which defines its geometric view as hyperbolic (for smaller planes), Euclidean (for the human H/c2 plane of informative spin time cycles and light space-rods).

Let us elaborate a bit more on this essential component of the Human World – NOT the only worldview on space-time there is.

Now in the post on astronomy we explain ‘why’, light is a c-constant rod of space, and hence c² the factor of Lobachevski/Gauss definition of the form of the geometry. It follows that because H-Planck, the constant of spin or minimal quanta of information of the human Universe is exceedingly small, and the ‘degree’ of Euclidean nature of a world depends on r²/L², in this case h/c² to be infinitely small so the ‘radius’ of a circle and the ‘angle of perception’ of a triangle becomes pi, and 180/3, we humans see an Euclidean light space-time.

But a bigger radius of perception (smelling minds with atomic pixels, i.e. insects) will be hyperbolic; or a larger rod of measure (action at distance in gravitation), will be elliptic (quark/black hole? perception).

Now of course, a ‘science’ as the human ego-centered science, which is fully unaware of its ego-centered paradox, and confuses the world of its mind with the entire Universe will ignore organic and perceptive explanations of reality, but they are THE ONLY ones that explain the whys of reality.

Lobachevski’s pangeometry. Where is the 5th dimension: @ FRAMES OF REFERENCE

Now it must be understood that the universe is made of dimensions of space and time, which are opposite but equal in number and merge lineal and cyclical motions into bidimensional, 4-dimensional and 6 dimensional super organisms.

they are the six motions of time of Aristotle, and we have grouped two, dissolution and aggregation, a dimension usually studied by Analysis, as the generic name of the 5th dimension. As the arrows of present space or 4 dimension, ct, or vt, the immersion of past and future form a 6 dimensional structure. This would be the part of ‘¬Ae not treated here.

Let us now consider the 3 branches of mathematics, which are 5 Dimensional analysis, algebras of time worldcycles, and spatial geometry of space quanta.

They study obviously the 6 Aristotelian motions of space-time, and as analysis is a branch of algebra, we divide our upgrading into i-logic Geometry (after the Non-Euclidean), and ¬æ, Non-Aristotelian, Non Euclidean algebra, which studies not in terms of continuous world-minds as geometry does the ‘Maya of the senses’, but the whole of the discontinuous Universe of infinite worlds of the 5th dimension.

Thus algebra (and analysis specifically concerned with the processes of social numbers that sum and emerge or rest, and divide and plunge down the scales of eusocial love of the 5th 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’).

Now, it will surprise the reader to know that 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. It was discovered by a German adequately named Bach-mann, for its musical architectonical rigor.

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.

Indeed, 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 5th 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.

The 3 geometries of points of views in the symmetry of the 5th dimension.

Screen Shot 2016-03-20 at 20.44.22Screen Shot 2016-05-14 at 20.49.31

In the graph, the world of human perception was in the work ‘the World’ of Descartes not the perception of the entire Universe, but htat of the Euclidean human mind, which in Absoute Geometry, defines its linearity according to the factor that measures the relative ratio between the ‘cyclical eye radius’ of the informative observer and the lineal ‘space distance’ it measures: Tƒ/Sp ≈ r2/k2 ≈ h/c2 ≈ 0 for the human lineal Euclidean Universe, a expression that we will latter clarify in depth when studying the geometry of the 5th dimension.

The Universe thus has different worldviews, with different geometries, according to the size or length rods we use to perceive it with in human beings are the very large rods of ultra-fast light speed, and the size of the ‘quanta’ of information or cyclical form of the observer, which in the human case is extremely small, given the fact that we are electronic eyes, which perceive H-Planck quanta (spins) as our unit of information.

However there are different other worlds of relative sizes of information and length, as we move down or up in the scales of beings of the 5th dimension. All of them can be defined as either elliptic in its perception (larger beings) or hyperbolic (smaller beings), since as we move up and down, the Tƒ/Sp changes, growing for smaller hyperbolic beings and diminishing for elliptic larger beings, departing from our single Parallel Euclidean world.

Screen Shot 2016-04-11 at 19.32.015th postualte

Mathematics, as a language that represents reality with simplified symbols, has a limited capacity to carry information. Its symbols, geometric points and numbers simplify and integrate the fractal, discontinuous reality into a single space-time continuum, the Cartesian Space/Time graph, made of points without breath.

However the points of a Cartesian plane or the numbers of an equation are only a linguistic representation of a complex Universe made of discontinuous points with an ‘internal content of space-time’. In the real world, we are all pieces made of fractal cellular points that occupy spaces, move and last a certain time.

When we translate those space-time systems into Euclidean, abstract, mathematical ‘numbers’, we make them mere points of geometry void of all content. But when we look in detail at the real beings of the Universe, all points/number have inner energetic and informative volume, as the fractal geometry of the Universe suddenly increases the detail of the cell, atom or far away star into a complex complementary entity.


So we propose a new Geometrical Unit – the fractal, Non-Euclidean point with space-time parts, which Einstein partially used to describe gravitational space-time. Yet Einstein missed the ‘fractal interpretation’ of Non-Euclidean geometry we shall bring here, as Fractal structures extending in several planes of space-time were unknown till the 1970s. So Einstein did not interpret those points, which had volume, because infinite parallels of ‘forces of ‘Ðimotions’ and information’ could cross them, as points, which when enlarged could fit those parallels, but as points in which parallels ‘curved’ converging into the point.

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

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

Euclid 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:

Abstract, continuous, one-dimensional point:

. ____________

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.

             Real, discontinuous, n-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. The maths are the same, the interpretation of reality changes, adapting it to what experimentally we see: a cell-like point enlarges and fits multiple flows of ‘Ðimotions’ and information, and yet it has a point-like nucleus, which enlarges and has DNA information, which seems a lineal strain that enlarge as has many point-like atoms, which enlarge and fit flows of forces, and so on.

Curved parallels are isnot meaningful, because if such is the case parallels which are by definition ‘straight lines’, stop being parallels. So we must consider that what Einstein proved using Non-Euclidean points to explain the structure of space-time is its fractal nature: points seem not to have breath and fit only a parallel, but when we enlarge the point, we see it is in fact self-similar to much bigger points, as when we enlarge a fractal we see in fact self-similar structures to the macro-structures we see with the naked eye.

That is in essence the meaning of Fractal Non-Euclidean geometry: a geometry of multiple ‘membranes of space-time’ that grow in size, detail and content when we come closer to them, becoming ‘Non-Euclidean, fractal points’ with breath and a content of ‘Ðimotions’ and information that defines them.

So each point is in fact a 3-dimensional point, and if we go to the next scale, a 3×3=9 dimensional point and so on. Yet those dimensions are the so-called fractal dimensions, which are not ‘extended to infinity’ but only within the size of the point.

In Euclidean geometry, a point has no volume, no dimension, but string theorists say that even the smallest points of the Universe, cyclical strings, have inner dimensions that we observe when we come closer to them. That is the essence of a fractal point: To be a fractal world, a space-time in itself.

‘Any Non-Euclidean point is a fractal space-time with a minimal of 3 internal, topological, spatial dimensions and an external time motion in the st+1 ecosystem in which it exists’

The Universe can be perfectly understood when the human mind and its subjective vision of reality from the perspective of its limited ‘Aristotelian’ and ‘Euclidean’, logic, (temporal and visual, spatial perception of the human mind) is considered only one of the infinite points of view, performed by particles and heads that gauge information, move ‘Ðimotions’ and constantly create the events of the universe. Each of those particles and heads create its own perspective and mind view, or mapping of the Universe to which they act-react accordingly.

The fundamental particle of the Universe is not a physical form but a logic particle: a knot of time arrows, which in any scale of reality, from physical particles (quantum knots of ‘Ðimotions’ and information) to biology (knots=networks that absorb ‘Ðimotions’, information and reproduce and evolve into bigger knots) act under a single mandate: to maximize those time arrows, a fact that we formalize with an equation, the function of Existence: Max. ∑Sp x Tƒ; which is the fundamental function of both, logic and mathematical languages. In the graph, all such points of view, will define a system of relative perpendicular coordinates, through which it will enact its time arrows, departing from a central knot of information.

In the graph, apperception of the Universe happens by reducing the cycles of time into a single spherical point, according to the topological properties of all spherical systems, which can diminish in size without deformation.

This is called the Poincare Conjecture, and his recent proof was the most important finding on mathematics for decades. It simply speaking only n-spheres can reduce without limit an n-dimensional Universe into a fractal mirror without deformation and tearing. And that is why there is a fundamental dominant membrane-point system in all the organisms of the Universe, which creates the order of reality. The paradox though is that the membrane, the time cycle is moving very fast, and the point in the center is a static mind.


Fractal, Non-Euclidean Points of view. Upgrading the foundation of space-mathematics.

All in the Universe are thus complementary systems made of networks.

Now this might sound absurd to the anthropomorphic reader that thinks humans are different from the rest of points of the Universe, but it is a fact that those points obey in their actions and communicative flows within a network the same isomorphisms: humans and electrons behave the same when they move through slits or in herds, the geometries of social groups are also the same and the ultimate purpose of those points, to feed on Entropy and information, whatever kind, is also the same in all networks of the Universe. And so that group of isomorphisms of networks becomes a primary why for all beings of the Universe.

A fact the leads us to the final element needed to understand the why of the Universe: ‘non-Euclidean points’ organize networks that become points of a higher scale, which reproduce and organize new networks; and so the Universe keep growing in fractal scales, from particles that organize networks and become atoms that organize networks and become molecules that organize networks and become cells, that become organisms, that become planetary societies and planets and stars form gravitational networks that become galaxies, organized by dark matter into Universal networks.

Because all entities have motion reproduction is merely the repetition of a motion with form. Because each entity has 4 time arrows, all of them trace multiple trajectories in search of those arrows, hence they realize multiple time cycles.

For example, a human feeds on Entropy and information with body and head, reproduces through multiple social cycles and evolves into societies. Our actions are more complex but essentially the same of those of any particle.

So the unit of reality is a space-time cycle, and many of them create a knot of time cycles or entity of reality, which will be reproduced by repeating those formal cycles with motion in other region of space-time; and many of those knots of time cycles, which are self-similar, since they are born from the reproduction of a first form, come together with self-similar beings into networks.

Some of those networks are spatially extended with a lot of motion (fields in physics, bodies in biology) and some are very tight, formal, with a lot of in/form/ation, (particles and heads or nuclei in physical and biological jargons). Both types of networks together then create a complementary organism, which is fitter to handle both Entropy and form; hence it survives better, it ‘exists’.

Today scientists of measure scorn philosophical and logical analysis of causality in time because it cannot be easily put in numbers. But numbers are only one of the languages of information in the Universe, and many of its properties of bio-logic nature are better described with logic words. In that regard, we can now fusion philosophy and science answering the fundamental question, ‘why we exist’; since once we realize that we are ‘made of time cycles’, knots of time cycles and networks of knots of time cycles, an intelligent, informative, eternal universe of motions and wills of existence makes the dogmas of deism and mechanism, childish myths.

‘What is existence’ cannot be revealed from the simplex point of view of a mechanical world, which cannot explain the fact that we are made of motions with form that leave a trace on space but are essentially actions in time that have a social finality – to create more complex networks, chaining knots of actions into systems. This social will of every point and entity of the Universe is completely at odds with a mechanist, fixed, solid, senseless, dumb Universe.

Motion in Time and social evolution are concepts that require the capacity to gauge information and interact with other self-similar points to create those organic networks. Further on, dual networks tend to evolve and reproduce new points through exchanges of Entropy and information.

The result is the creation of 3rd network/system: the reproductive network. And so most systems of the Universe are organic, ternary systems made of points (which can be anything from atoms to cells to human heads) organized in 3 networks. We exist as organic networks, to sense flows of Entropy and gauge information. Existence justifies itself.

Those Non-Euclidean points crossed by infinite parallels are able to gauge information, which implies a perceptive, intelligent Universe in all its fractal, self-similar scales of reality – a world in which even the smallest atoms can act-react to the environment, ‘aperceiving’ light and gravitational forces.

Aristotle and Leibniz, the 2 foremost predecessors of the 4th paradigm of biological whys distinguished conscious perception from vegetative and mechanical perception. It means perception has degrees of complexity. So the simplex particles of the Universe act-react in a mechanical way; yet they Still gauge information, reason why quantum physicists called their theories gauge theories, and they Still have 2 complementary networks of Entropy and information, reason why quantum physics is based in such complementary principle.

Fractal i – 1 points gauge information with infinite parallels.

The intelligent, active, temporal, informative Universe can be described with the formalism of logic and mathematics because its fundamental unit – a spacetime cycle – can be explained with ‘feedback’ equations, used in system sciences to explain the back and forth interaction between two poles or elements of an equation. Se<=>To where Se is a component of spatial Entropy, a motion element, body or field and To is a cycle of time that carries information, particle or head, becomes the syntactic, logic, minimal unit of reality.

These simple first elements of reality – points with volume that exchange Entropy and information, creating waves of Entropy with form (no longer lines – 2nd postulate), according to a set of isomorphisms based in their self-similarity, (no longer equality – 3rd postulate), that makes them evolve into different topological networks (no longer planes – 4th postulate) – make mathematics an organic language able to describe the logic of creation of all systems of the Universe made of infinite fractal, organic networks intersecting and creating when we put them together under those isomorphisms and topological restrictions, the puzzle of reality that the simpler, 3rd paradigm called the space-time continuum and becomes now a General System of Multiple, Fractal Spaces with vital Entropy and Time cycles with information, the two substances of which all beings are made.

And so with those 3 scales of ‘existence’: time cycles, knots of time cycles and networks of knots of time cycles (Non-E Points) we can explain all the ‘actions’ and systems of reality made of those cycles, knots and networks; and describe a complex Universe that exists ‘in time’ more than in fixed space, since it has always motion; it is also dynamic, made of cyclical, feed-back equations whose causal relationships, forms and trajectories are the essence and purpose of existence. We thus consider a more complex analysis of time arrows, beyond the duality of Entropy and information, which combine creating a reproductive arrow, exi, and further on socialize, ∑exi, creating networks. And so the universe has also an organic will: to create networks of self-reproductive points of Entropy and information.

Yet the most astounding property of those points is to be points of view, points with will, which perform actions with the purpose mechanical or not, but probably felt in all scales as the inner freedom of the point, of obtaining Entropy, information reproduction and social evolution.

The 4 wills or whys of the Universe are indeed embedded in the postulates of i-logic geometry. The point to exist has to be complementary, to feed and gauge Entropy and information and to last beyond its wearing it has to form part of bigger social networks or reproduce itself to last beyond death.

This simple program self-selects those species that reproduce and evolve socially even if that contradicts the primary individual arrows of the point. Thus the engine of the contradictions of behavior of points is that tug of war between the Galilean paradox of all points which gauge bigger his nose than Andromeda but need to hunt in herds and control the forms of the Universe with self-similar minds, joined in networks, this eternal duality of freedom vs. order, individual ego vs. collective spirit.

To exist is to act with motion and form, trying to achieve the ‘arrows of time’ or will of the Universe – feeding your Entropy network, absorbing information for your informative network, reproduce your system and in doing so, starting an external process of social evolution with self-similar entities to yourself. Those processes can be described with mathematics but we have to accept an intelligent, perceptive, fractal, self-similar Universe of infinite points of view gauging reality in a mechanical, vegetative or conscious way to explain why it happens. Their mathematical description stems from the duality between geometric form and logical function (hylomorphism).

Thus, the postulates of fractal i-logic geometry define also the basic arrows= cycles/dimensions of the Universe: the 1st and 5th postulate define a point as a system whose inner parts are able to transform and emit Entropy and information, E>O<e; the 2nd postulate defines an exi wave of communication that reproduces Entropy and form between 2 fractal points; the 4th postulate defines the social evolution of a herd that creates a fractal plane – a network with dark spaces; and the 5th postulate explains a point mapping reality, as it absorbs Entropy and transforms it into information through its small apertures to the Universe. Since even a minimal quark, as Einstein affirms, should be crossed by a relative ∞ number of strong forces.

It is the organic will of all systems that search for those 4 arrows what makes the quark to exist as a knot of such flows of time arrows: a physical particle traces energetic cycles described by the principal quantum number; shapes the form of its trajectory, a fact explained by the secondary quantum number; iterates along the 3 coordinates in similar shapes, an act described by the magnetic number, and gauges information to evolve in social groups, a fact described by its spin number. And those 4 numbers define it as a Non-Euclidean quantum knot of complementary Entropy and information with a 4D will of time. And as it happens they express the 4 arrows of time: E->Principal number; I-> secondary number; Re-> spin number; 4-> social evolution: magnetic number.

Further on, we can reduce all those topologies of social numbers and networks to the canonical 3 topologies of a 4-dimensional Universe, proving that those 3 topologies have the properties of Entropy, information and reproductive events. And so we talk of 4 ‘arrows of time’ or dimensions of change that create the future: energetic and informative systems and events, which reproduce a wealth of self-similar beings that organize themselves into social networks, creating bigger wholes – new scales of reality.

And this simple game of complementary beings that in favorable conditions reproduce self-similar beings, self-organized into bigger social networks becomes the why of all realities. Even the simplest particles, quarks of maximal information and electrons of maximal spatial extension and motion ‘decouple’, reproduce, when absorbing Entropy into self-similar forms, and associate in complementary networks called atoms, made with a central informative mass of quarks and an energetic, electromagnetic, wider body of electrons.

The 3rd paradigm of metric measure is not at ease with such ‘dynamic, spiritual concepts’, even if they can be described with the same mathematical formalisms as the previous example of the quantum numbers show. Those apprehensions however are dogmas, which stem from anthropomorphic beliefs.

Fact is that even the simplest complementary systems (quarks and electrons) interact together and if they can absorb more Entropy=motion they are able to repeat=reproduce the cycles of its system.

And so we talk of a 3rd reproductive system: from quarks and electrons, the fundamental particles of the Universe that decouple in new particles when they absorb new Entropy to living organisms, the fact that all is motion with form makes easy to reproduce those formal motions in an organic way.

Thus the new concept of a world made of formal motions brings about also a more complex philosophy of reality – organicism.

Organicism and its mathematical units, fractal points, that gather into social networks called topological spaces substitute the restricted concepts of Euclidean points, continuous spacetimes and mechanism, explaining why all those time cycles exist, guided by 4 time arrows:

Entropy feeding, information gauging, reproduction and social evolution. Those 4 categories are the so-called drives of living beings, the quantum numbers of particles, the 4 dimensions of our light space (electric-informative height, reproductive magnetic width, energetic length and social colors). Thus, there is a ‘Universal Plan’ with an existential finality: to create organic systems, departing from Entropy bites and information bits evolved 1st into social networks, then into complementary systems and finally into organic systems, news points of a bigger fractal whole: particles become atoms that become molecules that become cells, organisms, planets, galaxies and Universes. It is the 4th organic why that completes the adventure of science and this work explores in all its consequences.

Reality can be resumed in 2 words: networks, whose flows of exchange of Entropy and form create the patterns and events of reality and organicism, the philosophy of reality based on them. Organicism means reality and all its fractal parts are made of vital spaces (bodies and forces) and time cycles (informations).

We do not exist in an abstract background of time and space but we are made of time cycles and lineal spaces, cyclical and lineal strings if we were to use the restricted jargon of physics, a specific case of the wider jargon of general systems, which evolve socially to create the complex systems of each science.

Those wider, more complex definitions of time and space will substitute and absorb according to the Principle of Correspondence that makes each paradigm a particular case of the new, wider view, the limited concepts of a single space-time continuum and a mechanist description of the Universe, proper of the age of metric measure, which the pioneers of systems sciences and complexity have wrestled with throughout the XX century.

Recap: the minimal unit of the universe is a Non-Euclidean point/number, which classic mathematics defines as void of inner form and organic properties, to simplify the networks of numbers and point-like entities of the Universe for its geometric study. In reality though, points have breath; that is, they are real entities with Entropy and information parts, and so we have to upgrade Euclidean postulates with the new tools of Fractal and Non-Euclidean mathematics to make the language of geometry closer to reality.

Frames of reference.


The graph also explains the natural tendency of men to be self-centered and consider the world and the Universe around its ego. This is in fact natural to the way the Universe constructs points of view of measure, as Still centers of perception, in which the I is bigger than Andromeda Galaxy. But science must account for those distortions of perception. Religions and tribes, nations and castes are NOT scientific points of view. But they do enter the mind of people who do science. And it is important to remember it, because the 5th dimension brings a much bigger jump into objectivity and lack of ego to our vision of the Universe.

And so a  definition in brief of the 5th dimension illustrated in the next graph will be:

‘The 5th dimension is the dimension of spatial size (abb. Se) and speed of temporal clocks (Abb. To). Both parameters are inverted: when systems grow in size the speed of its time cycles slows down and vice versa. Smaller clocks tick faster and bigger ones tick slower, as it happens in galaxies, human beings or DNA.’

Indeed, if information can only be obtained from a fixed point of view, it follows the Universe is made of egos. Or as Aristotle put it, God is the unmovable ‘ego’ that moves all the Entropy around himself and there are infinite Gods, particles that gauge information, heads that see, minds that smell… This must therefore be included in the next paradigm of science to build an even more objective, less anthropomorphic reality.

Now in the 5th paradigm, this again changes and we have even a larger point of view. As we affirm there are infinite clocks of time, and infinite broken spaces and we adopt the point of view of all of them.

This is illustrated in the graph: In the right side we see our Universe, which Descartes affirmed correctly in his book ‘the world’, it was truly the ‘mind of man’ and Kant realized it was a light-mind.

Physicists however, limited in their logic and philosophical reasoning, never cared that much for the ultimate meaning of what they measure. That is why philosophers called their theories ‘naive realism’ and Hawking affirms that philosophers of science criticize him because they don’t know mathematics

It is not that the case, but rather the fact that a mere mathematical description of spatial measures, is NOT enough to explain the Universe. And the graph shows why. What physics has achieved is a rather perfect description of distances and motions from the perspective of light.

This has been done increasing human accuracy with metal-eyes, telescopes and microscopes, LHC-like accelerators and electronic devices. But it Still is only a light-mind. In reality in the Universe there can be infinite different minds and sciences of each mind. A dog’s mind will map the universe with smells – big atoms as pixels. It will be simpler than the pixels of instruments looking at light but Still a valid mind.

So there are ∞ mappings of the Universe from infinite points of view and multiple possible pixels.


2) A line is now a wave of communication of energy and information between points.

Now all this is formalized by the 2nd postulate of i-geometry which studies the flows of communication between 2 i-points that create a wave, and then as multiple i-points communicate, a network (4th postulate). A social gathering of points into herds, and then the gathering of several planes, creates an organism:

That is the game. And understanding how to exist in balance, in the golden mean, the best way to play it:


In the graph, the maximal creative function combines energy and information from two polar beings, establishing a balance of to and fro transformation of energy  into information that make the system stable, ‘existential’, to last in time. When the E and O components of the event are unbalanced, it is a predatory, darwinian event in which the pole with maximal exi force will absorb the other as relative energy extinguishing the entity. Thus in death (max. O or Max. E) an unbalance breaks the ‘ties’ of existence between body or field and particle or head and the system becomes extinguished.

A line is a wave-like event which communicates 2 st-points through a herd of fractal micro-points- a lineal action, exi, of energy that carries a frequency of information in which a message is encoded. The language of information is highly invisible to points outside the network that emit those messages as a flow of micro-points, self-similar micro-replicas of the mother-point that travel in waves across the external Universe transferring energy and information.


In most events those flows balance one point with more energy, Eo, that each science defines with different slangs, (‘a white hole’, ‘energizer’, ‘past form’, ‘male’, ‘body’, ‘yang’, ‘moving field’, etc.); and an informative point, Ot, the smaller form (‘a black hole’, ‘codifier’, ‘future form’, ‘female’, ‘head’, ‘yin’, particle or ‘center of perception’, etc.) Both become united by a dual wave that transfers energy from Eo to the informative point of relative future, Ot, and information from the future point, Ot, to the relative past point, Eo, creating together a cycle of temporal energy. The description of those points and cycles, which are common to all beings of space-time, creates a fractal, i-logic geometry common to all sciences and Universal species.

Generator Equation of SpaceTime fields. Vital dimensions: forward bodies & top mindsScreen Shot 2016-05-14 at 20.38.28

In the graph, the interaction between fields of energy and particles of information, either in Darwinian events in which the most complex form normally absorbs the energy of the bigger one, or in complementary events, which create a stable system of energy and information, explains most events on time and forms of space of the Universe.

In the graph, taken from more detailed analyses of the interaction of knots of time arrows in their processes of creation of a certain topological plane of existence=organism, we can observe how arrows of time interact, creating flows of energy and information that shape fixed cycles, which seem to us (paradox of Galileo), stable structures that anchor those points of view into a stable region of existence.

The equation that defines those events between energy and information arrows is common to all sciences, known as the principle of conservation of energy and information. It explains the 2 simplex Time Arrows of the Universe, which as the graph shows are at the heart of most events of the Universe.

‘All what exists is a type of energy that trans/forms itself back and forth into a form of in/form/ation’.

This Law fusions the principle of conservation of energy and information, the 2 main laws of all sciences. We call the creation of energy the arrow of energy, one of the two primary arrows of time in the Universe, or ‘entropy’; and we call the creation of form, the arrow of information, the second primary arrow of time, which in physical space happens in masses under the ‘informing’ force of gravitation that creates ‘bidimensional’ and ‘tridimensional vortices’ of space-time, according to Einstein’s principle of Equivalence between mass and gravitation – hence systems with more formal dimensions than simple, lineal forces; or complex formal 3-dimensional warping in life systems (DNA, protein, dimensional warping which store the complex form of life). Thus, again we see that systems apparently so different as physical mass or biological molecules do use ‘formal dimensions’ to store the information of their systems; while in other events employ lineal forces and lineal ‘fat’ molecules or lineal limbs to store or display energy. All what exists are processes that create information or entropy or its complex combinations (reproduction and social evolution). Since Reproduction, e xi is born from those simplex arrows, and social evolution follows by the self-organization of self-similar reproduced beings, it follows that the main Law of science is also the proof of the Law of Existence and its 4 arrows. Since this law is the main Principle of all Sciences:

‘All energy becomes trans/formed into information: S E<<=>∏I’.

This can be expressed with a Feed-back equation of energy and information:  E<=>I   or E x I = K (dynamic/ static)

Since all is Time=Motion=Change and space is just a static slice of time we call SE<=>I the generator, feed-back equation of Time-space: Those SE<=>∏I cycles generate the events and trajectories of each and every part of the Universe. Where its 4 elements describe the 4 arrows/motions of time:

S (social evolution) Energy < (Reproduction)> Information.

Further on, those 4 elements become the parts of all physical or biological systems, when we perceive them in a static ‘dharma’ or moment of ‘present’:

S (cells/waves) of Energy (bodies/fields) X (particles/heads) of Information =Complementary system.

Thus the Generator Feed=back equation of Reality also represents the species of the Universe. Its complex study, carried out by General System Sciences, requires the use of Non-Euclidean Geometries to define the topology of each part of those ‘knots of Time Arrows’ that act as ‘reproductive bodies/fields’ and ‘informative heads/particles’ of any exi, complementary system; and the development of a complex causal, ‘non-Aristotelian’ logic to define the order and interactions of those arrows. Since the ‘parts’ of each whole knot of time will have functions defined by the needs of any system to gauge information, absorb energy and reproduce.

It is also clear than in all those organisms the fundamental element, the center of power, is the brain-head and its languages of information, which control the body of energy and shape the form of the organism. They set a selfish dominant time arrow towards the growth of information, which is the ultimate cause of the cycle of life and death of all organisms that end up in a 3rd age of excessive form/information, as they warp all the energy of the system.

This fact also explains the relationship between information and future – the most difficult dimension of time to understand by mechanist science since we do not see the future, as we see spatial geometry with our machines. Yet the future already exists in the realm of complex, bio-logic thought, as we will all warp our energy into information and die; and as a species we will always evolve into more informative beings in the future that will feed on the energy of those entities that don’t evolve and die. Let us then consider those systems of time knots, adding the concept of a causal order between its arrows of time from past to future.

The creation and extinction of bidimensional, herds of energetic space and networks of temporal information explains the dynamic events of all scales of existence: As time curves energy, spatial planes acquire informative height and vice versa, the destruction of informative dimensions creates planes of energy. Particles of information are small, spherical forms/cycles, like your eyes and brains or an atom’s proton or a black hole in a galaxy. They are on top of the system, where perception is ‘higher’ and show convex topologies of maximal form.

Bodies of energy are bigger planes or lines that store energy to move the organism, like your body or an energetic weapon, moving forwards in the relative, diffeomorphic dimension of length. And so both, high information and long energy combine to reproduce a system in the z-dimension of width. The Universe has 4 time motions, which perceived in stillness (Galilean Paradox) create the 4 vital dimensions: energy is length; information is height; width, its product, is reproduction and time brings the arrow of  organic evolution, as ‘points of view’ organize in bigger social organisms that ‘survive better in time’, because they have more energy and form than the individual cells.

Thus, each of the main time arrows is defined for each local space-time as a diffeomorphic dimension that only reaches till the limits of the organism, but extends its action’ further into the ecosystem or plane of existence in which the organism resides. In complex algebra, all this can be modeled with Partial equations of the total function of exi=stence, which define each of those 4 arrows, in a scale of increasing complexity and logic causality:

e, i, exi, ∑e<=>∏i.

Recap: The holographic principle (bidimensional energy and information) allows the constant transformation of energy into form and vice versa, EóI, creating the fundamental principle of science, the principle of conservation of Energy and Form: ‘All what exists is energy that trans/forms back and forth into information’. The formalism of those 2 arrows of time gives birth to the generator equation of time, S EóI, which defines all species of the Universe as self-repetitive fractals of energy and form Let US then consider the basic representations of the function of existence, from where most disciplines extract their particular graphs and differential equations.

 3rd postulate: Social networks

The scales and ST¡ences of reality.

As the widest one is the concept of a topological network-plane, of the 5th dimension we can consider it in more detail.

  • From the perspective of the central point, which branches into a fractal network (the standard Geometry of elliptic spherical forms, ‘not explored by geometers of the XIX c. because fractal mathematics did NOT exist) the world is an elliptic geometry with him at the centre, connected to all other ∆-1 parts. So networks are elliptic Non-E Geometries where there ARE not parallels as the brain, black hole or Wall Street knot of the organism galaxy or economic ecosystem is connected through its informative networks (nervous, gravitational and financial systems) and its quanta of information (electric messages, gravitational waves and money) to all its ∆-1 ‘slave cells’ . Yet those cells are unconnected and so they exist in a hyperbolic geometry of infinite parallels.
  • Thus any ∆-super organism is a group of n-1 cells joined by energetic, informative & reproductive networks that communicate them. Those ∆-1 cells are also superorganisms made of small i-2 molecules joined by Sp, ST=exi & Tƒ networks and so on and so on.

Thus we define any system as an ∆-superorganism made of smaller, similar ∆-1 super-organisms. And each ∆-scale of superorganisms & its ecosystems are studied by a human science but all of them follow the same Invariances & emergence Laws & Galilean Paradoxes of 5D Metric formalized with the tools of Existential Algebra & Non-AE=i-logic Geometry.

We unify all Natural Systems as superorganisms using a single template definition, since they differ only by the i-scale or ecosystem in which they exist or the specific types of energy & information their networks are made of:


In this old graph (where n is old notation for an ∆-plane), we classify all the scales of the 5th dimension as super organisms  according to the combined 3 geometries which put together create a network plane:

nism) is a population of iterative (name a cellular species), related by informative (name a language or informative force) and energy networks (name a kind of energy), which combine into a reproductive network that iterates the organism.’

Fill the gaps with a specific species, language of information and force of energy and we can define any network-organism, which will be a ‘part’ of a whole world-plane or ecosystem, composed of several species that occupy different ‘vital spaces’ but interact through the same language of energy & information:

An world-plane or ecosystem (name a specific world-plane) is a population of several (name the species), related by informative languages (name their languages or informative forces) and energy networks (name the energies).

I.e.: The ‘world-plane of ‘zoology’ includes all beings of relative size, i=6, that use light as information, called ‘animals’. The world-plane 8, a galaxy, includes as parts, all celestial bodies of size i=7 related by gravitational networks, etc.:

∆-3:  An atomic organism is a population of (electronic) energy and (nucleonic) information, related by networks of (gravitational) information and (light) energy.

∆-2: A molecular organism is a population of atoms, related by networks of gravitational energy and networks of electromagnetic information (orbitals, London, Waals forces).

∆-1: A cellular organism is a population of molecules, related by energetic networks (cytoplasm, membranes, Golgi reticules) and genetic information (DNA-RNA.)

Screen Shot 2016-05-14 at 20.46.18

∆=o: A human organism is a population of DNA cells, related by networks of genetic, hormonal and nervous information and energy networks (digestive and blood systems).

∆+1: An animal ecosystem is a population of different carbon-life species, related by networks of light information and life energy (plants, prey).

Screen Shot 2016-05-14 at 20.46.30

∆+1: A historic organism or civilization is a population of humans, related by legal and cultural networks of verbal information and agricultural networks of carbon-life energy.

∆+1: An economic ecosystem or nation is a population of human workers/consumers and machines, related by networks of digital information (money, audiovisual information, science) & energetic networks (roads, electricity)

An economic ecosystem differs from a historic organism because they use different languages of information (civilizations use verbal or ethic laws while economic ecosystems use digital prices) and include 2 different species: human beings and machines.

∆+3: A galaxy is a population of light stars and gravitational black holes, related by networks of gravitational information and electromagnetic energy.

∆+4:  A Universe is a population of galaxies joined by networks of dark matter and energy.

Now, each super organism is a network because it has an enormous amount of dark space-times it does NOT see, the cat alleys which the mind does not neeed.

Discontinuity is essential to pan geometry and the continuity axioms no needed to build them.

The i-logic of Non-Aristotelian, Non-Euclidean geometry, of the ternary causality of a π cycle, made of 3 diameters in its perimeter means that the diameters of the protective membrane blind the system, who only sees what its connected sensors see.

Thus the 3 diameters in the simplest 1-sphere system (a disk) cover 96% of a pi-perimeter surface, letting the point of view in the center, see only a 4% of light, through the holes of the membrane, leaving π-3/π 96% of dark matter for the logic mind to see without the glaring of light.

This is in fact what we do NOT see of the Universe across the halo membrane of our spiral galactic disk (proportion of dark energy and matter).

So we can consider that our electronic eyes miss 96%, which is the volume beyond our perceived scales of the 5th dimension, which however exists in the larger russian doll of gravitation.

Poles of communication

It is now when we can consider what those points do beyond the couple formation: the answer is obvious, the create social groups, networks, which seem plane but are quantised as flows of entropy and form communicated between two poles (perceived as energy and information, as they ‘transit’ through a relative ‘active support’ of mass-charges-vortices of time).

Those poles that create through a relative support or ‘field of energy and information’, flows of entropic lineal motions carrying cyclical frequencies of form, which communicate between points codified languages that express a common will to absorb energy and information, entropy and form. This is what points do: to absorb entropy and form, to move and perceive, to enact their will of existence, and repeat, reproduce their existential momentum, their existential force:
Max. Se x To < ∑se x to (se≈to).

Why, who, when just by survival chance? this program was imposed we do not know. But the natural enactment of the program is to increase the network flows between points to create bidimensional waves, that become rlative S-T planes susceptible of being excited by, and carry and reproduce a certain wave of energy and information, STi=se x to, an existential being.

In the subtle commanding flows of reality, it remains to be understood if the larger ST field forces the existence of certain smaller seto beings codes the smaller being or vice versa. In any case, as the points share actions of energy and information, similar to their selves, an ∑n-1 dense flow of energy and information surrounds a network, which starts to pop up self-reproductions of itself, which will constantly share energy and information as they keep reproducing and tying up networks of communication. To that aim all growing network of similar cells must have access to a simpler field of quanta of energy which they can absorb and mild into themselves, and for that their larger existential force, as wholes will ease the task.

As the networks become denser they will finally self-organise themselves socially into a whole, with a hierarchy of ‘organisation’ along S=10 scales of social growth, in which cells will specialise in the ternary 3 x 3 sub-systems of entropic, informative and reproductive tasks, and emerge as ones of new decametric scales.

Networks thus grow both in the 3rd and fourth dimension of space-time, filling up a space-time plane but in the social growth of its elements, and its fractal diminution into smaller cellular scales. This ‘invagination’ of an initial being through the program of absorption of energy and information, replication of form and social evolution: ∆o->∆a->∆e->∆i->∆U, is what we call the program of existence:

Points perceive to orientate (∆o) its motions with accelerations and decelerations and change of orientation, ∆a, which allow the being to feed on energy, ∆e, in order to reproduce its systems either individually or sexually to iterate itself, ∆i, and then evolve socially into ∆U niversal new single planes of existence. The program of existence of the being thus make it grow constantly, and by growing, its ‘Generator Equation’, which represents its complementary systemic nature, its ternary elements and on the whole its ‘existential force’, that is its Spatial entropy, temporal information and reproductive capacities, ∑ Se x To = STi, becomes more efficient, and makes the whole ‘networks system’ stronger.

Law of inversion of scales.

Finally to mention the inversion of ‘scales’ if we use the metrics of information, which increase as we become smaller, or the metrics of size which increase as we become bigger. Yet the product of both become invariant:

Sp x Tƒ =∆±4, where the 9 planes of existence of the logarithmic 10¹º scale of the Universe, are equivalent as each one has more and less energy and information. So we should write in the logarithmic scale, each plane as the equal sum of its logarithmic capacity to carry information and its relative size:

Sp-4 +Tƒ+4 = Force = Sp-3 +Tƒ+3=Atom=Sp-2 +Tƒ+2=Molecule=Sp-1 +Tƒ+1=Cell/Matter=Sp +Tƒ=Human Scale=Sp+1 +Tƒ-1=Planet=Sp+2 +Tƒ-2=Star=Sp+3 +Tƒ-3=Galaxy=Sp+4+Tƒ-4=Cosmos.

We however simplify this equal value of all Sp x Ti=world-planes of existence for easier analysis, considering only the relative Ui scale of growing planes, with a cardinal that starts in the smaller:

Forces: ∆i=1, atoms:∆=2, Molecules ∆=3, Matter/cells ∆=4, human organisms, ∆=5, Planets, ∆=6, Stars, ∆=7, Galaxies, ∆=8, Cosmos, ∆=9, between the invisible beyond human informative perception dark energy and dark matter.

It is thus clear and we shall use the term STience, to differentiate this philosophy of science, and perspective on reality, that the Universe is a game of space-time, and each science the study of one of its Universal Planes.

It has failed though for centuries to explain the whys and ultimate structure of that space-time puzzle. It has been ‘Science’, that is only a Science of space. STience is a wider concept where space is submissive to the flows of time and its actions, where beings are space-time beings and so all sciences are stiences that study varieties of space-time beings of different Universal Planes.

Science has been corrupted in its ultimate ‘quest’ for the whys of the Universe by its own success in a rather more pedestrian task, to find the laws of measure and equations of motion in space and time, the hows of reality. This shallow description of motions in an external world today is often confused with absolute knowledge. It is not. Because while we know we move in space and time, we have yet to solve what time and space is in its deepest sense.

Stience will not be completed till we know the whys of those motions of space and time; and specially the whys of existence in space and time. In other words, till we do not find an epistemology of ‘Stiences’, which accepts the fact that the Universe is made of relational space-times distributed among ∞ relative Space-time systems and species.

Thus there should be a philosophy of science, we shall call ‘STience’ which is dedicated to the study of all entities as made of space and time parameters.

Now each of the new postulates of i-logic geometry has an enormous range of phenomena to study with them. Let us consider merely a theme related to the waves of communication of the 2nd postulate, locomotion as reproduction =exchange of field forces and forms.

In the first case the fundamental law of quantum physics, Tƒ (Fermion) < Sp (boson ) > Tƒ (Fermion) allows the reproduction and communication of information between 2 relative mind-particles. In the second case we solve the paradox of Zeno and understand the true meaning of motion in a 5D fractal space-time.

Thus, the line, as distance or dimension, D, or as motion, or speed, v, is in fact a wave, a bidimensional space-quanta, and we shall then find that indeed, H, K, and c², the quanta of space of the 3 main quantum, thermodynamic and gravitational scales of physical systems are bidimensional.

We can though consider a perspective based on the relative ‘number of parameters’ or dimensions needed to define a being.

The point is a scalar cycle, which can be defined with a single parameter of length or frequency in its relative space or time states.

Then with 2 parameters, the definition is either a wave that communicates two points in a harmonic oscillator or a clock of time in motion with angular frequency, ƒ or w, which needs also two parameters, because it is indeed twice as complex as the point.

You can define a line with a single dimension; you need 2, to define the cycle.

And so by a mixture of restrictions of spatial topology, time cycles and i-plane ‘mathematical structures’ and a will of surviving and enacting the game reinforced by various methods in the zero-points, the astounding wealth and variety of forms of the sentient, eternally moving Universe and its ‘existential forces’ keep evolving:

 In the graph, the planes of existence and some particles of the human being – with an informative scale, Max. Ti=Min. Si. The human being interacts along all those planes to extract its ∆a actions of motion ,∆o, perception, feeding, reproduction and social evolution. We does co-exist in all those scales, and our components act in all of them allowing our motion, etc. Our actions, our program of existence, which is not different from any other being. And in each plane each point enacts actions which are synchronised into waves of actions, of larger ‘existential forces’ that group into new cyclical waves of actions, emerging once and again in an upper scale.

How all those scales co-exist together? The answer is an interesting but difficult concept to grasp:

In the smaller world there are more time cycles than in the larger world. How does we adjust this ‘time longitude’, larger in smaller worlds respect to bigger simpler worlds

In a strictu senso, if we consider all the planes to have departed from a single past-point, T, the upper scales will have a ‘time delay’ similar to the delay of any web respect to its harmonic initial focus, only though we are measuring time delay. Thus the smaller world will be in a ‘relative future’ to the upper worlds. On the other hand, the upper world must be done into the future, with the smaller worlds already sustaining them as past forms. Here there is a fascinating first contact with one of the most complex elements of 5D-ST theory – the relativity of past present and future states, that co-exist in quite mixed orders.

In brief, the smaller worlds have more ‘time content’ than the larger worlds, which are both, slower and increasingly lagging in a relative past to the smaller worlds actions which code them, but both are from their relative frame of reference futures, as the larger worlds see the smaller as their sustain, and the smaller code the larger as their past.  Consciousness of what is time past or time future is thus relative to the point of the scale of planes we occupy.

Unification of all stientific planes: topi-geometric Definition of the Universe.

Now, we shall complete this introduction with a definition of the Universe from the perspective of the philosopher of science, which we will elaborate and explain in great detail in the rest of this post.

“The Universe is the sum of all the symmetric, ternary super organisms (spatial synchronous view) performing a world cycle of 3 ages between generation and extinction (diachronic point of view), as they travels through 3 relative planes of the 5th dimension, growing in size and diminishing in a balanced, Sp x Tƒ= ∆±1 survival form, performing from its centred point of view, the 5 space-time actions of absorption and emission of energy and information, (active meaning of space and time), ±∆e,i,∫u, which ensures their survival.”

As each science in fact merely studies space-time cycles and relationships around a given fractal scale of the 5th dimension:

5th dimension best scale

 How to include metric spaces. Topology: informative, energetic and reproductive systems.

According to the Principle of Correspondence, each new wider, more comprehensive model of reality must include all the cases of the model it substitutes. So while a 4-dimensional description of multiple space-times suffices in itself to give meaning to reality, it appears unconnected with the previous paradigm of metric spaces, reason why we must achieve a more detailed analysis of those cycles and give them specific mathematical operations, as to be able to connect them with the 3rd paradigm of metric spaces, its geometries and mathematical algebras. This is done at two levels:

-By describing them with a higher form of geometry, topology.

-And by describing more precisely the 4 arrows of time, subdividing them in more specific types of events and adding precise algebraic operations to each of those cycles.

Let us consider briefly those 2 elements that will be developed in depth in other works.

Some initial precisions though are needed. Today information is not understood as ‘form’ but measured, as it corresponds to the science of metric spaces, since Shannon, by considering frequencies and patterns in one dimension. But here in/form/ation as the name indicates is given by form. So Shannon’s analysis of information is correct but explores only patterns of information in one dimension (such as the information carried by the frequency of a wave). If you have though 2 dimensions you can square the volume of information you can store and transmit. And in 3 dimensions you get a cubic quantity. And so we observe that most complex systems have at least 3 ‘levels’ of complexity in the creation of information. So lineal proteins fold into bidimensional membranes that fold into complex 3 dimensional patterns, which are in fact the active information.

The symbiosis berween function and form is evident: The line is the shortest distance/motion and so it is the main form of energetic organs, from cilia, to legs to light fields. The cycle stores the maximal information and so it is the usual organ of information, from cameras, to vowels to eyes.

Yet when we consider more complex topologies of information, we talk of hyperbolic spaces that store information and are basically a complex ‘sum’ of chained cycles, often forming a tube of height, and so your head is at the end of your height and antenna is at the top of height. And height becomes a dimension of information.

When we consider energetic systems, normally they are external membranes that protect with its strength and filter the energy of the external world. And so because they enclose the system, they are normally made of tiles, squares, hexagons that put together cover totally the space; they are a sum of planes even if the total sum might appear sometimes as a spherical form and in topology they are call spheres. Finally the cycles of reproduction are toroidal cycles that come and go from the informative center to the energetic membrane, combine both and reproduce the system.

The 3rd type of topology – reproductive topologies that combine the other 2 arrows – become the 3rd complex arrow of time.

Yet if those 2 simplex arrows shall explain it all, we must combine them further, realizing that ‘from 2, yin=information and yang=energy, comes 3’, since ‘the game of existence combines yin and yang into infinite beings’ (Cheng Tzu). Indeed, philosophers have always known that reproduction combines energy and information into self-similar beings. And the 4th paradigm will show how all complementary systems of the Universe, from the simplest particles, quarks and electrons to the more complex, humans and perhaps universes, reproduce their form by combining their energy and informative organs and systems, repeating them in another discontinuous location of space and time.

So there are not only 2 simplex arrows of energy and information but also a complex arrow that combines both, Energy ó Information: the Reproductive arrow. And again, while there are many different ways to achieve that arrow; we observe always that an energetic, lineal, topology (since the line is the shortest distance/motion between two points, the simplest energetic systems are lines, or planes), and an informative topology (since cycles are the perimeters that store more information in lesser space, informative organs, are cyclical) mix to reproduce. So men are lineal in form and are the energetic sex, and human are cyclical and are the more perceptive sex, and both combine to reproduce. Machines are reproduced by humans which are the cyclical, informative component that forms the raw materials or energy to make them. And so on. Because it is an obvious logic consequence of the discontinuity of vital spaces which are finite and the limited length of a time cycle which always ends, that to survive species must reproduce or else its logic form perish.

So the Universe is ultimately an organic system of reproductive systems of energy and information.

Thus, once we establish the 3 topological regions of any system, which is their why we can add more detailed measures and convert each topological space in a specific species of reality connecting the why and the when of the metric paradigm, fulfilling the Principle of Correspondence.

Recap. To fulfil the principle of correspondence multiple time-spaces must be able to connect the why of the cycles/arrows of space-time with the precise geometries and algebraic measures of the metric paradigm. The 3 dimensions=arrows=cycles of space are the perpendicular 3 topologies of the Universe: the function/form of energy, the function/form of information and the function that combines them, e xi, of reproduction. Those 3 dimensions define all topological spaces. Absolute space is its sum; it self a Non-Euclidean system.

HOW DOES the 3rd postulate connects with classic mathematics? Easy. It is called topology and we have already mentioned those connections in a previous paragraph.

4th postulate: the logic of communication

The law of behavior in the organic Universe is 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, perpendicular  manner. This means robots will perceive us as different and eliminate us with its faster, higher mathematical minds. They WON’T be our slaves for too long.

A case of Darwinian devolution among men that perceive each other as different and enter into a perpendicular, Darwinian relationship and a case of social evolution between 2 forms that perceive each other as equal and enter into a parallel relationship of social love. The fourth postulate of non-Æ topology thus ‘vitalises’ the laws of mathematics, establishing the three fundamental geometrical≈behavioural relationships between T.œs, according to its:

  • Informative (Particle-head) communication, possible in cases of relative similarity≈parallelism (which determines parallel herding and social evolution).
  • Perpendicularity≈difference in particle-head, which will determine if one system is related to the other and there is body-similarity, hence can be used as energy its darwinian destruction,
  • Or 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.

Indeed in three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. It follows that 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.

The different degrees of Parallelism, Perpendicularity and skewness are thus essential concepts of vital Non-Euclidean geometry.

We have introduced in the graph with a bit more of complexity the logic dimensional laws that define how systems IN ANY scale of the inverse, 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, 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, so you understand that in the Universe organic, geometric and scalar relationships are symbiotic to each other. 

All this said, warning including, mathematics is fun and the biggest pleasure of my youth, 30 years back when I discovered 5Ð was to upgrade maths. And then mathematical physics, till I ran into the total lack of Live memes=eviL behavior of military physicists and gave up scholarship.

MATHEMATICS AS WE JUST SAW ARE VITAL AND SOCIAL,ETHIC and the pretension of being a formal abstract language coherent in itself without mirroring reality is just more of the tiresome egocy of man.

But there is will in the Universe. Communication can be darwinian or collaborative, between 2 or more than 2 in networks. Who decides that? The will of survival of each system. And we will elaborate on that. But basically systems exist to absorb more space and more time, more energy and information in the common language for their limbs/fields bodies/waves and particles/heads to survive, as they ARE made of space and time of energy and information.

So there is a set of rules in the brain-point to decide what ±E, I to absorb or emit to survive. So 4 quantum numbers, 4 genetic letters, 4 drives of survival, feeding on energy, moving expelling energy, perceiving with information and communicating information with a limited scope to create social groups, new scales of the 5th dimension, through a common language, or full in sexual reproduction, become the 4 strategies of survival of systems.

For the sake of arguments, we shall resume them in 5 letters, a, for accelerations (locomotion studied by physics), e for energy feeding, i for informative communication, œ, for an offspring that reproduces the being, and u for social communication that creates new networks and super organisms.

But in a mere mathematical geometrical way, we can talk of two type of motions, parallel motions among systems that communicate to evolve socially together, and perpendicular motions, in which systems ‘cross’ other systems, penetrating its vital space with the usual consequence of darwinian fighting and annihilation of the information of a system.

Thus perpendicularity and parallelism acquire in T.Œ, also an organic nature, which is essential to classify the way systems relate to each other both in mathematical logic and biologic terms at the same time, and for that there must be a will of survival, automatic, apperceived, vegetative or subconscious in all beings, imposed by the simple fact that only systems who perform those 5 actions of survival, a,e,i,o,u, ‘exist’ as time goes by. And those who don’t feed, inform, reproduce and evolve socially (with the geometric motions we must interpret in organic terms) die and so the program of survival imposes itself.

We need the 3rd man, the point of view in the middle of the cycle. He who measures. And he who measures is not as the admirer of Newton Mr. Blake painted in his famous portrait, ‘God’, a geometer with a compass (though he does trace cycles in this he was right, and he was an artist, a creator, a painter in 5 dimensions, in this he was right too). But he paints with a logic brush, the brush of time cycles, ‘of a logic higher than that of man’ (Saint Augustine).

3) Two points are equal depending on their internal and external similarity (a logic, behavioral postulate with a corollary that connects non-e geometry with social evolution: 2 similar points will associate themselves, in parallel networks, depending on similarity; 2 different points, will use each other as energy, in Darwinian, perpendicular events).

– 4th Postulate: Equality is no longer only external, shown in the spatial perimeter of any geometrical form (Congruence in Euclidean Geometry) 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 3rd 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:

1) Reproductive functions in case of maximal self-similarity or complementarity in energy and form. For example, in the body, the Max Sp x min. Tƒ (male) < => Min. Sp x Max Tƒ (female) form the complementary couple

2) Social evolution, when points share a common language of information, they superpose their combined momentum. In our notation:

$t x ðƒ= $t x ðƒ -> 2$t x ðƒ.

This behaviour is standard in most ‘hyperbolic’ infinitely parallel systems and bidimensional Sp+Sp or Tƒ+Tƒ equal systems. For example it is the superposition of waves, the bidimensional layers of liquids, the superfluid properties of quantum systems, etc.

3) Darwinian devolution when forms are so different that cannot understand each other’s information and feed into each other, establishing a relationship of perpendicularity akin to an elliptic geometry, in which the predator intersect functionally the prey and devours it.

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 Spe=Spe).

Finally they will simply not communicate; when there is neither a common information to evolve socially nor a common energy to feed on.

In this case the pan geometry is ‘discontinuous’, without need for complicated Geometrical symbols – a geometry of ‘cat alleys’, of ‘dark spaces’ and parallel Universes, where we ignore completely the other being as it does not invade of spatial territory nor has anything to ‘tell us’.

Yet because any point absorbs only a relative quantity of information from reality, self-similarity is relative and it can be faked for purposes of hunting, allowing biological games, such as camouflage and capture, or sociological memes that invent racial differences, allowing the exploitation of a group by another.

The geometric complexity of the 4th 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 maximised.  The same phenomenon between cells with the same inner information /DNA originates the ‘collapse’ of waves into tighter organisms.

The 4th 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).

Screen Shot 2016-05-13 at 15.03.40

In the graph, the 2 essential forms of communication between dual systems, attractive and repulsive, according to the orientation and external or outer location of its foci. An interesting question of astrœphysics is the analysis of orbital systems, which MUST by definition have a second focus. What is the meaning of the 2nd focus of the sun?

If we add the parabola and other exponentials, which are the essential ∆=sxTn, curves of the 5th dimension, to the canonical cycle of a single point of view, we obtain the conic, which is the canonical point of view oriented along two different geometries of the 5th dimension (equal when the orientation is in the same plane):

4) a plane is a network of points with ‘dark holes’ between them (hence the 96% of dark matter of each points outside its ‘plane of existence).


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