**SUMMARY**

**FOREWORD**. THE DIFFERENT CONCEPTS OF SPACE

**PART I.** THE 3 AGES OF GEOMETRY

I. EUCLIDEAN. GREEK BIDIMENSIONAL GEOMETRY. THE HOLOGRAPHIC PRINCIPLE

II. CARTESIAN. MATHEMATICAL PHYSICS

III. NON-EUCLIDEAN TOPOLOGY.

**PART II.** ∆ ST GEOMETRY

••••••

**FOREWORD: DIFFERENT SPACES**

“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.

**∆-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:

**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 Tƒ, according to the metric of 5D: Sp x Tƒ = 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 Tƒ = k, the smallest mass is actually the greatest/densest/heaviest*.

**The rashomon effect of truth. Epistemology of absolute relativity.**

IN THAT REGARD, we must also 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 / Tƒ: 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 (: Oh, well, only to tell you such geometry was my first true breakthrough on 5D theory, somewhere in California 3 decades ago… Of course nobody gave a fuk so to speak… ha, ha, likely they thought i was a madman and a trouble-maker, which I am, as only madmen, children and saints always say the truth (: and i have a bit of all of them).

**PART I.**

**ð. 3 AGES OF GEOMETRY**

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:

**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.*Without this realisation nothing of the XX c. explosion of abstract mental spaces to represent reality would be possible.*- Further on, he understood relational space-time in space is defined in a first incursion in topology by the concept of ‘
**adjacency**‘,*which completed the 3 fundamental ‘modes’ of relationship through geometrical space of t.œs – complementary adjacency, perpendicular darwinism and parallel social evolution – hence a concept*essential to the organic structure of the Absolutely relative Universe defining for the first time**topological transformations**are those in which motion does NOT deform the fundamental*properties of reality in space,*starting a trend culminated by Hilbert’s foundations of geometry (yes the guy we criticize so much – he did also do some work of merit :), with his emphasis on some key abstract concepts such as betweenness, congruence, continuity, incidence, separateness…*which are clearly relative concepts concerning***scale and symmetry,**the mind elements that allow a singularity or point of view to ‘construct’ a wor(l)d-view over and ‘stiffen’ the motions of reality to make a mental mapping of them. - 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 i**s 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, f*orming an absolute enclosure.

And ultimately as ALL points are in fact ‘two strong’ points, two poles, which are equivalent, *it establishes a fundamental property of Nature, the bilateral symmetry with inverse properties self-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:

**Adjacency**(forms that are pegged, hence forming part of the same time§paœrganism).**Perpendicularity**, (forms that penetrate and disrupt its inner systems, basis of darwinian events.)**Parallelism**(things that maintain its distance and allow communication through a common medium or network, basis of social evolution –*studied in affine geometry*.)

In non-æ geometry they will be extensively studied as the fundamental modes that define the relationships of ST, complementarity and ‘symbiosis’ (adjacency), darwinian struggle (perpendicularity) and ∆§ocial evolution (parallelism) of all systems, *becoming the essential qualities to understand how spatial relationships define temporal events among all systems and scales of nature, studied by the fourth postulate of ‘congruence and similarity’. *

IT IS THEN ESSENTIAL TO understand the ultimate meaning of parallelism vs. incidence/perpendicularity also AS MENTAL descriptions of two logic states – one of parallel social evolution and one of darwinian colliding ‘tearing’ by the perpendicular, incident line, taking the concept out of its spatial representation, as Lobachevski’s ‘first great insight’ did for all of future findings of 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 o*n 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.*

**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.

EUCLIDEAN, BIDIMENSIONAL GEOMETRY OF POINTS WITHOUT PARTS. 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.

CURVATURE, SURFACES, DIMENSIONS. VECTORS. GEOMETRIES WITH OUTER LOCOMOTIONS.

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

GEOMETRY WITH INNER WAVE-LIKE SPACE-TIME MOTIONS.

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’.

∆-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 f*ractals, which will be the natural next step to the analysis of THOSE wholes made of point networks.

T: And the analysis of singularities with the ad on of chaos theory and the formation of ‘attractors’.

So finally all those organic, scalar *properties of mathematical space-time, becomes complete now,with:*

∆•ST: Non-Euclidean Vital Geometry.

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 5^{th} dimension make survival possible. Geometry is then the study of the spatial form that the functions which dominate the vital, sentient Universe, adopt in their existential actions.

And as such is the best method to 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):

Max. Tiƒ: Human ego: HILBERT ‘IMAGINES’ POINTS & PLANES.

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…

**I. GREEK BIDIMENSIONAL GEOMETRY**

** ARISTOTLE’S GODS≈ P.O.V.S**

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.

**BIDIMENSIONAL STILL GEOMETRY.**

The first age of geometry is the greek bidimensional age. AND IT bears proof of gst and its holographic principle that most theorems of geometry can be proved in a plane.

Of them, we shall deal here with a few, adding some new discoveries, specially regarding the ‘postulates of non-E’, needed to fully grasp bidimensional geometry and why their theorems matter.

- 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.

**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:

In the graph, taken from a physical wave, a particle reproduces its forms as it moves as a wave of adjacent particles one after another. This is the definition of motion, which solves Zeno’s paradox. Proper motion does not really exist, but reproduction of information along a path, with limits for each world and geometry (in the Euclidean human space, with the limit of c-speed for transfer of energy-form). So continuity can be defined in a single plane with the postulate of adjacency.

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:

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 3^{rd} 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 3^{rd} 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.

Indeed, one key question in the whys of physics is ‘why’, systems move in a relative field, its ∆-1 scale of the 5^{th} dimension, towards the gradient region of maximal density of force – so we move towards the attractive 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 5^{th} dimension, as then they will be above or below but not ‘into’, lay ‘on’, therefore is NOT a real event but a parallel event. A ‘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 A^{2}xioms of Greek Geometry (Archimedes, Aristotle’s axioms; in my Leonardian notebooks, written with shorthand incomprehensible Spanglish, i-logic weird symbols, which perhaps in the future some robot will try to decipher, he will find my abbreviation of those 4 Axioms of continuity and angular perception, written A^{2}c^{2}ioms, ab. A^{2}c^{2} 🙂

They are concerned with the perception of size and its comparison from a given point of view. And again, as always in the dual/ternary Universe, as in the case of the lineal and circular continuity principle, we have one axiom dealing with lineal sizes and the other with circular/angular perception of sizes:

*
*ARCHIMEDES’ AXIOM. If CD is any segment, A any point, and

*r*any ray with vertex A, then for every point B

*≠*A on there is a number

*∆*such that when CD is laid off

*∆*times on

*r*starting at A, a point E is reached such that

*∆*x CD ≈ AE and either B = E or B is between A and E.

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.

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__

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

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 3^{rd} 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__.

- Of any three points on a straight line, just one lies between the other two.
- 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.
- 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.)

- A motion carries straight lines into straight lines.
- Two motions carried out one after the other are equivalent to a certain single motion.
- Let A, A′ and a, a′ be two points and half lines going out from them, and α, α′ half planes bounded by the lines a and a′ produced; then there exists a unique motion that carries A into A′, a into a′ and α into α′. (Speaking intuitively, A is carried in A′ by a translation, then the half line a is carried by a rotation into a′, and finally the half plane α either coincides with α′ or else it has to be subjected to a “revolution” around a as axis.)

The axioms of motion 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 1^{st} and 2^{nd} 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 3^{rd} 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 (3^{rd} 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 4^{th} 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.__

- 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).__

- Only one straight line can pass through a given point that does not intersect a given straight line.

These axioms, then are sufficient to construct Euclidean geometry in the plane. All the axioms of a school course of plane geometry can in fact be derived from them, though their derivation is very tedious.

The axioms of Lobachevski geometry differ only in the axiom of parallelism.

V′. Axiom of parallelism (Lobachevski).

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

We shall in the 3^{rd} and 4^{th} 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:*

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 5^{th} dimension, which is not needed in a Universe of absolute relativity of scale. The lengths of segments are changed in the process and so are the angles, the outlines of objects are visibly distorted.

What then remains?

Immediately we see, of the essential qualities that the property of a number of points lying on one straight line is preserved; and those are as anticipated in our i-logic axiom of a line, 3 of them, for such lines to be fully straight. So the projection on the 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.Œ a modality of Spatial Reproduction.

The other action related to projective geometry is obviously perception, in the inverse arrow to its spatial reproduction, when *the plane of energetic space is projected back into the o-point of perception that gauges information. And this gives birth to an interesting field when we compare the two ‘different directions of time’, 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’*

*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 4^{th} 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:Where it is assumed that the determinant is different from zero.

It turns out that every affine transformation reduces to a motion, possibly a reflection, in a plane and then to a contraction or extension of space in three mutually perpendicular directions.

Quite a number of properties of figures are preserved under each of these transformations. In fact affine geometry is remarkably extensive, showing ultimately that *growth in size through lineal increase of and 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 5^{th} dimension.

In other words, the lineal affine growth of a system and affine geometry on the whole is justified by the ∆(Sp x Tƒ) = a K process of growth of the system, within the 10^{x} growth region , which is the region in which the metric of the system is lineal before its Lorentzian regions of emergence or dissolution in the ∆±1 scales.

Now modern mathematics obviously does NOT consider this 5D realist interpretation of the reasons of existence 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 3^{rd} age of information of any system – not a value but a loss of classic realism, the perfect age in which language and reality are in mirror correspondence to each other.

We shall not consider that metalinguistic approach, which completely ignores the outer world reflected by mathematics, in the obvious opposed realist philosophy of i-logic mathematics. As we hold truth Gödel’s proof of the incompleteness of any categorical definition and proof of existence based only in an internal metalanguage or internal logic.

*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*

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).

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.*

**III AGE: **

**SPACE AS MENTAL INFORMATION**

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*

*E*uclid affirmed that through a point external to a parallel only another parallel line could be traced, since the point didn’t have a volume that could be crossed by more lines:

*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 shouldobvious, 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.

**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§paœrganism, 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 1012.

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 BASICALLY 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).*

*creating in this manner the*‘force’ membrain of gravitation.

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…*

**XIX-XX C. **

**EXPLOSION OF MENTAL SPACES**

**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.*

**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.

POSITIVE VS. NEGATIVE OR NEUTRAL

An important first difference though from the ordinary coordinates, originated in *locomotion analysis, where we have inverse timespace directions,* consists in the fact that color intensities cannot be negative, as *we are using here pure formal space*. When x = y = z = 0, we obtain a perfectly black color corresponding to complete absence of light – a theme, which is essential to understand WHY imaginary numbers do exist for certain dimensional spaces but NOT from others, which we can resume in a simple statement, called ‘horror vacuum’:

*Negative values exist only in ternary cyclical ‘π’ time§pace zero sum worldcycles, as it is merely the inverse 4D (∆-1) vs. 5D (∆+1) arrows of form, self-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.*

CONTINUITY

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.

TERNARY EMERGENCE.

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.

So ALMOST ALL WHAT WE KNOW ABOUT REALITY COMES FROM MEMBRAINS, WHICH HIDE ITS INTERNAL REGIONS, EVEN IF MOST OF THE TIMESPACE OF REALITY COMES FROM THE VITAL ENERGY THE FRACTAL POINT ENCLOSES, AND ALL OF ITS VIRTUAL MAPPING INFORMATION COMES FROM THE MIND SINGULARITY.

Let us then introduce another huge field of modern mathematics – the study of the membrain, called intrinsic differential geometry of surface, where our rule of relative form according to size also applies:

*‘1D $mall measurements do NOT measure the whole world cycle of the being, so they are lineal. long-lasting measure bring the whole worldcyle or enclosed super organism so they are ‘constrained’ into a zero-sum or limiting membrane and appear as curved geometries’. *

**Application of to the ‘surface’ of a sphere, such as the earth gives us also the Intrinsic, differential geometry of surfaces:**

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. F*or example, a family of parallel straight lines does not have one.

**General ‘laws’ of emergence: o->O->•->@**

All this lead us to understand that ultimately as all departs from ∆•s≈t laws, geometry requires always a first ∆±i distinction between what ‘pros’ call, the geometry “in the small (parts)”, which is clearly dominant and “in the large (wholes)”. The main of those dual theories should then follow the obvious ∆st law that *wholes are more resistant, efficient and stable than parts; hence small/parts are easier to deform, while wholes are far more stable full T.œs – the ultimate reason why wholes and new scales keep happening.*

For example, in 1838 Minding showed that a sufficiently small segment of the surface of a sphere can be deformed, and this is a theorem “in the small.” At the same time, he expressed the conjecture that the entire sphere cannot be deformed. This theorem was proved by other mathematicians as late as 1899. Incidentally, it is easy to confirm by experiment that a sphere of flexible but inextensible material cannot be deformed. For example, a ping-pong ball holds its shape perfectly well although the material it is made from is quite flexible – laws those akin to the laws of ‘surface tension’ of soap bubbles with wide application in physics.

Another example, is the tin pail; it is rigid in the large, thanks to the presence of a curved flange, but separate pieces of it can easily be bent out of shape. As we see, there is an essential inversion between properties of surfaces “in the small”, ∆-1 and “in the large”, ∆+1.

A 1D t vs. 3D ð wider 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.*

Let us then move to 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.*

**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.

**MIND SPACE OF ELECTRONIC LIGHT **

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:

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, r^{3}/3, the human measure is the external surface, r^{2}/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:

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!: hc^{2/}e^{2}=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:

*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:

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.*

**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.

**THE ELEMENTS OF TOPOLOGY**

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.

**THE GENERATOR’S TERNARY SYMMETRIES AND ITS S=T 1, 2, 3 DIMENSIONAL ANALYSI**S

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.”*

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.

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. And the best way to generate by transposition a line that grows into a plane that grows into a cube (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§

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:*

= reproduction of motion through the $-length dimension, causing….**$**≈t≈$… Translation*∆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.**

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) – essential definition of ∆§topology:

*“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.

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.

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.*

Where the genus is a number less than the number of cuts NEEDED to break the form in its component parts, eliminating its adjacency.

**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 A*pollonius 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 i*ts content of ST vital energy is the same; but as the dodecahedron dies, it changes first its tight surface, which as we know in death processes grows in excess detaching itself from its volume – from 12 to 20 elements (another key number, with many vital interpretations, being indeed the number of amino acid variations on the protein surface of living solids, and so on)… and then reached its maximal form on the stable surface, it can only switch arrow of time.

As death is exactly that process: when all the vital energy is consumed and the skin is fractured ad maximal and no more motion can be extracted from the body, the system changes arrow and decomposes back to the past.

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 ∆º middle scale, to deduce both the upper properties of paths and the lower structure of its points.

Since the group of paths can be defined not only for surfaces but also for arbitrary sets of points, provided only that we can speak of paths in these sets and of their deformations.

**RECAP: ∆º PATHS and ST-TORUS.**

Now to resume all said, with the minimalist Rashomon effect (considering the ∆º-plane element and the Γst present form, which is the most synoptic form to define meaningfully a Toe), we can consider:

**∆º: ****Paths** as the ‘fundamental action’ element of the 3 scales of Topological transformations and as such the understanding of its laws in mathematical physics (actions, law of least time, etc.) are the knot and bolts of existence on topological T.œs.

And the same concept applies to the Torus *which even if it can be written in terms of its ternary generator:*

**Γst:** **TORUS:** $-long circle < ST: Combined path > ð-short cycle…

It is mostly the ST-function in 3 dimensions, equivalent for that reason to the flat plane in 2 dimensions, which can become easily a cylinder with a single cut, or a sphere with a single handle. *As such 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’.*

**Knots**

It is for that reason that the study of paths in its purest sense, knot theory, has become so relevant as *the most synoptic of all topological analysis to represent the entire Universe.*

A knot is a closed curve lying in the ordinary three-dimensional space. Let us then remove from space the points that belong to the given knot and consider the fundamental group of the remaining set of points.

As figure shows, its position can be very varied. Two knots are called equivalent if one of them can be deformed into the other by a continuous process without breaking the curve and without self-penetration.

This group is called the group of the knot. It is immediately obvious that if knots are equivalent, then their groups are isomorphic. Therefore, if the groups of knots are 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).

**TOPOLOGICAL STUDIES OF TIME MOTIONS**

We HAVE covered thus most of the themes of geometry in a very synoptic manner, enlightening them all with new insights born of GST, according to the purpose of this web, which is to show the organic, space-time nature of all toes and languages, unified by those principles and the capacity of GST to further new insights on all stiences.

It only rests to consider an example of the galilean paradox – which allows to use pure geometrical SS dimensions of form to study equivalent problems of TT-dimensions of time motions.

We already said that Paths in that sense in ∆st must be treated with the duality of ST dimensions, one of motion and one of form, complementing both the topological space-only view and the view of points moved *through curvature forces, proper of physical studies of topology. *

Since the mind fixes motion into form to ‘make sense’ of motions, order them and understand its general laws, something which topology does with its…

**S=T@. Topological methods: motions becoming forms…**

which allow to resolve complex motion s=t n-dimensional processes transforming them into topological forms – but this is an 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*.

In that sense many-dimensional manifolds, the so-called phase spaces of dynamical systems that take into account not only the configurations that the given mechanical system can have, but also the velocities with which its various constituent points move are treated by huminds with hidden topological methods, *which prove the Galilean paradox or equivalence between space-forms and time motion*. As we have shown in those posts often with the simplest example:

So goes with the other fundamental, ‘angular motion’: Suppose we have a point that can move freely on a circle with an arbitrary velocity. Every state of this system is determined by two data: the position of the point on the circle and the velocity at the given instant. The manifold of states (the phase space) of this mechanical system is, of course, an infinite cylinder (a product of a circle with a straight line).

And this indeed what reality does by *converting an Simple up and down, Harmonic motion into a rotary motion into a wave, giving us a fundamental ‘Generator Group’ of physical motions for all kind of systems:*

**Γst:** $: SHM (up and down vibration) < ST-Wave motion > ð§: circular motion

Thus, if a single ‘lineal inertial motion (above) is the fundamental motion of Nature, for ‘loose’ open ‘strings’ (topologically speaking), the previous generator resumes the 3 fundamental motions of Nature, for 2 Dimensional *systems (connected to two fractal points). *

And those are really the whys in ∆st of all the regular motions in Nature; since when we move to the more complex 3- body problem, regularity disappears in a Universe which is really about bidimensional poles, and its combinations=communications of space-time.

Of course the number of dimensions of the phase space increases as we increase the number of degrees of freedom of the given system. Many of the dynamical characteristics of a mechanical system can be expressed then in terms of the topological properties of its phase space, which *fixes its regularities somehow in a more all-encompassing view of the whole motion, *so in fact modern topology started with the famous Poincare’s use of such ‘comprehensive’ mappings of the whole world cycle of a motion to study and partially resolve the 3 body problem.

Yet we shall not go into further complications, as our blog is really about proving that the foundations of all laws of science depart from the Disomorphisms of ∆@S≈T.œs. So it follows that more complicated systems deduced in classic science from those primary forms and motions are also product of those Disomorphisms and need no further analysis.

Instead, we will introduce barely the ‘future of the future’ of geometry, which will happen once ∆st is truly established and used to teach from the scratch the absolute geometry of reality, departing then NOT from corrections of classic geometry as we have done above, but from the 5 postulates of i-logic geometry…

**PART II. ****∆@ S≈T GEOMETRY **

In this second part 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.

**ABSTRACT. 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)

**III. i-LOGIC GEOMETRY: FRACTAL POINTS**

**: WORLD’S PANGEOMETRIES and ITS ∞ MINDS**

‘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 5^{th} 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 5^{th} dimension’, in which the 4^{th} 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 5*^{th}*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.Œ does 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 5*^{th}* 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 5**^{th}** 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 5^{th} 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 5^{th} dimension), is a larger subject, still not fully developed by the only human world-point, which as Boylai on the view of non-E spaces, can only exclaim ‘I have discovered (not invented, as he said, the ever arrogant human ego) a new strange world – and not out of nothing as he said, but out of everything’).

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 5^{th} dimension, first explained by Lobachevski, as we see smaller beings with a hyperbolic geometry, which multiplies its ‘fractal forms’, and larger ones with an elliptic geometry which converges them into single, spherical ones. Hence the hyperbolic geometry of quantum planes, the elliptic geometry of gravitational galaxies, and the middle Euclidean geometry of light space-time, in which the Lobachevski’s constant of time and space is minimal, since our quanta of information H-Planck is minimal compared to our quanta of space-light speed.

__The 3 geometries of points of views in the symmetry of the 5__^{th}__ dimension.__

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 ≈ r^{2}/k^{2 }≈ h/c^{2 }≈ 0 for the human lineal Euclidean Universe, a expression that we will latter clarify in depth when studying the geometry of the 5^{th} 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 5^{th} 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 Eucidean world.

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*.

THE 5 NON-E POSTULATES OF VITAL GEOMETRY.

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 ‘—motions’ 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 ‘—motions’ 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 ‘—motions’ 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 ‘—motions’ and information) to biology (knots=networks that absorb ‘—motions’, 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.

2) A line is now a wave of communication of energy and information between points.

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 2^{nd} 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.

3rd postulate: the logic of communication

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 3^{rd} 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).

– *3rd Postulate:* Equality is no longer only external, shown in the spatial perimeter of any geometrical form (3rdEuclidean Postulate) 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:

Sp xTƒ= SpxTƒ -> 2Sp x Tƒ.

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 3rd 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 3^{rd} 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).

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 2^{nd} focus of the sun?

If we add the parabola and other exponentials, which are the essential ∆=sxT^{n, }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 5^{th} 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).

** ** **RECAP**

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:

- A first young, Greek age of static bidimensional space-geometry
- 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.
- 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).

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.**

**I GREEK AGE**

**HOLOGRAPHIC BIDIMENSIONAL STILL GEOMETRY**

**I.NTRODUCTORY SAMPLE: PYTHAGORAS REVISITED**

So let us start enlightening 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 **M**agnitude 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 pas 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:

The tetrarkys. ∆-1: THE 10 DIMENSIONAL INNER PERSPECTIVE.

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.*

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 ‘Spe’ Subsystems lymphatic, digestive (not portrayed the urinary/excretor system).

**Now connecting the graphs with elliptic geometry and the bilateral nature of the Universe and its expansion of space, ***IT MUST BE NOTICED THAT REGULAR FORM DISTANCES MATTER NOT; the convoluted bilateral networks that connect the singularity brain with all its antipodal points 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.*

**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:

The 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.

**1ST & 5 ^{TH} POSTULATES. FRACTAL POINTS.**

**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.

**THE SECOND POSTULATE OF I-GEOMETRY. WAVES OF COMMUNICATION**

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.*

**The 3rd and 4th postulate: Social networks**

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.

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.

**Fractal points: the fundamental particle**

T.Œ does show the scaffolding of mathematics by defining those terms as the Postulates of i-logic geometry, based in the structure of cyclical time, fractal space and the 5^{th} 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.

**Dimensions.**

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 bidimentional, 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 5^{th} 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 5^{th} dimension), is a larger subject, still not fully developed by the only human world-point, which as Boylai on the view of non-E spaces, can only exclaim ‘I have discovered (not invented, as he said, the ever arrogant human ego) a new strange world – and not out of nothing as he said, but out of everything’).

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 5^{th} 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 Eucidean world.

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 ‘—motions’ 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 ‘—motions’ 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*

*E*uclid affirmed that through a point external to a parallel only another parallel line could be traced, since the point didn’t have a volume that could be crossed by more lines:

*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 ‘—motions’ 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.

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 ‘—motions’ 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 ‘—motions’ and information) to biology (knots=networks that absorb ‘—motions’, 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.

2) A line is now a wave of communication of energy and information between points.

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 2^{nd} 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.

3rd postulate: the logic of communication

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 3^{rd} 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).

– *3rd Postulate:* Equality is no longer only external, shown in the spatial perimeter of any geometrical form (3rdEuclidean Postulate) 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:

Sp xTƒ= SpxTƒ -> 2Sp x Tƒ.

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 3rd Postulate is caused by the topological forms created by any event that entangles Multiple Spaces-Times. Since it describes the paths and forms of dual systems, which connect points: Self-similarity implies parallel motions in herds; since equal entities will maintain a parallel distance to allow informative communication *without interfering with the reproductive body of each point*.

Darwinian behavior implies perpendicular confrontations, *to penetrate and absorb the energy of the other point*. Finally, *absolute, inner and outer self-similarity* brings boson states, which happen more often to simpler species like quarks and particles that can form a boson condensate as they do in black holes, *where the proximity of the points is maximized. And indeed, the same phenomenon between cells with the same inner information /DNA originates the ‘collapse’ of waves into tighter organisms.*

The 3^{rd} 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).

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 2^{nd} focus of the sun?

If we add the parabola and other exponentials, which are the essential ∆=sxT^{n, }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 5^{th} 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).

__The scales and ST¡ences of reality.__

As the widest one is the concept of a topological network-plane, of the 5^{th} 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.)

**∆=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).

**∆+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.

**Unification of all sciences and 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:

**I AGE: CLASSIC TOPOLOGY. ****CONSTRUCTION OF ORGANIC, FRACTAL NETWORKS**

When we start in a more professional way to understand the 3 topological forms of the Universe, we immediately confront the fact that a topological plane is made of points, joined by lines, and so enter into a more real description of the scalar universe as forms which are networks of points joined by flows of energy and information. The concept of an organism arouses immediately as an organism is a system that co-exists at least in two levels or scales of size, joined by networks=flows of energy and information.

In the graph, the 3 canonical forms of space-time, the sphere, the toroid and the fractal plane, which in close analysis are always networks of points. Indeed, topology at professional level however is not a continuous geometry but a sum of points that put together at a distance seem to be not a network but a continuous form. Hence the existence of scales in the Universe, in which each point of a topological form is in itself a world in a lower scale. Since the 3rd leg besides space-time symmetries of the GST philosophy of science is the fractal, scalar structure of the Universe, and how those scales co-exist and create organic systems.

We can then 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.

**Triangulation.**

*GST relationship: One of the fundamental discoveries of GST is the ternary structure of all what exists as a whole. This is shown everywhere, in geometry from the recent theory of causal triangulation that shows how to construct a space-time Universe with only 3 ‘points’ and a causal time algorithm between them, to the earlier topological discovery of this section: most topological laws can be reduced to the study of its triangular elements in the ∆-1 scale of the whole form.*

The most important case is when all the faces are triangles and then we have a so-called triangulation (a division of our surface into triangles, rectilinear or curvilinear). It is easy to reduce the general case of arbitrary polygonal faces to this case: It is sufficient to divide these faces into triangles (for example by drawing diagonals from an arbitrary vertex of the given face). Thus, we can restrict our attention to the case of a triangulation. The combinatorial method in the topology of surfaces consists in replacing the study of such a surface by the study of one of its triangulations, and of course we are only interested in properties of the triangulation that are independent of the accidental choice of 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:*

In the graph a balanced simplex system is composed of a tubular digestive Spe 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.*

(to be cont’ed)

**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:

*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.

IN THOSE PERIODS, *THE BEING EXISTS IN A SMOOTH MANNER, AS NOTHING TEARS. *

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.

**VITAL DIMENSIONS**

** **** Generator Equation of SpaceTime fields.**** Vital dimensions: forward bodies & top minds**

*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 *S**E*<=>*I *the generator, feed-back equation of Time-space: Those *S**E*<=>∏*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 3^{rd} 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.

R*ecap*: 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* E**ó**I, 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.*

** The antisymmetry of temporal information: Death**

A consequence of the 3 ages of time is its antisymmetry as opposed to the bilateral, mirror symmetry of spatial forces: Time is antisymmetric as the old age is the inverse of the young age with the parameters of energy and information inverted. Youth has maximal energy and minimal information while the old age has minimal energy and maximal information. And yet many attitudes and processes might seem self-similar, if the researcher doesn’t know how to differentiate informative and energetic parameters, since those parameters might be quantitatively the same, only that inverted in value. Space is symmetric as left and right reproductions are mirrors of each other with the same parameters of information and energy. This explains why spatial reproduction happens only in the E=I, balanced classic age of the system, when both parameters are identical and a self-similar reproduction of a bilateral, spatial being is possible.

The understanding of the antisymmetry of time vs. the symmetry of spatial processes has wide applications in the solution of questions pending in all fields of research, from the understanding of the temporal, weak, informative force that has no bilateral symmetry to the ‘chiral’ processes of evolution of life forms vs. the symmetric shapes of self-similar left-right, spatial bodies to the meaning of ‘antisymmetric’ gender and the 2 ‘antisymmetric’ sides of the informative brain.

But the most important of all antisymmetries is that of death, which could be considered in the jargon of physics, a ‘local antisymmetry’ of time. Indeed, if life is the arrow of information towards the future e->I, death is the inverted arrow, i->E, that erases form back into energy; but this time travel to the past is always local, affecting only the limited world of the species, so antiparticles are the local antisymmetry of time that kills particles, death are the local antisymmetry that kills life and the big-bang the local antisymmetry that killed a previous Universe. And both directions are different, since ‘death’ lasts min. time and causes a maximal space expansion and life lasts max. time and expands minimally in space. Both balance reality in an eternal present of past to future to past existential fluctuations:

*Life-future: Max. I x Min. E x Death-past: Max. E x Min. I= Immortal present: E=I*

The understanding of that local antisymmetry is the key to resolve problems of physics such as the weak force, why we see less antiparticles or dead people than particles and life people (much longer in time), or the non-evaporation of black holes (as antiparticles are the same than the particle, albeit when moving to the past they seem to co-exist), and so on.

But its implications in biology and philosophy are much larger: we are ultimately ‘back and forth’ vibrations of space-time, virtual existences, dust of space-time that always revert into a zero-sum, which makes the Universe and its game eternal.

*Recap*. Old age has inverted exi parameters to the young age.

** Graphic representations of the 3 rhythms of existence**.

*In the graph, a classic, Taoist representation of the 3 ages of life and its inverse parameters of youth (max. energy) and old age (max Information) represented by the triads of the I Ching, and a modern graph of duality showing those parameters as a semi-cycle, which in certain simple beings like light are in fact both the ages of time of a physical wave and its form in space, as light quanta, h=exi, is indeed both our basic cycle of time and surface of energetic space of which all are made.*

Humans perceive simultaneously the 3 ages of some species too small and fast to differenciate them. As particles in bubble chamber or cars in the night, we see the long time of the species – an entire spiral vortex with the 3 ages of a space-time being. In the next graph, such long world-line, or life-graph is in fact a spiral in terms of spatial population and cyclical speed: The organism looses mass and accelerates inwards its form as it becomes slimmer in space. The first age of the spiral is its energetic mouth that brings in a galaxy the interstellar dust, which will give birth to stars, in the middle, reproductive age and will collapse in the center, the informative black hole:

But any form can be represented in a spiral graph of space-time. For example, the evolution of the FMI complex with its 800-80-8 cycles of increasing informative evolution is an spiral of 3 horizons in a decametric scale: 800 years civilizations, 80 years nations and 8 years ‘decades’, in which the product is evolving faster and now becomes integrated by networks. This 3-horizon evolving ternary network, central to social sciences can be also represented in a spiral of accelerated birth and death of human civilizations destroyed by wars. Any mass or charge is in fact a natural spiral, through which photons and gluons become denser till collapsing in the bigger particle, as interstellar gas does in a galaxy. All those spirals can be seen as equivalent to the world-lines of lineal physics, now world-cycles of a living organism both in time and space, as a whole or as a wave of evolving particles.

The equivalent graph in lineal time will give us a bell curve – the lineal version of a cyclical life-line: The speed of time/information changes during the ages of life, shaping a bell curve, such as the st-1 age is the fastest age of informative evolution, which diminishes in the young age to a halt, when the speed of cellular reproduction and energetic growth becomes maximal, both reach a steady state in the mature age, of eóI, rhythmic back and forth beats; and then in the 3^{rd} age energy decelerates first and then information decelerates and collapses till death provokes a maximal explosion of information into energy. Those phases of different speeds of energy and information can be generalized to any system and event that will always go through 3 ages: an accelerated seminal and young age of increase of form and energy, a steady state of constant speed and a decelerated age till the system comes to a halt and the event dies away.

One of the key quantitative elements that connect the 4^{th} and 3^{rd} paradigm of metric measure is the fact that not only each species has a different rhythm of change and clock of time, but the same species changes its time rhythms and speeds through its 3 ages. This feature is fundamental to understand all processes that involve evolution, from the different masses of ‘gravitational vortices’ of information, ruled by the equation of a vortex (V_{t}xR_{s}=k in simplified notation; thus the closer to the center, the faster we rotate, the faster the frequency of information of the vortex and the shorter the space we need to close a frequency cycle), to the evolutionary process of information in this planet, which can be mapped out and its frequency shown to follow a logarithmic process of acceleration, which now reaches its zenith with the creation of informative machines. In that regard we can consider the following ‘speeds of time’ of any event or self-similar ternary topology of energetic youth/membrane, reproductive maturity and informative, 3^{rd} age (hyperbolic center of a system):

*-st-1: Seminal age:* Informative evolution is maximal, since the system occupies minimal space (Black hole paradox: Min. Se=Max.Ti). It is the palingenetic age.

*– Energetic Youth:* Evolution of information decelerates after the ‘landing’ of the species in the higher St-plane. Now acceleration of energy takes place.

*-Reproductive, mature, steady state:* Both energy and information maintain a constant speed in EóI feedback cycles, which will last depending on the access of the event to new energy and directional information.

*– 3 ^{rd} age:* Now both energy first and then information follow a decelerated process of collapse; first energy becomes exhausted and then…

– *Death:* Information collapses, exploding into its st-1 scale of cellular energy in a relative zero time (the minimal unit of time of the st-scale). Death is the fastest motion of the Universe.

*Recap*. In physical terms the 3 ages of biological change can be considered the positive, accelerated youth, the reproductive, present age of constant speed, in which humans repeat themselves and the 3^{rd} negative, decelerated age of temporal age, or 3^{rd} age, which ends in death, when the balance of the Universe is restored.

** Universal Constants are ratios of energy and form.**

The previous analysis of the generator equation of the Universe brings an essential element of all formalisms of physics, the meaning of Universal constants and the key complex parameters of science, measures that combine energy and motions such as speed or momentum, which are conserved as complex ‘time arrows’ and show certain constants in each specific species, which are the vital and universal constants of biological and physical, complementary beings.

We thus define Universal constants as ratios between the energy and information systems of a certain ‘space-time membrane’ (the bigger organic structures of the universe, the light membrane and the gravitational membrane, origin of most physical constants), or complementary being (the constants of mass-information, momentum, the social constants that measure the number of particles and field quanta put together to create a physical entity; the vital constants that are ratios and proportions between the energetic reproductive and informative systems of living beings, etc.):

*‘Existential constants are ratios between energy and information parameters of any system. Universal constants measure those ratios in the more extended systems, the membranes of the Universe, its quanta and fundamental particles. Vital constants measure those ratios for specific biological beings.’*

Let us consider the main of those constants:

*Simple, social numerical constants.*

-Social constants of space, which define the number of cells that form an efficient spatial structure. They tend to follow pairs, as 2 is the natural symmetry of left-right of bidimensional space.

– Social constants of time, which define the number of events needed for a transformation or completion of a time cycle; the most common of which is the number 3 of past-energy horizon or youth, present, reproductive state of the event and future, informative state.

– space-time constants, which combine both spatial symmetries and temporal events; Pi belongs to this concept as it is a self-similar number to 3, with ‘small’ apertures between the 3 cyclic elements or phases of the cycle, which taken as a single picture of space-time form the spatial shape of the cycle when perceived in space.

Yet the most important of those numerical constants is the number 10 or tetrakys, the perfect social number that creates a unit of the higher scale with 3 x 3 energetic, reproductive and informative elements, and a 10^{th} central point that ‘emerges’ to become a unit of the next scale.

Among those decametric scales, made famous by Eames in his film ‘potencies of 10’, it is natural to consider the ‘scale of all perfect scales’ S^{t } to be (9-11)^{9-11} systems; that is groups of 9 to 11 spatial units organized in 3×3=9 to 11 scales of growing social complexity. And indeed we find that number to be extremely common to the point that we could consider it the ultimate number of scales of the complex universe between the fractal entities humans perceive (atoms and galaxies, unified in fractal, complex physics as self-similar forms of the quantum, electronic and gravitational, cosmic membrane): We find it as the difference of force between those 2 membranes (electromagnetism is 10^{4×10} times stronger than gravitation). Then 10^{11 }atomic ties make a DNA molecule; 10^{11 }neurons make a brain; 10^{11 }humans have lived in this planet; 10^{11} stars form a galaxy and 10^{11} galaxies make up the cellular Universe.

In that regard, Einstein said that the ultimate Nature of Universal Constants could not be ‘physical values’ but special numbers, which would be ‘relationships’ between substances that constantly appear in Physical equations, ‘like pi and e’. And he was right.

*Existential parameters: reproductive speed and momentum*

On the other hand, the simplest algebraic operations between energy and information parameters become the essential ‘complex parameters of the Universe and all its species’:

*Speed = V=s/t=spatial energy/temporal information.*

Speed is in complex science no longer a mere measure of the translation of a form but in detail, all forms (paradox of Zenon) move not, but reproduce its form, step by step, as a wave of light does. The wave affects a lower scale or network of relative energy, making it to adopt its formal configuration. So form becomes reproduce, imprinted along a previous scale of reality and motion becomes reproduction. Light reproduces over the simple energy of gravitational space and the wave displaces. A wave in the water does the same imprinting one after another the form of the network of atoms and so on. So it follows that the systems which are simpler with less information move faster because they have to imprint less form. And so V=s/o information=infinite, which is the perceived speed of gravitational space whose information we do not perceive, while the more complex life structures move slow as they have to keep its huge amount of stable informative shapes unaltered. And each 3 months we change all our atoms to reshape our form.

So the new parameter should be called ’reproductive speed’ as we consider that motion is a manner of reproduction of form.

*ExI= Momentum or existential force.*

This is the second key ratio of reality which multiplies the energetic and informative strength of the system. In physics is equivalent to momentum, mv where M(i) is a measure of cyclical form, and V(e) a measure of motion, the definition of energy in this work. In biology measures the top predator power of the system which will dominate its ecosystem when it has the strongest energy body and the more intelligent informative brain. And each species will have a certain existential force, reproductive speed and energetic, ∑ and informative ∏ social parameters, which will be its 4 ‘main numbers, needed to define the form (from the quantum numbers of physics to the vital constants of living beings – metabolic constant, brain/body ratio, brain volume, physical strength, etc.)

*Irrational constants.*

Because arrows of time are back and forth transformations of energy into information or its combination to reproduce together a self-similar form, and certain numbers define certain ‘efficient’ social geometries, there are Universal constants, which define for each species of the Universe a given ratio of transformation of energy into form or a reproductive ratio that combines two simpler energetic and informative bits and bites to reproduce a new form (h constant, c-constant, etc.)

The dynamic relationships between the 2 motions of the Universe, energy and information are invariant in form, motion and scale; therefore fluctuating around fixed equilibrium values, which is the ultimate meaning of Universal Constants.

Thus, all of them can be reduced to the generator equation of the Universe, the feed-back cycle of energy and information, from where all laws of reality can be derived:

*Energy <=> Information; ExTi= Irrational Constant*

A Universal constant cannot be a perfect number, because it will create a fixed Universe; thus Universal constants are irrational numbers, which show a minimal fluctuation. Consider for example the main constant of the Universe, pi. If pi were exact then the spiral made with 3’14 lines would not be a vortex but a perfect, static cycle. Yet if pi is either +pi or –pi, the cycle will not close by defect or close in excess. What this means is that the cycle will be a bit more curved inward, and so it will be an informative cycle; or it will be cured outward by defect and so it will be an expansive, energetic spiral.

We know, for example, that the orbits of planets are decreasing by a few centimeters a year, so they will finally fall into the sun. They are, if we consider a dynamic, temporal view of them, inward, informative spirals. Yet an antiparticle, which is exploding information into energy, ‘dying’ in a big-bang that annihilates it, is bending outward.

So irrational numbers are the absolute constants of the basic exchanges and transformations of energy and information of the Universe. The main ones are:

– Pi, the formal constant of creation of in/form/ation. Since pi transforms 3 lines of energy into a ternary cycle with one more dimension of form: a string of 3 lines with 3 dark apertures for a total 0.14. Within those 3 lines there is a 2^{nd} dimension of height, or information and a volume of space. The entity has grown

– Phi, the Golden Ratio, which is the constant of reproduction that multiplies an organic system into self-similar forms.

– e, which is the constant of extinction of form back into a lower scale of energy that devolves a formal being into its cellular subspecies. Its most common ternary form is et=3=20.

We find those constants, both in physical and biological processes related to those transformations of energy and form – showing the fundamental equality of all Universal Systems.

For example, *e* appears in the decay of radioactive atoms that release energy; phi appears in the organization of a sunflower spiral; pi appears in the h-constant of transformation of light flows into electronic actions.

How many Universal constants there are for any system? We advanced 3 basic U.C. at the beginning, pi, the ratio of creation of information, phi, a reproductive ratio and e, an extinctive ratio of destruction of information into energy. And indeed, all systems have at least those 3 basic constants.

*Vital Constants**: proportions between brains and bodies.*

A final type of constants expresses quantitative proportions between the reproductive body elements and informative particles/heads of a complementary system. As a general rule the commonest proportions of energy/information are:

– The particle/head of information is dominant in information parameters, (dimensions, mass-weight or number of network connections of the informative system, cellular density, etc.), in a 3 to 1 proportion with the body/field of energy, both in time (so in genetics, the dominant, informative element has a 75% chances and the recessive element, 25% chances); and in space (so the Universe has 76% of dark mass).

– In terms of spatial, energetic parameters however the body is dominant (spatial dimensions, cellular numbers, energy volume etc.) usually in a proportion of 90-80% to 10-20%. This is due to the fact that the informative element tends to be the central 1-element of a body tetrarkys, so we have a captain every 10 sergeants and a sergeant every 10 soldiers; 9 glial cells that give energy to every neuron, a 10% of taxes that go to the Middle age Priest, which directs the herd of believers, and so on. Yet another simpler, very common dual structure is a spatial square with a central knot, which gives us a 20% of informative elements (the center) and an 80% of energetic elements (the vertices of the square).* *

*Recap*. The generator equation of space-time and its 4 main arrows of time, understood as symmetric transformations of energy into information or reproductive combinations of both, coupled with the invariance of topological form, scale and motion of the universe explains for the first time the meaning of universal constants.

**Inverted constants: The chip/black hole/mouse paradox**

The inverted properties of energy and form, shown in the law of Range, apply also to any complementary system of the Universe. So smaller animals have faster metabolic rates because its energy /form cycles are faster. Their ‘clocks of time’, we could say move faster. Further on, in complexity this implies that paradoxically the smaller beings have more information (chip/black hole paradox). So the smaller the chip is the faster it calculates. This paradox is essential to understand the dangers of black holes. Precisely because they are so small they will reproduce faster and accrete faster, in the same manner a smallish virus reproduces much faster and it is more dangerous for an organism than a bigger bacteria.

If we adopt according to Galileo’s paradox a static point of view, universal constants are NOT only algebraic values, but invariant geometries that repeat in all scales of reality. And this is the ultimate meaning of General Relativity, since Einstein made a precise, simultaneous, present measure of the ‘static form’ of those vortices of mass obtaining G as a measure of the relative curvature of the gravitational force in each point of the vortex.

The central concept of a Fractal, scalar relativistic Universe is obvious: the same invariant game, the same forms, the same motions, happen in all the scales of reality. And so the Universe is relative and invariant in its energetic motions (original Theory of Relativity), in its forms (cyclical forms of information and lineal energy) that repeat in all scales, which therefore are also invariant.

We have seen now how that invariance is played as a ‘ratio’, ExTi=K, which allows smaller beings to live shorter but live faster. As we have seen the properties of energy and information are inverted. So the smaller we become the faster we rotate, the faster we live, the faster we beat. For example, we know that a fly sees 10 times faster than a human being, reason why we cannot catch it. Yet the ant who lives longer lives 7 years × 10 times faster=70 years of inner, subjective existence.

The same concept applies to a physical vortex of information, V(t) × R(s)=K than to a living being that processes energy into form (a mouse beats its heart faster than a human; a cell divides and reproduces faster than a mouse, every 24 hours, etc.).

The entire cosmos and all its scales are related by that simple paradox: the smaller we become the more information we process. It is the Moore Law: the smaller the chip the faster it thinks. The reason is obvious: smaller, faster systems, close ‘logic cycles’ of information faster. In complex beings it means faster thoughts in smaller neurons, packed in tighter groups. In the physical world, the bigger rotational motions of cosmic masses are slower than the cyclical rotation of particles, but their product remains constant. And we can write this fundamental law of the Universe, with multiple self-similar applications in any entity made of fractal space-time, again as a general case of the generator equation:

*Universal ExTi = Universal Space Extension × Time-frequency = Constant Entity = K*

An expression, which appears in all scales of reality (Heisenberg Principle, Vital Constants, etc.)

*Recap*. The Universe is just and harmonic: small beings are more intelligent, faster than big ones. It is the paradox of David and Goliath; the paradox of the chip, the paradox of the black hole…

** How to include metric spaces. Topology: informative, energetic and reproductive systems.**

However, 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 3^{rd} 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 3^{rd} type of topology – reproductive topologies that combine the other 2 arrows – become the 3^{rd} 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 4^{th} 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.