**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**

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

**Fractal, Non-Euclidean Points of view. Upgrading the foundation of space-mathematics.**

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. 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 is the most important law of the 4th paradigm, foreseen by Leibniz in his Monadology, 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 isomorphisms of the previous paradigm of a single metric space-time continuum.

Further on those points must be described always in 4 dimensions, with motion. This should have been obvious, but abstract mathematics 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 that 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.

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 cycles5.*

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, Riemannian 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.

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.**

Yet this 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 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.

**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.

**(to be cont’ed)**

**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. *

Now, my apologies, i shall try not to be so pedantic… but it is unavoidably in this section, as we have been quite simple in the first line, and we must connect GST which is scientific jargon (even if ideally all of them would abandon

**to be cont’ed**

So what we learn , a system that goes through its 3 cycles of life, ma

**to be cont’ed**

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

** (to be cont’ed)**

(to be cont’ed)