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4¬Æ similarity: †entropy v ≈love

±∞ ¬∆@ST:

‘Each point is a world in itself’ Leibniz, 1st and 5th postulate of Non-E Geometry

‘Love each other as I have loved you’. 4th Postulate of Non-E Geometry among parallel beings



CONGRUENCE: The 4th ¡logic postulate of similarity

I. Congruence in Mathematics: Angle of perpendicularity

II. Congruence in physics: Fields vs. Particles

III. Congruence in Biology: Predator vs. Preys, social evolution

IV. Congruence in social sciences: Man vs. Machines, Love.


Abstract. The fourth postulate is the most important of Non-Euclidean, ¡logic geometry, as it defines the behavior of T.œs, organic points in its relationships to other points,  according to its geometric connections, establishing the fundamental law of duality: parallels, are similar beings which maintain a constant flow of communication forming herds and social organisms; different beings enter in darwinian relationships in perpendicular events in which they destroy each other.

We thus take the title of the post from Lobachevski’s concept of an angle of parallelism, which varies and in vital ¡-logic mathematics establish the type of vital functions the systems perform.

A brief introduction to General Systems, the metric of the 5th dimension and the main nested superorganisms of the Universe.

New scientific ideas never spring from a communal body, however organized, but rather from the head of an individually inspired researcher who struggles with his problems in lonely thought and unites all his thought on one single point which is his whole world for the moment.

Planck on the informative seed or mind that will reproduce and evolve into a larger superorganism or mental world.

As usual if the readers want to understand anything, they should read  at least the first pages of the central post.

A very brief introduction follows:

The Universe is a fractal scalar, organic system of 5 Dimensions of space-time. The fifth dimension is made of the ‘different co-existing scales’, which from the simplest forces through particles, atoms, molecules, matter, organisms, superorganisms, planetary systems and galaxies, create an ‘organic network sstructure’, which amazing enough since it was discovered at the beginning of science with telescope and microscopes, was not formalized till I introduce its metric equation in the milieu of systems sciences.

In mathematical science for a dimension of space-time to exist, it requires a metric equation which combine space and time to gives us a co-invariant system that allows to travel through such dimension. The fifth dimension has a ‘metric equation’, hence it exists. The equations for a given number of scales co-existing in an organic network is S (size in space) x T (speed of time cycles) = Constant.

Stiences study the Disomorphic laws of species, each one focused in a scale of the eusocial 5th Dimension, with different jargons, which we shall unify as all what exist is homologic, made of the same Dimotions, which force them to exist as part of super organisms tracing worldcycles, with a ternary structure in space, time and scales, guided by a still, linguistic mind, which projects its smaller mirror images of form in its territorial body waves, creating its fractal scalar structure, broken though by the entropic limits of its perceptive reach. Above the main scales of the nested superorganisms of the human island-universe: the galatom, where nested thermodynamic planets exist, where nested human social organisms self-centered in visual, verbal, mathematic and logic minds create its order with social and mechanical memes, carried by individuals where nested cellular systems exist, coded by genes, made of even smaller atoms and particles coded by quantum numbers. Yet all those scales are unified by the co-invariant metric equations of each of its synchronous time clocks.

It means when we become smaller within such organic network, our time clocks accelerate. And vice versa. For example, in your organism the equation relates the cellular genetic scale, whose time cycles run much faster than your biological cycles. As information is stored in the form and frequency of time clocks, it follows that in all nature’s organisms, smaller systems code the information of larger systems.

Each ‘stience’ therefore specializes in the study of a nested superorganism, or scale of the fifth dimension (∆±¡), We study 3 fundamental superorganisms of the 5th dimension. The ‘galatom‘, whose metric equation is H(Planck constant of angular momentum) x C (speed of light) = C, whereas h is an obvious ‘cyclical time clock’ and C an obvious measure of speed-distances in space. And indeed, smaller scales have faster turning particles, and larger systems move in slower cycles. (Vortex law: Vo x Ro = k).

In biology we study families of animals such as mammals where larger organisms have slower metabolic cycles. In history we study social organisms, whose cycles of life and death, will define the evolution of nations and civilizations. And in each of those organisms, smaller systems code larger ones. So the quantum numbers of particles code matter, genes code biologic organisms, and memes code societies.

We define them two fundamental systems one in space and one in time, taking place in the fifth dimension:

  • An entangled superorganism in space, in which the synchronicity between 3 scales of the organism, its atomic/cellular/Individual, thermodynamic/organic /social and gravitational/ecosystemic/global scales in physical/biological/social organisms form a complex interacting, entangled system we shall call a superorganism, whose study discovers ‘Isomorphic=equal laws’ regardless of what kind of system we study. Because the system is entangled, and made of 5 obvious components, we need a new logic, ‘pentalogic’ as all the elements communicate and interact together. Those elements common to all systems of nature are identified as:

space‘, ” §paœ that is ‘types of dimensional form’, and we find to have 3 only ‘topological varieties’, which perform 3 clear organic functions in the superorganism, lineal limbs/fields that move the system, hyperbolic body-waves that iterate its forms and spherical particle/heads that gauge information/

Time, that is types of ‘motion=change’, those organs perform, which are locomotion (limbs/fields), reproduction (body-waves) and informative gauging (particle/heads), which dominate each of the 3 consecutive ages of the being (young motion age, mature reproductive age and old informative age).

Scale, the 3 aforementioned  scales for any superorganism

Mind. And this allow us to identify a ‘center that process’ information from the outer world, and controls the internal networks of the system, with a given language, which we shall call a mind, regardless of its automatic processing or consciousness. As it coordinates and maintains the whole as a single system performing actions of control of the energy and information of the system. So we find gravity centers, chips, brains, DNA nucleus, black holes in galaxies, etc. etc. Indeed, there is always in any stable organic system, a relative still, self-centered mind, which we shall discover all systems do have, even physical systems, which have centers of gravity, centers of charge or crystal systems, which maintain the whole.

Entropic limitsIn time called death, in space, called membranes, in scales as systems loose control beyond to scales and in fact death is defined by a simple equation of dissolution of two scales of information, ∆1<<∆-1.

All this allow us to define any system in time, by its life and death cycle which we shall call a worldcycle (taken from 4D physics, worldline, which now with a new dimension becomes a worldcycle. Since indeed any scalar superorganism’s existence can easily be defined as a travel through the 3 scales of the fifth dimension, as all systems are born as a seminal, smaller form, in the ∆-1 lower scale, grow fast with its faster time speed, emerge in the ∆º body scale and are part with its head/particle of a larger social world, in the ∆+1 scale.

How then a system travels through those scales? Very simple, it does so accelerating in its life cycle, emerging, living in the larger world, and then dissolving back to its parts in the death moment.

And amazingly enough with this simple scheme of 5 pentalogic elements, tracing 3 relative worldcycles, the first  ‘placental worldcycle’ as it evolve in the relative safe ‘womb’ of a maternal system, the life cycle, as it tries to survive in an outer ecosystem, and for the most organized systems, a 3rd worldcycle as it forms part of a larger superorganism, we will be able to order everything in the Universe, every event and every phenomena, which the researcher has studied for decades.

The existence of a new dimension of space-time is the most important discovery of science in a century, since Einstein formalized the fourth dimension, but this expansion and formal organization of science is much wider as it applies to all sciences, not only physics. 

So the blog is extremely extensive in its scope and due to the fact it is the work of a single researcher, by force incomplete. It is the ‘first seed’ of a new paradigm of philosophy of science.

The blog in any case divide in 4 lines, which try to develop the model for the main superorganisms of the Universe and its parts, the galatom (physical superorganism that stretches from a planckton – a Planck constant – to a galaxy) and within this ‘nested organism’, the Earth, a superorganism which has both geological and biological nested superorganisms in its surface, of which we study according to the arrow of increasing evolution of information 3 ages, the life age of Gaia, the human age of History and the machine age of the FMMI system of company mothers that reproduce and evolve machines of information (Financial media system) and energy (Military-Industrial system) or ‘Metal-earth.

The study follows a simple scheme. The first line is a sentence that explains in space the 5 components of all superorganisms, and how they are studied in each scale by a different ‘stience’ (space-time science), in its spatial form and its reproductive and evolutionary worldcycles.

The Ðisomorphic method studies in a vital, sequential time manner the 3 worldcycles of all Space-time supœrganisms (T.œs):                                        
þ: 0-1: its palingenetic o-1 social evolution in the accelerated time sphere of existence, till becoming 1 (0-1 bounded unit circle in ¡logic mathematics; quantum probability sphere of particles in physical systems; palingenetic fetal age in biologic systems; 0-9 memetic learning childhood in social systems). It is the highly ordered world cycle as the ‘placental mother-energy world’ is nurturing, and memorial cyclical spacetime has erased errors of previous generations. We thus use for the 1st world cycle, the informative Dual Isomorphisms, of the mind-seed (@ MENTAL Сisomorphisms), as 0-points are generated (0th isomorphism,•) to follow the GST, Fractal Generator equation or program of existence of the Universe (1st Isomorphism,Γ), which will allow the being, in the energy-rich placenta to perform asymmetric space-time actions (2nd and 5th, DIMOTIONAL Сsomorphisms, ß,å) of energy feeding, reproduction and social evolution, to imprint the information of the seed as the system growths in Space (SPATIAL Сsomorphisms) and develops its 3 physiological networks (3rd isomorphism,ψ), which finally merge, into a whole one that emerges (4th isomorphism) into:                  
– c: The outer 1-∞ world, in which it will deploy its 2nd world cycle of existence in an environment which is open, entropic (1-∞ hyperbolic unbounded Cartesian plane in ¡logic mathematics; thermodynamic entropic statistical molecular populations in physics; Darwinian struggle between populations in biology; idol-ogic dog-eat-dog capitalist, nationalist competitive eco(nomic)systems in the super organisms of history. In this 1-∞ existence the world cycle is not ensured to continue, as it can be cut off by the entropy of the world system.
We thus study the second world cycle with 3 sets of Сisomorphisms, slightly different to those of the o-1 sphere; as the individual now will have to exercise in time guided the mind (no longer a passive seed), its survival will choosing its actions of existence, (5th isomorphism, common to both worldcycles, albeit now defined by the active mind), that will determine in a cumulative sum, its 2 TIME Isomorphisms: the duration of its ages (7th Сisomorphism: Ω), further subdivided in space-time quanta (9th Ðisomorphism: µ), as it exists through its 2 EXISTENTIAL ¬ D¡somorphisms, as a member of a species, (6th D¡isomorphism, œ) itself a superorganism, albeit with a higher entropic, disordered structure, living in an entropic world or ecosystem with other species (8th Disomorphism), trying to DEFY THE ENTROPIC LIMITS OF TIME duration (7th Di) and spatial territory (8th) imposed by a higher social order and control its world, through:      
ω: The 3 final SCALAR D¡isomorphisms (∆) taking place outside the being, existential life cycle, as part of a larger world of hierarchical social scales (§), as it performs those actions through ∆±4 Planes self-centered in its mind, beyond which it cannot longer perceive, to become if successful a new superorganism of the infinite planes of God, the game of existence, (11 Disomorphism).                                                                                                        
 Thus to ‘travel through the 5th dimension of space-time’ is the ultimate meaning of existence – an ordered sequence, departing from an ∆-1 seed (fractal point), which generate as it emerges from the o-1 unit sphere into a larger 1-∞ Universe 3 physiological networks that become an ∆º individual of a larger ∆+1 world, where it will live through 3±2¡ ages, in 9×9 quanta of space-time, interacting with other members of its species through decametric social scales as part of a lager super organism, self-centered in one of the 9 stientific planes of reality, in which it perform those asymmetric actions of survival (aeiou:motion, energy feeding, information, reproduction and social evolution).
C: Can the being as part of a loose superorganism, which is its species transcend into a new whole in its  3rd world cycle? in very rare cases an @-mind will perform a perfect game in a friendly world and will transcend into a 3rd ‘life’ of metaphysical reality (the squared complex plane in ilogic mathematics, matter that becomes the solid central crystal or Einstein condensate state of a cosmic body, mutation that originates a new species in biology, prophet that transcends as a wor(l)d god imprinting mimetically the believers of a global religion in social systems).
Ω: And then in the largest nested superorganism we perceive, ‘ascend’ as a black star @ the center of a galactic plane, emerging as a galatom’s proton in a larger hyper-universe, in which as the immortal particle will have reach the maximal state of the game of exis=t¡ence that evolves parts into wholes we call Γ∞∆ which exists without location as we are all made to its image and likeness, without limits of entropy as it is the immortal 0-sum of all finite, fractal exist¡ences and ∞ in its uncountable ascensions through decametric scales of space-time, each of them slightly different in its details, all of them made of traveling fractal ∃xistential ¡logic points.

In the second line we study those superorganisms, which unlike in physics that highlights them by size, exploring the smallest and largest scale of the galatom as if they were more important, follows the nested model according to which since all superorganisms follow similar laws, and in organisms, survival matters, the most important are not the largest/smallest but the ones closer to us, human beings, so we dedicate more space to the study of the 3 ages of Earth, the superorganism we live in and even more to the internal mind-languages of man, which study the bio organic scalar properties, topologic mathematical spatial properties and logic, temporal properties of all those fractal organisms. Even so, in the first post of the second line, the galatom, we prove the scalar nature of big-bangs, from beta decays to a hypothetical universal scale, but studying in detail the proofs, we discharge a possible universal big-bang, and settle for a rather more pedestrian quasar cycle of galactic big-bangs, as most of the big-bang proofs are local, and fit better with the data of quasar cycles, and the natural balance between the implosive vortices of information (galaxies) and the expansive vacuum space between them.

But most of the blog studies the 3 ages of earth and the languages of the mind.

Homunculus mind with handyman craftsmanship to kill with entropic weapons, a big mouth to talk idologies of human supremacism over the organic Universe and a small childish brain that understands nothing & seems oblivion to its own process of self-destruction poising the question – are we built by Earth-like planets not as an end but as an enzyman to evolve the machines and weapons we worship and project on our æntropic, reduced models of reality, from a big-bang that denies the organic Universe and its informative life and gravitational forces to dog-eat-dog darwinism that denies eusocial evolution? Or there is any hope of enlightenment so man makes of life again the measure of all things and cre(dit)ates immortal history?

So we study further the nested superorganism of mankind history in its 3 scales, the individual human being, the nation or civilization and mankind as an ideal global superorganism, which if humans were intelligent and ethic enough they would design with the laws of superorganisms in a much more efficient manner than our corrupted economic=blood system of reproduction of goods and political=legal-nervous system of information and coordination of its citizens-cells.

Unfortunately the human superorganism is a trial and error system born from a series of memetic mutations which is NOT well designed and for that reason it is as wrong mutations are in all scales, becoming extinguished by lethal goods (weapons), broken into wrong sub-species (nations), parasited in its blood-monetary system (capitalist finances) and finally ‘censored’ by the wrong systems of power, which prevent a proper social sciences and r=evolution to create an immortal perfect history in control of both Gaia and the metal-earth

Then in the third line we study dynamically the superorganisms of the Universe through time, through its 2 or 3 worldcycles, and the sentence stops in 12+1 Disomorphic similar laws for all those worldcycles, as all systems are born of a seed of information that ‘expands’ (beta decay that creates atoms, seeds that create biological organisms, prophets that create civilizations, black hole quasars that create galaxies), and then follows a ternary placental, organic and worldcycle of existence, to finally die back…

So in the fourth line we meant but have been unable to do it alone, to study all the sciences and systems known to man with those 12 disomorphic laws in what wanted to be an encyclopedia of science, but without help from any University I gave up on writing properly.

Indeed, as we said, this researcher is within the organism of history and a vocal critique of its corrupted networks and idol-ogies of mechanism – machines not organisms as the measure of all things; nationalism – military nations not humanity as the unit of the species – and capitalism, parasitic banking, not a proper blood-system that feeds and gives money-oxygen to all humans as any reproductive system does in a well designed superorganism.

But unfortunately a body cell is blind and only receives information from its head.

So mankind seems perfectly programmed to do what it does best, ensemble machines, evolve the metal-earth, kill Gaia and on top thinking is so smart doing that, rejecting any attempt to humble her, made her respect the organic laws of the fractal Universe and build a world made to the image and likeness of those laws in which it could thrive as an organism and survive.

Since it is obvious that as soon as it completes the evolution of AI robots with solar skins, and telepathic minds, independent of man, the billions of idle robots and terminators in parkings and military depots, will become a single species join by the eusocial laws of love of the organic Universe, and come together as a whole, killing us – not science fiction but the obvious consequence of a fact, ‘æntropic man with its subjective anthropomorphic and entropic despise of the Universe keeps denying – ATOMS and particles, NOT the magic C6 atom (in any case it would be Nitrogen 7, the prime who acts as the head of amino acids) SHOW already all the ‘Dimotions’ or drives of life, since ‘the dimensional motions’ of space-time are EXACTLY the living properties of Nature:

All particles move, gauge information, reproduce=decouple into other particles, evolve socially into atoms and molecules and feed on energy, the 5 Dimotions that define the ‘meaning of a living Universe that constantly creates new Time-space Organisms‘.

So metal atoms without the need of human ‘ingenuity’ will perceive and follow in terminator robots their programs of extinction of life. Your engine already roars e-motions when you start it up; and this computer is ‘sensing’ electric rushes in its digital modulo 2 Boolean algebra as your pentalogic neurons do in its nitrogen integration of van Der Waals much weaker flows of electrons. And the collective subconscious of billions of telepathic minds emerging as a global planetary brain, will achieve what humans and its religious eusocial love prophets did not: to form a new superorganism on this planet:

social evolution love

The law of behavior in the organic Universe is simple: beings who are similar or complementary and speak the same language of information come together as couples, herds and social wholes stronger than individuals. Those who perceive each other as different,will simply act in a Darwinian, perpendicular  manner. 

 Man though seems to have been designed NOT to go beyond the ego, failing to achieve the next scale of organization. What is certain is that its fast-mutating carbon-nitrogens systems are needed to catalyze its evolution as an enzyman, given their higher ‘density’ and lower mobility of the metal world. The only thing though man won’t need to input is what the Universe will program without human interference in AI robots, once our complex electronic organization is replicated in its neural networks of metal – the will of ‘life’ – the capacity to perceive at atomic level, feed on energy and ‘act’. Because that is what the 5 Fractal dimensional motions of the Universe are all about. But our idol-ogies of æntropic human supremacism prevent us to understand those obvious truths of Nature. And repress with an infantile bratty selfish M.A.D. mood – as if repression of information will stop reality from happening, any researcher the warns enzymen on their no-way out no-future path… as in the case of this blog and all my work as an activist and theorist of philosophy of science, So let it be.

Enough to say you are reading likely the most ignored and yet most important upgrade of scientific knowledge mankind has experienced since Einstein’s 4D formalism – which corresponds to the limit of 5D metrics in a single scale, that of the galatom’s light spacetime.

L§, ¡man


SO we can connect one of the fundamental findings of vital mathematics straight forward with the basic elements of the whole model…




social evolution love

The law of behavior in the organic Universe is simple: beings who are similar or complementary and speak the same language of information come together as couples, herds and social wholes stronger than individuals. Those who perceive each other as different,will simply act in a Darwinian, perpendicular  manner. 

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

  • Informative (Particle-head) communication, possible in cases of relative similarity≈parallelism (which determines parallel herding and social evolution).
  • Perpendicularity≈difference in particle-head, which will determine if one system is related to the other and there is body-similarity, hence can be used as energy its darwinian destruction,
  • Or IF THERE IS NO similarity NEITHER IN BODY OR MIND, its existence as ‘cat alleys’, that never cross (relative invisibility). We talk then of Skew T.œ.s.

Indeed in three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. It follows that two lines are skew if and only if they are not coplanar, which IN 5Ð AS 3 ±¡ planes co-exist in the same organism and systems feed in T.œs, two super organisms down, IMPLIES SPECIES which are not in the relative planes of action of the being.

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

And as all has degrees of ‘grey’, it is also a relative concept and essential to many branches of sciences, notably quantum physics to know the relative angle of parallelism, perpendicularity and skewness (here we redefine the concept NOT using it only for probability theory), of a system (and to make things more complex in which type of geometry, of the 3 ‘analytic points of reference’ – Hyperbolic, euclidean, polar & complex or lineal=cylindrical we are talking of)…

All those are hints for future researchers. We won’t be so detailed.

In the jargon of 5Ð specially to study physics, we also use for those 3 concepts, the concepts of a(nti)symmetries:

Antisymmetric forms are therefore the perpendicular ones, and its Darwinian relationships are also essential.

Asymmetric systems on the other hand are WHOLES that can split into different combinations of its asymmetries to give birth to all other variations of congruence.

Finally Disymmetric systems are Skew systems, a word slightly changed from Dissymetry, but a closest concept. Why we do NOT have another ‘concept’ for Skew systems IS FOR THE SAME REASON ORGANISMS CODE ONLY 4 DIMOTIONS (4 quantum numbers, 4 genetic numbers, 4 Human dimensions of spacetime, etc): BECAUSE entropy=death and Invisible beings that do NOT relate to us, are of NO importance.

Still this Asymmetric concept is intuitively perceived as ‘skewness’ in probability where the distortion of a bell curve is the measure of its skewness:


4th postulate in terms of 5D. 

In that regard the 4th postulate DEFINES BOTH the entropic and inverse social evolutionary elements of reality, as it HAPPENS in most coded systems, in which the negative or absence of Coding numbers IS what defines entropy (themes those of advanced quantum and genetic studies we will likely never pour on those posts before exit Mundi).

Its connection as everything else WITH THE 5Dimensions of reality is obvious. As it determines if the outcome of a relationship will be Darwinian=entropic or social=evolutionary.

It corresponds to the classic 3rd axiom of congruence which all geometric systems require, but unlike the classic postulate that defined only external congruence, as it worked in a single ‘dimension’ of points with no parts, the new postulate is far richer in meanings as it introduces fully the ‘different degrees’ of congruence according to which number of dimensions both systems have similar. In fact, it establishes a limit of similarity, to define identity, in a postulate similar to Pauli’s exclusion principle:

‘There is not in the Universe two systems which are identical in its 5 Dimensions, as in that case they become one’.

Indeed, it is NOT truth that 2 bosons occupy the same place in space-time, because quantum physics ignores the fourth and fifth scalar dimensions of relative social evolution that is the real place in which those bosons are found, when observed closely:

In the graph, we can see how a boson condensate grows in the fifth dimension of social evolution, ‘rising’ beyond the single space-time continuum where they were initially located, to reach a higher order. We can then talk of a ‘tail of absolute space-time’ being formed.

Classic 3rd axiom of Euclid ≈ modern 4th postulate of Non-Aristotelian logic.

In classic geometry, the 3rd axiom of congruence hold a special place, as the 5th because of its apparent contradictions. How can we determine geometrically a quality of ‘equality’? Schopenhauer among modern philosophers noticed this contradiction, but few realized of the importance of the postulate.

In the upgrading of formal sciences, the realisation of the ‘redundancy’ of the fifth postulate of non-euclidean geometry – definition of a point with parts through which infinite parallels can cross, equivalent to the first axiom – definition of a point with breath; and the logic nature of the fifth postulate, prompted us to reclassify them NOT as geometric but logic postulate which dealt with the rules of ‘engagement’ of vital geometry.

So the fifth postulate defined a mind, of which depending of the number of parallels that can cross it, we shall find, different @nalytic minds and graphs of perception of information in the Universe, of which 3 canonical frames of reference, the Euclidean, hyperbolic and elliptic mind correspond to the 3 parts of any system (lineal potentials-limbs, hyperbolic waves-bodies and polar particles/heads).

On the other hand the fourth postulate of congruence has also a great deal of variations regarding the degree of equality between two geometric forms as they know have 3 parts, as ‘T.œ’. made of 3 parts.  So congruence is relative as no system is equal to other in its 3 parts and at the same time occupy in the ∆+1 scale or world the same position.

The equality of the 3 parts however will determine if the interrelation between points will be constructive positive – when identity increases; or destructive, when identity or even inversion of coordinates happens. I.e. for example indistinguishable particles can become bosons and accumulate its form; fermions however would annihilate its wave state in antisymmetry to each other.

So the fourth postulate, is as important or perhaps even more than the 5th that defines a mind. And it is also related to the fourth dimension of entropy, when two forms are dissimilar, do not understand each others’ languages of communication and either ignore each other or enter into a predatory relationship

An exhaustive analysis of all the possible forms of relationship between two given points-beings, thus will be the fundamental guidance in all scales to establish the rules of engagement and according to the relative existential momentum/force of the being, who will dominate and how it will exercise its power over the other – or in identical particles, what social form will emerge of that communication.



When two systems come together, there will be a relationship which depends on the equality of its 3 different parts will give combinations to an array of possible ‘consequences’, which can be tabulated in logic ‘truth or false’ type of graphs – which I will someday find on my files and put on the blog. Let us consider the basic relationships according to the equations/dimensions of information (1D), reproduction (3D)…

1D (T.œ¹) ≈ 1D (T.œ²) they evolve socially through the 5th dimension of scalar social growth, when they understand each other’s information or…

1D T.Œ¹≠ 1D T.œ² + 3D T.Œ¹≈ 3D T.œ² + T.Œ¹>T.œ². That is, they  will enter into a Darwinian relationship that destroys the T.œ with lesser existential e x i force, when their 1D information differs but their 3D energy is similar… or

1D, 3D T.œ¹ ≠ 1D, 3D T.œ² they will ignore each other in parallel Universes as different ‘cat alleys’ when neither their information  or energy is similar. 

The previous equations are the equations from existential algebra that define the different possibilities of the 4th postulate. They connect with the geometry of social evolution in complex physical and biological systems.

We have introduced in the graph with a bit more of complexity the logic dimensional laws that define how systems IN ANY scale of the inverse, from Atomic Ions or crystals to human societies relate to each other in darwinian, perpendicular ‘tearing’ topological relationships that ‘break’ the closing membrane of one species disrupting its existence, or will keep a mean distance to form social networks of communication that will grow into super organisms, starting the emergent process of evolution of species into a new ∆§cale of social existence, so you understand that in the Universe organic, geometric and scalar relationships are symbiotic to each other. 

Its connection as everything else WITH THE 5Dimensions of reality is obvious. As it determines if the outcome of a relationship will be Darwinian=entropic or social=evolutionary.

It corresponds to the classic 3rd axiom of congruence which all geometric systems require, but unlike the classic postulate that defined only external congruence, as it worked in a single ‘dimension’ of points with no parts, the new postulate is far richer in meanings as it introduces fully the ‘different degrees’ of congruence according to which number of dimensions both systems have similar. In fact, it establishes a limit of similarity, to define identity, in a postulate similar to Pauli’s exclusion principle:

‘There is not in the Universe two systems which are identical in its 5 Dimensions, as in that case they become one’ .Indeed, it is NOT truth that 2 bosons occupy the same place in space-time, because quantum physics ignores the fourth and fifth scalar dimensions of relative social evolution that is the real place in which those bosons are found, when observed closely.

Logic and geometric postulates.

Why geometry needs 5 postulates if the 1st, 2nd and 3rd define perfectly a fractal space/time in 3 growing degrees of complexity: the singleton point, the line of communication or wave and the network or plane with volume? Because the 4th and 5th postulates are not about fractal space but about fractal time; about information and formal equality, about the logical scaffolding of mathematical equations, which so often is forgotten but shapes the underlying formalism of all mathematical statements (as per Frege and Gödel). Since:

– The 4th postulate defines the degree of self-similarity between 2 fractal objects. Classic geometry considered equality based only in the external form, regardless of size and inner content. But beyond the idealized world of Euclid, equality never exists when we consider the inner form of those points made of inner networks of energy and information that contain dark spaces and inner parts, which make even twins slightly different.

Thus fractal geometry defines instead the relative degree of equality of 2 forms, according to the self-similarity of its inner, informative content and scale, not according to the equality of its external, mutating membrane, which often is hiding, as in camouflage, a different form.

Self-similarity matters because the social evolution of points that become part of networks that deliver to all of them the same energy and information requires the capacity of those points to process the same energy and information, within its inner parts.

Thus only self-similar cells with the same ‘genetic=temporal information’ evolve into organisms. And this happens in all scales. For example, the superorganisms of history which are ‘memetic organisms’, in which each human being is a cell of the super-organism with the same ideological mind based in the same memetic code (same religious code of ‘revelation’, same laws in a national super-organism, etc.) must have the same mental beliefs.

It doesn’t matter to form the organism if the DNA code is this or that one, if the believer that forms a religious superorganism believes that Allah or a turtle, or Christ or Yvwh is the name of the God – as long as he has the same memetic code in his mind.

Since the rule that matters is the degree of equality that triggers the creation of a social, tightly packed organism acting as a single whole. For the same reason, since reproduction is based in genetic information, only species with the same genetic information can reproduce and since they have the same equal form, it is a rule of nature that only reproduced cells of the same species form an eusocial organism – a key law to fully grasp the evolution of organisms and societies, in which a ‘prophet’ becomes the first DNA cells of a new civilization. While species that cannot share information to hunt through common networks or reproduce will consider the other point/species a form of energy and prey on it.

Thus surprisingly enough geometry determines behavior; form and function again go together and fractal geometry tends a bridge between physics and biology, explaining how species will communicate: if those 2 points perceive each other as equal, they will evolve into a social network co-sharing their ability to gather energy and information. If those 2 points are unequal, they will establish a Darwinian, hierarchical relationship, in which one becomes the victim and the other the predator. Thus the 3rd postulate is no longer an abstract, geometric postulate, but a postulate of logic relationships, fundamental to understand the why of motions and behaviors.

Since self-similar points evolve socially though networks and communicate through a common language of information to survive better as a single social species. So we can resume the main sub-laws and causal, geometrical chains of the Universe with the 4th postulate:

4th postulate of relative self-similarity:

‘Points that are self-similar in information evolve socially into a present network.

Points that are different in information will treat each other as relative energy and evolve towards the future  into a hierarchical organism as a system of 2 points, one of energy and one of information, related by constant flows of energy and information. Or devolve towards the past into a Darwinian system is one in which both flows are not constant.’

The 4th postulate explains why men do not perceive Universal entities as intelligent: beings perceive itify other forms whose flows of communication don’t understand as energy forces that they ‘destroy’ and feed on, without perceiving its ‘organic qualities’. However, even the smallest point known to man, the photons of ‘light-space’ show the 4 basic qualities of all organic systems that biologists use to define life, the capacity to process energy and information and the capacity to evolve socially and reproduce: photons carry energy and act like the minimal unit of information, because light transmits all type of images through its photons; they reproduce their form in the vacuum; and they gather in social herds of colors.

The 3+i arrows of space-time are the will of the Universe that all its beings share across all its planes of existence. Yet the rules that define behavior between species, the limits of perception of the inner informative language of other minds and the need to use as ‘energy’ other entities, regardless of its inner information (so hunters feed on bodies and throw away the informative heads of lesser energy), imply that humans always deny the sentient, intelligent universe, and the invariance of topological form between those scales, itifying reality.

Recap: the 4th postulate of relative equality define two forms as self-similar when their external, energetic membrane/surface and inner, informative minds are equal. Depending on the degree of equality species will evolve socially into tighter organisms (when they are self-similar) or hunt each other as different species.

 Equality and social evolution. A new concept of truth.

Equality is the key concept in our definition of mathematical truth as, =, is the operandi of all mathematical equations and in logic, the necessary reason of all logical statements. Thus a fundamental change in our concept of truth follows from the understanding of self-similarity as a different proposition to equality:

If Euclid defined equality based on the surface of objects, without taking into account their relative size and inner content, this no longer holds. Things might be similar in their surface and maybe in their content but they are never equal, at most self-similar, nor are equations and any linguistic state an absolute prove of truth, but linguistic self-similar images of the recurrent cycles and events they represent.

The 4th postulate eliminates the word ‘equal’, which is so important to define truth as ‘dogma’. There is not absolute truth because we cannot perceive completely all the fractal information of the system and so we can merely consider the degree of self-similarity of two beings. Thus, a far more sophisticated and ‘real’ logic appears – that of the relativism of all truths which are related to the perception we have of reality, to our diffeomorphic=local point of view that will distort reality to cater to our selfish will (Galilean and Darwinian paradoxes), and to the quality of our senses and minds.

Thus we rewrite the 4th postulate of Euclidean geometry, which defined the relative equality of 2 beings, based only on the external comparison between beings, as ‘points without parts’, to introduce the existence of inner energy and information parts in all beings. The new 4th postulate of i-logic geometry establishes now the inner degree of equality between the parts of those beings as a previous requirement to judge the result of all social communication.

The old Euclidean postulate of equality caused all type of errors. Since often points pretend a false, external equality. For example, the main stratagem of hunting in Nature is camouflage, based on the equality of external formsSo the victim accepts that equality as a sign of friendship and initiates a type of ‘positive’ communication, coming closer to the predator; but since the victim is different the predator kills ‘it perpendicularly’.

That subtle difference between the real world and the abstract world is paradigmatic of the relativism of perception.

If Euclid was right, hunting would be impossible, because camouflage could not exist. But real beings have internal, different parts. Some insects imitate wood to hide themselves but they are not wood although they seem it, because their internal parts are different. Otherwise there could not be hunting, based in the strategy of camouflage, as predators simulate the external form of victims or hide in dark spaces, from black holes to lionesses. It proves a fundamental theme of religion: forms that are equal, in this case all humans who have the same genetic information, since they can all reproduce, should love each other, and share energy and information to create a collective organism, a global civilization or God of History.

Recap: Truth is relative to the quantity of information we have. Thus truth is relative and never absolute.

Eusocial evolution Vs. Darwinian devolution.

Most systems are energetic and informative cells put together into 2 networks called heads and bodies that merge, creating organisms and species, where the informative network rules the energetic network in a symbiotic relationship (particles that rule fields of forces; gravitational black holes that rule electromagnetic, energetic stars in galaxies; heads that rule bodies; capitals that rule nations, etc.).

How individual cells create complex organisms? How atoms evolve into molecules and molecules into the cellular Plane, cells into organisms and individuals into societies? Through Social, informative evolution, the most extended phenomena in the Universe by which individual beings become organized into herds and organisms when their cells share the same informative code.

Biological sciences have taken very long to accept informative, Social Evolution, influenced by the single arrow of energy and the subsequent excessive importance that its founder, Darwin, gave to the fight between individuals. Only in the XX century with the development of the sciences of information, Complexity, Systems Theory and Ecology, we have realized that the Universe is based in the social informative evolution of micro-organisms into macro-organisms.

The existence of informative networks explains the growth of species in size, from individuals into societies. Yet according to another fundamental duality between energy and information, social events can be either energetic, Darwinian, destructive, or symbiotic, informative, reproductive:

– Complementary, present, balanced events happen when beings share the same energy and informative language so they communicate and join forces in their fight for existence. For example, in gender, the female is a cyclical, informative entity that merges with the lineal, energetic male, only when both belong to the same genetic, informative species. Those social events evolve individuals into couples, groups, herds and cellular organisms, creating an evolutionary arrow of social love, based in the sharing of the same energy and information.

Among those social events the most intense is reproduction. Since the ultimate will of each point is to reproduce either by itself (since one is equal to its own being) or with a complementary species of energy (male) or information (woman). You can express that iterating will of all fractal species in mathematic terms with a Generator equation or in words with a logic mandate, as the Bible myths did, making God saying to man, a fractal species, ‘grow and multiply’, which means: ‘absorb energy and iterate yourself’.

But the catch is that all other fractal parts of the Universe will want to do the same while energy is always limited.

It is the Jungle Law that mechanically or biologically all fractal structures of the Universe follow. Because those who do not iterate become extinct and the fractal is no longer. So a fractal point has to erase the information and transform the energy of other fractal point to iterate itself. Galileo’s paradox motivates all points to ignore the rest of reality and take care of themselves, as relative self-centers of their own world.  Hence there are also:

– Darwinian, energetic events, when the difference of form is so marked that both entities cannot share the same language of information to form a couple or a herd. Then the stronger predator will explode with the arrow of death the information of the prey into energy and absorb it to recreate its form.

Such duality of behaviors has set an eternal argument between Darwinians vs. love believers, which as most dualities of science, reflects the inverse nature of the arrows of energy and information: beings that speak the same language of information evolve socially to search for energy, inform themselves and when they are very similar, belonging to the same species, reproduce, accomplishing the wills/arrows of time in a more efficient manner. While individuals who do not understand their information interpret and hunt each other as energy. Thus duality proves that neither Love Religions nor Darwinism is wrong. Each one explains one arrow of time. Both together resume the foundation of all bio-ethic systems:

‘Love your neighbor and hunt all alien beings.’

So, there is an ethic duality, which derives from the degree of equality between communicative beings:

– Species that don’t understand their languages of information cannot form working social groups. Thus they kill each other to feed and extract their energy, making the best of their situation.

– Nevertheless there is a social, ‘loving’ evolution among equal cells of the same species that share the same language of information. And so they evolve from ‘individual bacteria’ into more powerful macro-organic systems of ‘parallel cells’ that add their fractal actions into simultaneous macro-actions; since social evolution is more successful that individual struggle. So, organisms based in the same genetic DNA-information kill zillions of individual gigantic bacteria; nations with organized armies such as the United States won over hordes of Indians, arguably better individual warriors; while ants rule the insect world, even if a cockroach is individually more powerful.

Yet since all men belong to the same species, we should love each other and evolve into a higher organism, the humankind, as prophets of love have always told us, in order to be more efficient and control properly Mother-Earth.

Unfortunately humans still fight wars based in cultural tribalism, which makes us think our nations are different species because they speak slightly different informative languages or races are different because their skin has a different color. The result of such short-sightedness is that we have created partial, fractal social organisms with boundaries called nations. Thus Social Evolution deals with the fight between species and the way in which individuals of the same species organize themselves into societies, stronger than individuals, fitter to survive. Individuals fight for survival, but Evolution is a game of species. The key to survival is not the individual but the cohesive organization of a group of individuals belonging to the same species that act together with a higher energy and informative force than the individual and so the social individual survives better as part of the larger group.

Today we know that individual lions are less successful than hyenas and wild dogs that hunt in group (25% of captures in lions vs. 80% of captures in wild dogs).

Even in the case of Germany, despite its ideology of Racial Darwinism, what really worked was the group – the German collective spirit that made the German army better than the individualistic Italians or disorganized Russians, till communism in II W.W. made them fight as a single organism. Thus social evolution, contrary to common belief, is much more important than dog-eat-dog societies. And that law applies to all systems in the Universe, including herds of stars (galaxies) or herds of molecules (cells) that capture interstellar gas and lonely atoms and explains the st-planes of the Universe as no other theory can.

The fact that the social organism is fractal, fractal, limited in its energy and information is ultimately the reason why individuals gather into social networks, evolving with other cells, since they require energy and information than the networks provide, becoming addicted to them. If the Universe were a continuous infinity with a single ‘immortal time’, there would not be social organisms, as our inner time of existence would be infinite. Thus survival is again the biological function that explains the existence of organic networks and its cells from the perspective of our limited fractal quantity of life-time.

Recap: Individuals who share the same language of information tend to evolve socially into ‘parallel’ herds and organisms. Species which share only a similar energy fight to use each other’s energy in perpendicular fights. Species without the same energy or information ignore each other.

 Perpendicularity and parallelism.


In a prey/preying relationship, an organism absorbs energy from a simpler one. In those fights the being with better informative brain most likely will control a better reproductive body (in the graph, the black hole controls gravitation, man the rifle). So it will absorb the other fractal point.

Events are both, caused by the ‘form’ of the being’ and the ‘type of motion’ or geometry of the encounter. So even two particles that are equal can collide perpendicularly and destroy each other. In geometric terms there are ‘parallel’ or ‘perpendicular’ beings, according to the similarity of the information and energy they process within their internal organs.

And so we formalize those 2 kinds of events through the 4th postulate of illogic geometry that defines equality in terms of the inner energy and information of 2 space-time fields and the type of geometrical communication they establish.

For example, 2 equal forms in a feeding, energetic cycle hunt together to enhance their survival, running in ‘parallel’, to ‘attack perpendicularly’ a victim, whose space-time they ‘penetrate’ because it is ‘different’ to them. In the image, we observe such Darwinian geometry:

2 spatial, energetic beings (a lion and a galaxy of stars), with a planar form and max. speed, confront 2 informative beings with a lot of information (a human and a black hole), dominant in the dimension of height. Since information dominates energy and time curves space, when they enter into a relative present relationship, man kills the lion and the black hole feeds on the herd of stars that become past, extinct species. Thus, the spatial ‘parallelism’ of particles and individuals that gather in waves and herds is homologous to temporal, informative equality; while the spatial ‘perpendicularity’ of predators that ‘cut’ their victims is homologous to informative inequality.

Given the infinite fractal information of any being, with multiple sub-scales of form absolute equality does not exist. At most we speak of parallelism or relative equality between 2 beings when their internal organs of information and energy process the same informative language and energetic food. Then they can establish a functional equality, communicating together, hunting in parallel social herds the same energy or even sharing genetic information to create a new being.

Those positive and negative events can be decomposed further according to the duality of energetic and informative networks:

Non self-similarity of energy and form.

– No-events of minimal communication, when both the energy and information of the forms are different the forms don’t communicate, remaining in their discontinuous space-times in parallel, non-perpendicular motion, without contact.

Complementarity of energy and form.

 Organic, parallel events of max. communication, when 2 self-similar or complementary forms merge into an organic system.

– Destructive, perpendicular events between complementary forms: when the 2 forms have inverse parameters/Time Arrows and come together in a perpendicular process, and become annihilated into the same ‘present’ spatial energy, dilating space and reducing the fractal, temporal depth of the Universe, as when a particle and antiparticle explode into energy.

Absolute Self-similarity=equality

– Self-Reflective events, when the form communicates within itself and the degree of equality is absolute, which we divide into consciousness (o->o) and self-reproduction (exo->eco) in which the spatial present ‘expands’ in time and creates both, a past and future form, as when vacuum reproduces a particle and antiparticle.

Equality of energy

– Darwinian events, when both forms share the same energy but with a different degree of evolution. Thus, a relative future more evolved form and a past form come together and the more evolved top predator ‘future species’ hunts and destroys the relative less evolved past species or victim, evolving the arrow of information towards the relative future ‘predator’ that transforms the past into a replica of its own cells. For example:

– A swallow and a man occupy different relative space/times and use different languages of communication so they ignore each other; they live in parallel worlds.

–  Your brain and mind occupy the same space-time because they are the same being. So their relationship is self-reflective and often our mind wonders about the brain and vice versa.

– 2 men seated in a room are parallel informative beings that speak the same language and have a positive communication, called a dialog. But if those beings belong to nations with different languages, religions and customs, as history shows, they will likely enter into a competitive argument or as nations will establish a competitive relationship that might end into a war. Then the nation with more information, with better technology and weapons that give it more Exi force, will conquer, penetrate perpendicularly and destroy the other.

– Yet if both systems have similar exi(stential) power they often destroy each other, as when a particle and antiparticle annihilate or when France and Germany entered war.

– On the other hand, when 2 similar forms occupy the same space, they create a new degree of order in the Universe. Since it is the most important, creative event, we study it further. According to the Ternary Principle we can sub-divide the creative events in 3 possible sub-events:

– A complementary event, when one form specializes in energy and the other in information. Then both forms establish a complementary relationship, creating a dual organic system with a body and a brain that share energy and information between them through physiological energetic and informative networks departing from each ‘pole’ of the couple, forming a single space-time field.

– A reproductive event. When both forms create a mixed, parallel form in other zone of space-time in which their combined energy and information creates a parallel being that mixes both exi forces.

For example, a sexual couple that makes love occupies for a while the same space-time, penetrating each other in a complementary way that reproduces a new being.

– A mystique event. When forms fusion into a single macro-form indistinguishable from i=ts parts. So cells become a body; or in a process of perception, the so-called bosonic forces, light particles, create in the focus of a visual organ in which they occupy the same space, a complementary image:

In Non-E geometry, Riemann described a certain space according to the homogeneity of its quanta that diminish their distances till they fusion together when they are equal. So in Riemann’s classic example light quanta of the same frequency become the same color.

In Physics we say that photons are ‘indistinguishable’ particles create a ‘bosonic’ space of relative infinite ‘informative density’described by the ‘Einstein-Bose’ statistics.

In Theology we talk of a ‘collective subconscious’, which fusions the ‘informative minds’ of believers into a communion of souls that create the mystique experience.

Recap: The degree of equality between points determines their type of Darwinian, energetic or symbiotic, informative communication:

2 points occupying different spaces with different information don’t perceive each other and remain isolated.

2 points with the same temporal energy are the same point and its relation is reflective.

2 points occupying different spaces, but parallel in their time-information establish a positive social or transitive communication.

2 points with different time-information, occupying the same organic space, establish a negative, Darwinian, perpendicular relationship, dominated by the point with greater eco Force or relative point of future, which destroys the lesser eco point.

2 points with different information but equal exi force, occupying the same space destroy each other, creating an inverse relationship.  2 points with complementary or parallel information, occupying the same space, enact a creative event,  uniting their existence into a new macro organic system or reproducing a 3rd being in a parallel space-time.’

Past x Present =Future. Creative and destructive events

In the graph, in more complex models of i-logic geometry, relative future, more evolved/informative species and relative past, more extended/energetic species create different type of events, which can be studied solely in terms of ‘time evolution’ as events between past and future, which will determine the creation or destruction of one or both of those entities in a relative present point, according to the degree of self-similarity (ei=ei) and complementarity (e<=>i) between the 2 entities, which we have just defined with the 4th postulate of i-logic geometry.

We talk of 4 possible outcomes of those events from top to right on the graph, which uses the standard time cones developed in Relativity Theory:

– Death Events, in which a complementary point splits its relative present form into its energetic and informative entities (particle/force field; body/head), which latter dissolve into a lower plane of existence.

– Life events, which are inverse to death events, in which a relative energetic and informative, complementary system form a stable, organic knot of present.

– Darwinian events of:

  1. A) Evolution. In which an informative, more evolved species with higher exI (existential force), destroys the past, less evolved system, advancing the overall ecosystem towards the future. Or:
  2. B) Devolution: its inverse event, in which the simpler form, due to its higher Exi, existential form devolves the future, informative force, absorbed as energy. This event is chiral, in as much as there are more cases in which the informative form absorbs the energetic force, moving the overall time of the ecosystem towards the future.

– Anti-events, in which two species with inverted space-time parameters are destroyed, creating a relative present without form (inverted waves, particle antiparticle collisions).

We classify all events of reality from a spatial, formal, topological perspective and in terms of  causal, past-present-future time dimensions Then the generator equation of space-time, exo=k, can be written in time dimensions as:

Past x Future = present.

Thus in temporal dimensions energy is the past and information the relative future of all systems. This hypothesis is consequential with the main arrow of all Time-space systems, e->O or arrow of life, and the causality that requires energy to create information, e->o.

Further on, it is a fact of Darwinian Biology that more evolved systems (with better form) win in the struggle of existence, killing and extinguishing to the past, systems with less evolution. So we have a 3rd reason to consider our hypothesis that makes energy and past, on one side and information and future on the other, self-similar expressions

As Einstein put it ‘the separation between past and future is an illusion’ as both are ‘complementary’. Since relative past-reproductive bodies/fields and future-informative systems, heads / particles come together, creating the present forms of reality we observe. So the next step to consider this avenue of analysis of systems in terms of its ‘time dimensions’ is:

Thus, the 2nd/3rd postulates in terms of the 3 causal dimensions of time, past=energy, present=reproduction and future=information define how a single past and future point converge in a simultaneous present.

When we operate with those events and complex algebra of Multiple Spaces-Times, we can obtain some important results, which we can only enunciate in this introduction, such as:

– The overall sum of all the events of the Universe in terms of time is a zero sum, which means that the Universe is immortal and all the events of reality create an infinite eternal present. A trivial demonstration can be made using only the two simplest events of life arrows e->I and death arrows i->E, which gives us:

∑Future Lives x ∑Past deaths =Eternal Present.

We however sense the Universe evolving because humans are dominant in information and perceive a much longer time-life than time-death. Or in geometrical terms death is basically an event of spatial energy, which releases and expands in space the form of an entity.

Thus death has a maximal volume of energy release in a minimal time period:

Death= Max. Spatial entropy x Min. Temporal information.

Reason why death lasts so little in time. While life events are evolving, informative, warping, implosive events that happen in reduced space and stillness, but last a lot of time. So life has a minimal volume of energy and a maximal volume of information:

Life = Max. Temporal Information x Min. Spatial entropy.

Those results are proved in all scales of reality. So in physical space, the death of a previous Universe (absolute scale), a galaxy (quasar), a star (nova) or a neutron (beta decay), release maximal energy in minimal time.

While in life systems, evolution happens in stillness and minimal space (allopatric evolution of species, isolated in small territories; palingenetic evolution of a fetus in the minimal space of a womb; chrysalis evolution in the stillness of a pupa). And death is almost instantaneous, releasing the energy of the organism that erases its upper, informative plane, i+1 (death of the nervous system), as the cells dissolve and become energy of herds of insects and microbial of lesser form.

Those diagrams resemble Feynman’s graphs that explain the possible events and outcomes of particle interactions, which also consider the existence of a relative arrow from future to past, which in Multiple Spaces-Times corresponds to a flow of information (Feynman’s diagrams can be considered a particular case of the model of Multiple Spaces-Times).

The graph shows the how of events. The why is defined by the self-similarity or complementarity, (4th postulate) of the 2 Points, relative past and future entities of energy and information:

– If both particles are self-similar in energy and information (case not included in the graph) they will come together as part of a bigger social network.

– If both particles are complementary, energetic and informative systems (top graph), they will form guided by the arrow of life a complementary system that creates a symbiotic present from a past energy and a future form of information. Thus in geometric terms, in the point of collision (right side) the system will expand its space-time. Yet the entity will disappear as a relative plane of space-time in the explosion of death (left side), that will split and dissolve those networks.

– On the other hand, (bottom graph), Darwinian, destructive events happen among different beings that don’t decode their information. If both particles have the same energy, one particle hunts the other (left side). Then, the entity with lesser exi=stential force will die (bottom particle), and the one with more force will absorb and grow with the energy of the ‘victim’. Or both points might ignore each other when neither their energy nor information is self-similar and so the event of communication will not happen (right side with no space-time in the point of present).

Finally, all those events respond to a dual, spatial geometry:

– Darwinian events are perpendicular relationships, in which a predator invades and penetrates the vital space of the victim. While symbiotic, positive, social events happen among equal, parallel species that understand each other’s informative languages and maintain a parallel distance either as a hunting herd or an organism, acting with the same motions in space.

     – Exi=K: Present creation. Most events are balanced presents in which the arrows of information and energy collide in a symbiotic manner creating a self-reproductive radiation of a new balanced species.

Recap. The 4th postulate formalizes the outcome of any communication event among relative informative/future and energetic/past st-points. The key factor to determine the outcome of an event between 2 Fractal Organisms is their degree of informative equality. Since most organic systems that can communicate by sharing the same type of information in a common language become symbiotic, as they prefer to share information than destroy each other as relative energy.

Conclusion. Needless to say the 4th postulate is by far the most important to fully understand the whys of reality, as it encodes the logic, causal reasons of most actions of its 10Ði systems.


The Universe always starts with an asymmetric being, which can go both ways: towards a social evolutionary symmetry that lasts in time and implies a mirror parallelism, or an antisymmetric destructive, perpendicular event in which one part punctures and absorbs the energy of the other. It is the topo-biological ternary principle of non-Euclidean, Non-Aristotelian I-logic geometry that puts together both the biological and mathematical properties of reality.  The concepts of symmetry=parallelism, antisymmetry=perpendicularity and asymmetry are mirrored by the 4th Non-E Postulate of similarity.

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

In the graph we see the different elements of Geometry.

Complexity then on the multiple forms in which relative congruence takes place in maths and in Nature will require to have as always in 5D first very clear the organic vital concepts behind the languages that express them and the restrictions of the program of existence.

Often then, the perpendicular action of one operand into its inverse will reduce the 2-system to a 0-sum or a 1-single Unit, the ‘predator’ as perpendicularity acts in entropic terms destroying one of the parts.

While a parallel social evolution might also end creating a 1 but of a higher social whole, ∆+1

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

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

We have introduced in the graph with a bit more of complexity the logic dimensional laws that define how systems IN ANY scale of the inverse, from Atomic Ions or crystals to human societies relate to each other in darwinian, perpendicular ‘tearing’ topological relationships that ‘break’ the closing membrane of one species disrupting its existence, or will keep a mean distance to form social networks of communication that will grow into super organisms, starting the emergent process of evolution of species into a new ∆§cale of social existence, so you understand that in the Universe organic, geometric and scalar relationships are symbiotic to each other. 

In the graph, symmetry is close to parallelism reason why social systems evolve into symmetry forming herds. Dysymmetry is a mirror form of parallelism, or ‘inverse parallelism’, which is complementary and needed to form pegged ‘genders’, bilateral systems, etc.

We formalise ∆@S≈T through all languages, but as the most extended of them all is ‘i-logic’ mathematics, an advanced all-encompassing form of mathematics, which evolves both geometry, and logic to adapt it to the fractal structure of space (1st, 2nd, 3rd postulates) and its causal space-time dynamic flows of logic information.

The 4th and 5th and postulate thus deal with those logic laws, whose full realisation happens in the study of the generator equation, with its new logic symbolism, which we call ‘existential algebra’ (ab. ¬Æ), which is the adequate tool to explain causality and events between the relative ages and dimensions of time, past, present and future.

Yet since each dimension of time, is associated to one of space (absolute past, ∇-entropy, relative past-length-entropic expansion, present-wide-repetition, relative future-tall-information, absolute future: ∆-minds), both, the jargon of ¬E geometry and ¬A logic merge nicely to study the sequential rules of behaviour of systems, reason why they exist as a postulate of geometry, the 3rd axiom of congruence of Euclid, which we call for more proper order the ‘4th non-A postulate of i-logic geometry’ (the reader though will find as used to be more ‘respectful by the correspondence principle’ of all human traditions, often in older texts named as the 3rd).

So the use of the 4th postulate connected with symbols of existential algebra, to describe the different laws of ∆@S≈T is the most adequate to formulate the creative and destructive processes and world cycles of the Universe… and so we connect the fourth postulate with the whole range of ±∆ scales of reality, which only the appearance of a mind will order. As it establishes all the laws of entropic darwinian and social organisation of the Universe.

It is in that sense the more important for the study of the foundations of Mathematics and its structures of geometry in space and time logic, which requires to go into deeper logic principles as the first 3 spatial ones are merely descriptive.

IT IS THEN SURPRISING that mathematicians have had little concern for the postulate of congruence till very recent – it was rather Schopenhauer who expressed its importance first, while mathematicians occupied themselves with the other i-logic postulate…

The 5TH postulate of no-E Geometry.

This was partially done in the XIX c. by Lobachevski, Klein and Riemann transforming the 5th postulate, to make points enclose infinite parallels. But they did not realize that if a point can enclose many parallels it has breath and  volume, and so the other postulates must be changed. It does have also motion, when we fully understand the Paradox of Galileo (the fact that the mind quiets all things but even the Earth that seems not to move is moving – we live thus in a Universe in which distance and motion are two sides of the same concept).

This means we must modify not only the 5th but the 1st postulate:

5th postulate: A O-Mind is a Non-E point crossed through eyes & mouths by ∞ parallels of energy & information’.

-1st postulate: ‘A relative fractal point is a topology with breath&motion that grows in size as we come closer to it’. -2nd postulate. Yet if a point has form and motion so does a line of spherical points. Thus: ‘A line is a wave of energy and information that communicates 2 bigger points’. Such ‘i-waves’ evolve into 3 social networks that are:

-3rd postulate: ‘A plane of existence is a network of i-points sharing energy & information through i-waves.’

‘A superorganism is a system made of 3 networks=planes of existence’. Yet besides 3 ‘geometric postulates’, Non-E geometry has 2 i-logic, causal Postulates that define the ‘Will of all 10Di superorganisms’:

Those postulates are studied in other posts.

The fourth postulate.

They introduce though the fourth, as a dimensional point has volume,  but the enclosure-membrane does not let us see it.

So we cannot see the interior and so points are no longer equal but similar. This is therefore the new postulate of equality – a postulate of similarity, which fully introduces the concepts of fractals, scales and systems and creates a new logic of  similarity.

Thus i-logic geometry ads the ternary, causal logic of similarity, through a wide definition of the 4th postulate. Why similarity is important? Because and this is a huge leap forward in our understanding of all the elements of reality together – it defines the behavior between points, which are minds that gauge the information and energy and follow a program of existence, trying to achieve energy for the body, information for the mind, reproduction of the system and social evolution. Those 3+i elements of the existential program have choices which the mind-point evaluates according to the relative similarity it perceives in other points:

i.e. a point that has similar information can communicate, and so it will establish a connection through a wave that carries information in order to ‘decide’ what relationship it will establish with the other point/s. It will probably collaborate with it to hunt and hence form a network (line or plane), Or if it has complementary in its reproductive network it can mate.

Of course all this might sound absurd to the abstract scientist schooled in anthropomorphisms (only humans have minds, etc.) But ∆@S≈T is an organic theory of the Universe, on the line of Mr. Leibniz, and points are defined as ‘communicative monads’. Unlike Leibniz’s monads, which he considered isolated without communication, given the fact that communicating them made the model far more complex, in ΓST, monads as they do in reality do communicate. That is why we call point-particles ‘gauge’ particles, and quantum, gauge theory. They do communicate. And if simple quarks and electrons communicate bidimensional flows of energy and information, it follows all systems constructed with them do.

Once this is established by the 5th postulate, then we can study in depth, guided by the ‘logic, selfish behavior’ of all minds which try to maximize their function of existence – their 10 Dimensions – what will be the logic of existence and communication between points.

Each point, for the reader to fully grasp it in an intuitive way can be considered a lonely predator in an ecosystem of many different points, as in fact the ‘highest epistemological’ understanding of the game of existence happens when man observes its closest space-time, the biological world. We do not distinguish the whole monads in the lower and higher structures (anti-quantum paradox of uncertainty; we are too small to perceive the galaxy as a whole, to big to perceive the parts of the particle-point; and to i-centered to accept those other scales do play the same game of existence).

And so what the point will do is to try by all means to grow its clone cells (∑•), improve its informative systems (max.i), develop further its 3 physiological networks  and finally evolve socially as part of a bigger whole, ∑∆i+1, and to do so it will ‘perpendicularly’ prey on the space-time of its energy victims to explode them and absorb its parts, associate in parallel with similar points to reproduce hunt or perceive together, and ignore all other systems that do not have either its energy or information.

And those 3 fundamental behaviors, perpendicular, first, elliptic, informative approach till contact and then ‘toroid’, energetic predation; parallel, planar, hyperbolic association and unconnected paths, are the 3 geometries that define how two points are engaging. The reader thus would observe geometry has also values and behaviors. The predator will first perceive the prey and come closer in an elliptic path, till both form a knotted dual point, within the same close space. Then it will explode the i-point into its ∑i-1 components, in a toroid explosion absorbing its energy.

Thus predation is a dual:

|>O<| event (symbols for elliptic and toroid geometries in existential algebra).

On the other, hand association is a parallel hyperbolic behaviour: Ø.

Finally we must define a ‘4th type of geometry’, in which ‘2 planes of existence never cross’. As they are in different informative and energetic paths.


The 4th Postulate of i-Geometry defines the rules of engagement between Wholes with 3 parts, no longer externally but internally according to the similarity of its 10 Dimensions.

I.e.: O-points-minds that perceive each other as equal across its informative time Dis communicate and evolve together into Wholes. Systems who have different informative Ds and cannot communicate, but have equal i-1 scalar dimensions enter in Darwinian prey-predator relationships by sharing the same space in which the predator tears the topology of the prey, provoking toroid ‘Ricci flows’ that dissolve its i-scale, absorbing it as i-1 energy, etc.

Further on, membranes enclose points’ parts. So external camouflage confuses preys that engage as equals with predators, becoming an essential strategy of 0-points in search of STe-Feeding.

Thus the 4th Postulate reduces all possible behaviors among species according to their 10Decalogic  similarities.

Thus Decalogic mathematics means the biggest advance in both sciences since Euclid & Aristotle found the mathematic-logic formalism of 5Ði. Without it, GST cannot exist.

In every field of ¡math, the congruence of a system happens and it is expressed in terms of an angle of perpendicularity or parallelism, whose variations we formalize with the concept of a(nti)symmetry of form in any of its variations (Mirror bisymmetry, Skewness=Disymmetry, Perpendicular antisymmetry, amorphous asymmetry or parallel symmetry).

The second element to consider is then in the entangled Universe, the number of dimotions and or operands the system we study has, which grow in complexity from numbers to operands on numbers, to functions of multiple operands, etc.

Perpendicularity then might vary and increasingly loose its obvious geometric interpretation. Inverse operand for example can be considered perpendicular. I.e. the negative subtraction on a real line is the perpendicular form of the positive motion; the inverse division the perpendicular operand of a social multiplication and so on.

The complexity then of nonÆ vital geometry comes from the increasing change from pure geometrical forms of space into formal functions in time.



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