THE 3D POSTULATE OF NON-AE GEOMETRY: THE POSTULATE OF EQUALITY.

Equality=parallelism=communication: Vs perpendicularity=extinction.

In the graph, the 2x2 Darwinian Vs Symbiotic events in which a relative form of past and a form of future come together, creating or destroying a relative present.

Darwinian, destructive events happen among different beings that don’t decode their information. According to Duality they are equivalent in spatial geometry to perpendicular relationships.
 While symbiotic, positive, social, events happen among equal, parallel species that understand each other’s informative languages. Which means that 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.
Thus, in terms of duality, 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.
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 3rd 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.
Thus we rewrite the 3rd 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 3rd postulate of Non AE 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 forms. So 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. However, given the infinite quantic, internal cells of any being, absolute equality does not exist. At the most we can 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 social herds the same energy or even sharing genetic information to create a new being.
So, we talk of an ethic duality derived from the relative degree of equality between 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 quantic 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.
Those positive and negative events can be decomposed further according to the Ternary Principle in 2x3 events, depending on the neutral, expansive or implosive nature of the event, giving birth to:
- Neutral events, when forms don’t communicate, remaining in their discontinuous space-times.
 - Neutral, Reflective events, when the form communicates within itself.
- Negative, Darwinian events, when a future and past form come together and the more evolved top predator ‘future species’ hunts and destroys the relative less evolved past species or victim, dilating the space-time towards the relative quantic future ‘predator’.
- Negative, self-destructive events, when a future and past form come together and become annihilated into the same ‘present’ spatial energy, dilating space and reducing the quantic, temporal depth of the Universe, as when a particle and antiparticle explode into energy.
- Positive, Creative events in which the spatial present ‘expands’ into time and creates both, a past and future form, as when the vacuum reproduces a particle and an antiparticle.
- Positive, Organic events in which the past and future merge into an organic system.
The 3rd postulate formalizes the outcome of any communication event among n Non-AE points:

‘2 points occupying different spaces with different temporal 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 TE Force or relative point of future, which destroys the lesser TxE point. 2 points with different information but equal TE 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, fusioning their existenceinto a new macro organic system or reproducing a 3rd being in a parallel space-time.’

Thus there are 3 x 2 possible sub-events of communication in time. Let’s consider some examples:
- 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 your mind occupy the same space-time because they are the same being and 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, which gives it more TE force, will conquer, penetrate perpendicularly and destroy the other nation.
- Yet if both nations or individuals have similar TxE power often they end up destroying 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 TxE 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 the 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-AE 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 colour. 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.