**SUMMARY**

**INTRODUCTION.**

**FOUNDATIONS**

**AI. ∆§ AGE:** **OPERATIONS & FUNCTIONS**.

NUMBERS. Its closure. Rashomon effect on complex numbers.

Dimensional growth – its 3 operandi: ±; x/; xª log:

Symmetric inverted growth/diminution around a neutral §œT.

Polynomial Functions. The fundamental theorem of algebra.

**2: ∆±i AGE: §ŒT. GROUPS, FUNCTIONALS**

Classic ∆st-symmetries.

Cantor’s §œ.T

The last dimension, functionals in Hilbert Spaces.

Old baroque age: Hilbert’s axiomatic method. Its incompletenesse.

Entropic death**.** Algorithms & Boolean Algebra: the metal mind

**3RD AGE. EXISTENTIAL ALGEBRA.**

Classic A-logic: single time arrow. Dogmatic truths.

The generator group. Ternary Non-AE logic.

Universal, ternary grammar of all languages.

The symmetries of 10 dimensional §œTS.

**INTRODUCTION. ¬ÆLGEBRA**

The Universe is a fractal super organism made of…

**A(nti)symmetries**between its 4 Dual components, which can either evolve together≈**Balance≈become symmetric or annihilate become Perpendicular/antysmmetric**:**S:**space; an ENSEMBLE OF**ternary topologies,****(|+O≈ ø)**… which made up the**3 physiological networks**(|-motion/limbs-potentials + O-particle/heads ≈ Ø-vital energy) of all*simultaneous**super organisms***∆: Planes of size**distributed in ∆±i relative*fractal scales*that come together as**∆º**super organisms, each one sum of smaller**∑∆-1**super organisms… that*trace*in a larger**∆+1 world…****ðime cycles:***a series of timespace actions of survival that integrated as a whole form a sequential cycles of existence with*each one dominated by the activity of one of those 3 networks: motion-youth, or relative past, dominated by the motion systems (limbs, potential); iterative present dominated by the reproductive vital energy (body waves), and informative 3rd age or relative future dominated by the informative systems, whose ‘center’ is:**3 ages**,**@: The Active linguistic mind**that reflects the infinite cycles of the outer world and controls those of its inner world, through its languages of information, which guide its 5 survival actions: 3 simplex,**aei**, finitesimal actions that exchange energy (**e-**ntropy feeding), motion (**a-**celerations) and**i**nformation (perceptions) with other beings, and two complex actions:**o**ffspring reproduction and social evolution from individuals into**U-**niversals that maximize the duration in time and extension in space of the being.

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

The study of those 4 elements of all realities, its actions and ternary operandi, structures *the dynamic ‘Generator Equation’ of all Space-time Systems of the Universe, written in its simplest form as a singularity-mind equation:*

* O x ∞ = K *

*Or in dynamic way, S@<≈>∆ð.*

So that is the game: 3 asymmetries of scale, age and form, which can come together or annihilate and each language=mind represents in different manners, those elements and its operandi.

In mathematics*, with the duality of **inverse operations, + -, X ÷, √ xª and ∫∂.*

The humind plays a key role in all this game, as we can only perceive through the languages of the mind, which ‘perceive in themselves’ as ‘still-simultaneous-linguistic mapping’. WE DO NOT PERCEIVE REALITY BUT LANGUAGES-MIRRORS, we often confuse with reality (Mind paradox).

**Languages express the elements of reality and its operandi**

It is then clear that what languages as synoptic mirrors of the mind will try to do is to establish the basic relationships between the space, time, scale of the being, expressing them through its operandi, DEPENDING on the degree of perception the being has of reality and its scales which might be reduced if the being is not fully aware of all the scales of existence, as most minds exist only in a plane of reality

So does mathematics, through combinations of:

Sum/rest->multiplication/division->potency/logarithm; point->line->plane->volume and so on.

And to do so, as a fractal can always be divided in sub-fractals, mathematical disciplines subdivide further at all levels in 5 elements.

**THE RASHOMON TRUTH OF MATHEMATICAL SYSTEMS**

It follows then from the definition of the 5 elements of all systems, an immediate classification of the five fundamental sub disciplines of mathematics specialised in the study of each of those 5 dimensions of space-time:

**S: ¬E Geometry**studies fractal points of**simultaneous****space, ∆-1,**& its ∆º networks, within an ∆+1 world domain.**T§: Number**theory studies time**sequences**and ∆-1 social numbers,which gather in ∆º functions, part of ∆+1 functionals.**S≈T: ¬Ælgebra**studies ∆º A(nti)symmetries between space and time dimensions and its complex ∆+1 structures… Namely is the science of the operandi <≈> translated into mathematical mirrors.**∆±¡ st: ∆nalysis**studies the motions, STeps and social gatherings derived of algebraic symmetries between functions and numbers (first derivatives/integrals), and the wider motions between scales of the fifth dimension (higher degree ∫∂ functions).**@: Finally Analytic geometry**represents the different mental points of view, self-centred into a system of coordinates, or ‘worldviews’ of a fractal point, of which naturally emerge 3 ‘different’ perspectives according to the 3 ‘sub-equations’ of the fractal generator: $p: toroid Pov < ST: Cartesian Plane > ðƒ: Polar co-ordinates.

To which we can add the specific @-humind elements (human biased mathematics) and its errors of comprehension of mathematics limited by our ego paradox **Philosophy of mathematics** and its ‘selfie’ axiomatic methods of truth, which tries to ‘reduce’ the properties of the Universe to the limited description provided by the limited version of mathematics, known as Euclidean math (with an added single 5th non-E Postulate) and Aristotelian logic (A->B single causality). This limit must be expanded as we do with Non-Æ vital mathematics and the study of Maths within culture, as a language of History, used mostly by the western military lineal tradition, closely connected with the errors of mathematical physics.

In this post we shall deal with the a(nti)symmetries and operandi of Algebra.

**Though**we cannot be exhaustive ias Algebra is the largest of all the sub disciplines of maths, concerned with time-space symmetries, but merely do a very fast analysis of its main themes, enlightened with ∆st insights to show the enormous power to ad new whys to reality and all its stience of 5D², since even the simplest truths scientists think are thoroughly understood have new insights observed when putting the Kaleidoscopic 3±∆ perspectives of the fractal organic Universe, what we call honouring one of my fav films, the ‘Rashomon effect’ (a truth can only be ‘judged’ with multiple ∆@ST perspectives.

**ALGEBRA AS A MIRROR OF**

**S≤ST≥T GENERATOR A(NTI)SYMMETRIES**

So WE redefine BOTH in terms of THE 5 Dimotions of time-space:

*“Algebra is the study of ST<≈>TS a(nti)symmetries BETWEEN space-time dimensions, hence focused in the Spatial p.o.v. and ITS OPERATIONS’ (we use often as synonymous the Latin term operandi, as usual trying to change slightly the jargon of stience from that of classi science).*

*Analysis is the study of small ST<≈>TS stop and go motions THROUGH space-time dimensions, hence focused in the Temporal=Change p.o.v. and ‘THE VARIABLES.’*

*dynamically substituted the main operandi of algebra, = equality, a mirage of the stop-formal-mind by the more accurate dynamic, ≈, symbol of feed-back and flow and similarity, that algebra mirrors the different steps=quanta of motion, S≈t≈t≈s≤t… of the GST universe.*

*since ∆nalysis deals precisely with ‘finitesimals’ that become whole wave-bodies of energy, integrated in space (∫∆-1=∆), and viceversa, with its ‘time steps=motion quanta’ ‘derivated=absorbed’ by the whole from a volume of space.*

So while the elements of algebra and analysis – equations of SŒTS is the same, the focus on spatial form (algebra) or temporal motion (analysis) makes them diverge. We can play with the acronym and say that Algebra deals with SŒTs (Space-dominant structures) and analysis with TŒS, Time dominant ones (in motion)

**Algebra operandi.**

It follows that more important than ‘variables’ are to algebra ‘operandi’, whose encoded meaning and ‘magic’ way of relating systems to get a ‘future or present’ outcome by merging them according to certain rules of ‘creative engagement’, truly gives the power to algebra to mirror the a(nti)symmetries of the Universe.

The key connector of T.Œ with classic science is the full understanding of the dual algebra operandi, ±, x/, ∂∫, √xª as part of the classic logic game.

It is immediate the correspondence of those operandi with the dimensional elements ∆st, as:

- The sum-rest are the inverse arrows of the simplest superpositions of dimensions
*between species which are identical in motion and form.* - The product/division rises the complexity of operandi a first layer, and serves the purpose, besides the obvious sum of sums, of calculating the margin of dimensions, as combinations which are not purely parallel between clone beings, most likely through the recombination of its ∆-1 elements, as the product of 2 Sœts inner elements give us all possible combinations. Ie. 5 x 4 = 20 IS also the number of connections between all the 5 elements and 4 elements of both sets. So multiplication ads either a dimension of multiple sums in the same plane, or probes for the first time in an inner scalar dimension.

- Then we arrive finally to the potency-root systems and integral-derivatives, which operate fully on the ∆§cales and planes of the system, which require two slightly different operandi. As §¹º ‘social decametric scales’ are lineal, regular, so we can operate them with potencies, roots and logarithms.
- ∂∫ But when we change between scales into new wholes and new planes of existence we are into ‘a different species’ and so we need to operate with the magic of finitesimal derivatives and analytical integrals, which keep a better track of the infinitesimal ‘curved’ exponential changes that happen between two planes, where linearity is lost.

In that regard the main difference between polynomials/logarithms vs derivatives and integrals is dual:

*Derivatives & integrals often transcend planes relating wholes and parts, studying change of complex organic structures through its internal changes in ages and form.*

*Polynomials are better suited for simpler systems, scales of social herds and dimensional volumes of space, with a ‘lineal’ social structure of simple growth.*

*as the Universe in social space has only 3 dimensions, before a discontinuity of ∆-planes is reached, ever since the infamous Fermat’s THEOREM we know that beyond quadratic x² equations, systems are not alway possible, solutions are not ever found and the humind wasted (as it does today with string theory) incredible amounts of ink to ‘invent solution’ to unreal 5D Universes (no solutions of quintics).*

*In the graph both in algebra and geometry there are ‘inflationary solutions’ of more than 3 dimensions in space, which are meaningless in nature, due to the discontinuity between planes, beyond 3 st dimensions.*

*Hence the importance acquired not so much by what can be resolved in Algebra but by what CANNOT, that is systems which do not reflect nature’s holographic, bidimensional nature, and hence have not automatic ‘radicals’, which occupied most of the historic development of classic algebra (resolution of polynomials, i>2).*

*reduction of a polynomial to products of Holographic bidimensional equations becomes then one of the strongest proofs of the Holographic principle of SxT dual dimensions.*

*which should be the true focus of algebra, as the operations allowed between space and time dimensions.*

*placing all the elements of the equation in one side, leaving the right side in zero, which ELIMINATES THE SYMMETRY between two possible sides of the equation.*

*as it gets further in depth into the dualities, paradoxes, symmetries, scales, hidden variables, multiple social groups that interact, interconnect, network, web and finally create reality; even if ultimately as a fourier transform, it can be decomposed in its minimal units of formal motion, the ststtss beats of reality, which NEVER ceases to move through s-t symmetric steps.*

**The search for ‘wholeness’ in a single equation.**

*find wholeness, and stop all motions into puzzled mappings that try to enclose it all, all the steps, all the variations, all the forms, and all the motions, in a single algebraic structure, a potential equation of all possible bifurcations of those steps, which humans think to have achieved with those ‘monstrous lie groups’ and other algebraic structures that try to be an impossible minimalist mirror that encodes the information of all the st-symetric steps available to reality.*

*0-finitesimal spatial mind x ∞ time cycles = Constant mind-world: §@<≈>∆ð*

**The rashomon truths: 3±i Asymmetries of operandi.**

*different uses, in terms of space, time, symmetry, scale and mind distorted pov. And so it will widen its applications but will require also an in-depth philosophical analysis of each of them.*

**The 3 scales of algebra.**

**ðNº≈§•:**, meaning that time numbers are put in relationship with spatial points, and time functions with space functions through the operandi of algebra.

*The sequential number*, which is the temporal version of the fractal Œ-point as its minimal unit, whose properties are ‘narrated’ in number theory.

*The function and its operandi,*which expresses normally a partial equation of the ternary fractal generator, most often of

*the space and time states, connected by the function and operandi, overwhelmingly the basic equation of mathematical physics (as in E(s)=M(t)c².*

*The structure,*which is the highest comprehensive analysis of full ‘blocks of time-space’ laws; and specifically encloses all its possible variations; today developed through

*the concept of Group*as a ‘receptacle’ of all the elements of the being, with a focus on its spatial symmetries and its similar expansion with the concept of the functional (function of functions) with a focus on its ∆§cale symmetries.

**S≈t. The main operandi of algebra as reflection of the st, topological, temporal and ∆§ symmetries.**

**≈, ≤,≥,**operandi IS the basic S≈t step connection

**∫∂**is the basic ∆±i connection

**±, x/**are the connection the basic ∆§ocial scales between planes operandi

**xª, log x**connect in a more ‘rudimentary’, less focused version of the ∫∂ ∆±i operandi

**(G, •,*)**connection as reflection of the variations of species, complex transformations between time, space and space-time (energy) states, and other symmetries of the generator; where we shall call G, no longer group but Generator.

**The 3 ages of algebra.**

**How algebra becomes so complex?** Let us understand that this is only a recent development.

**1st age of ‘arithmetics’.** Single ∆§ plane and ternary dimensions and S=T equation-symmetries.

In its original form, algebra dealt with mathematical operations on numbers considered from a formal point of view, in abstraction from given concrete numbers. So *it was really a way to calculate social numbers in their growth and diminution in herds (±); in its growth and **diminution in 3 spatial dimensions (X, Y)³. *

*A*nd* as such it could not go further because there are only 3 dimensions in a single space-time plane – **Fermat’s grand theorem, being a lateral case of the difficulty to create by superposition in a ‘higher dimension, a simple social sum, such as x³+y³≠z³.*

Hence the seflie teaser of some of this blogs’ post – that my fellow basque countryman Mr. Fermat might have known 5D², cause that is the only proof that fits in his Apollodorus’ margin (:

This impossible ‘simple’ problems are thus PROOF of profound LIMITS in 5D Universes – equivalent to the classic problem of doubling the cube in bidimensional geometry, also known as the Delian problem: Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first, using only the tools of a compass and straightedge. As with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible.

And it shows that geometry and algebra, points are numbers are ≈ similar but not =. So some numbers do NOT exist (√2, pi) as such but are ratios, which therefore can be found better in geometry and viceversa, some geometrical figures are neither pure forms or proportions and cannot be found with compass.

So while geometry was first born in a single plane of space-time ITS FUNDAMENTAL PROPERTY, arithmetics soon allowed to study at least the social evolution of numbers in a single plane of space-time. And this was the birth of Algebra, with its subtle continuous-discontinuous variations of geometry vs. arithmetics – NEVER UNDERSTOOD even to exist as Euclid had defined absolute equality in its wrong axiomatic method that has weighted so heavily on human thought and its ego-trips of absolute truths.

Equality does not exist only similarity. In any case the first age of algebra ended with the limiting study of a single space-time social evolution in 3 dimensions and its decametric scales.

**II Age of ‘Algebra’**: development of letters as parameters, mathematical physics and ∆-nalysis. The age of functions.

Things got interesting when algebra started to focus ON variable parameters which represented T.œs and its Disomoprhisms and symmetries, flows, and ST-eps of the different motions of beings in space-time.

This abstraction found expression in the fact that in algebra magnitudes are denoted by letters, on which calculations are carried out according to well-known formal rules.

Algebra now considers “magnitudes” of a much more general nature than numbers, and studies operations on these “magnitudes” which are to some extent analogous in their formal properties to the ordinary operations of arithmetic: addition, subtraction, multiplication, and division, adding *inversions of 4D-5D, ∫∂ and log/xª*.

And it adds dimension of motion to the original geometric calculations, hence the need for ‘variables’, operandi of change, and finally vector magnitudes, which include both form and motion dimensions, even if they can still be represented in metal space with geometric rules, as the well-known parallelogram rule of addition.

#### Functions and operandi.

In simple terms, a function *f* is a mathematical rule that assigns to a number *x* (in some number system and possibly with certain limitations on its value) another number *f*(*x*). For example, the function “square” assigns to each number *x* its square *x*^{2}.

The common functions that arise in analysis are thus definable by formulas, *which are related to the ∆s and ∆T duality of functions,* such as:

**§:** Polynomials of the type, *f*(*x*) = *x*^{2}. The logarithmic function log (*x*); & the exponential function exp (*x*) or *e ^{x}* (where

*e*= 2.71828…; and the square root function √

*x*.

**T:** Trigonometric functions, sin (*x*), cos (*x*), tan (*x*), and so on.

**∆: Differential functions.**

It is then also when having algebra completed its main task – the creation of a ‘translative method’ of reality and its dimensions through operandi, the fundamental 4,5D ∫∂ duality takes off as analysis, which becomes the most important field of mathematics, as the scalar Universe do include all other elements within its folding.

So Analysis will be algebra and also number theory and also @nalytic geometry… But not so much geometry which by definition is a spatial, single plane view that only with topology extends to an ∑∆-1 points-parts>Whole geometric forms.

And for that reason as Analysis becomes the natural evolution of algebra into the realm of wholeness and scales, Algebra enters a third age, in which the essential element of algebra, which is NOT polynomials – a rough approach to differential changes, approximated by Taylor/Newtonian binomial methods – but OPERANDI, the study of the relationships between the symmetric sides of the equations, takes a flight of its own, focusing now into the structures created according to the rules of engagement of those operandi.

While a new complete fundamental field as it ‘brings’ a new mathematical species into being, ‘vectorial calculus’ takes a flight of its own, thanks in great measure to the work of the most underrated genius of mathematical physics, and one of my favs for his absolute despise of wealth, power, fame and penpal peership (: mr. heaviside.

**3rd Age of sets, groups and ‘Algebras’.**

Complexity then arouse in the 3rd age of ‘Algebras’, when the jump of complexity was fully realized as usual in the Universe by adding more dimensions of ∆st to the discourse, in this case, trying to create ‘structures’ that embodied all the possible symmetries of space-time of the being, that is, *trying to find the complete equation of the Generator:*

*∆±i, $t≤ST≥§ð.*

Yet as humans ignore the proper focus of existential algebra (the Generator, all its sub-equations and embedded Ðisomorphisms) Algebras became complicated (unfocused complexity = complication), and instead of marvelling at the clarity and synoptic power of the fractal generator, they became increasingly pedantic and the ‘pest of group theory’ (Weyl) settled in, requiring a huge memorial effort (the lack of intelligent understanding bring memorial repetition). And so we enter into the Von Newman age (: the age of nerds, which thanks to computer thought – still thanks god, or else we won’t be here, in their memorial repetitive cyclical algorithm).

And there they are all, trying to find the fractal generator of Everything Organic (T.œ), with ever more complex algebraic Lie groups and statistics (you have lies, damned lies and statistics economists say of their go(l)d religion, we should paraphrase, you have ‘lie algebras, damned groups and statistics 🙂

This said the age of §œT and Group theory *must be modelled as the present foundation of Mathematics in terms of ∆s≈t; as §*

**ocial Generators=Groups of Time Spacœrganisms: Γ§œT.**

And why with its abstract, convoluted reasons, they do actually mirror quite well reality.

The answer is simple. A set is an ensemble of mathematical entities, hence ultimately plugs in directly from the roof down with the social nature of numbers and geometrical points. A group is further on, a fascinating *symmetry between time-motion or space-form. Let us consider this key feature to understand the Universe in formal terms.*

Along all the pages of this blog we have shown an enormous array of examples proving the duality between time-motions and its space-forms. So basically the fundamental symmetry of reality is that we *can express anything as a series of motions in time, which will be related symmetrically or inversely through one of the fundamental operandi of algebra, or any ‘verb’ of language, or any of the logic symbols of space-time flows of 5D (< ≈ >) forming a time-space event.*

And hence *FROM ONE OF BOTH SOLUTIONS WE WILL BE ABLE with the proper methods to extract the other mirror-solution, which we express in the fundamental duality/equation of space-time symmetry:*

**∫@≈∆T**

This is what makes Algebra, once *we apply the Rashomon effect (multiple 2, 3, 4 or 5 D povs and functions, depending on detail) so powerful to mirror real it as it is basically all about S operandi T symmetries. *

But the main task to do in algebra is to fully account for the meaning of all its operandi and establish its relationship with space and time symmetries. I.e. we talk of sum, product, integrals, logarithms and numbers, complex, irrational and so on with very little understanding, happy just with the pedantic, axiomatic method and its pretension of absolute truth – which by Godel is by all means incomplete, and reveals very little on the ultimate disomorphisms of space and time.

So the explanation of operandi and Space-time dualities=symmetries+inversions is what will take us to the very deepest levels of understanding of the Universe.

And as it happens, mostly by chance and with little understanding, this *is what Group theory enables in great measure due to its most remarkable features, one of structure and the other on method:*

STRUCTURE

To define 1 or 2 operations, loosely called ‘product’ and ‘sum’ but which *can be anything as long as it truly defines one of such symmetries because it requires, the 2 fundamental features of a space-time symmetry: inverse elements (where inversion is also loosely defined) and a neutral element, which can be considered the symmetry axis, in a close but quite general symmetry with the 3 elements of the Generator – Sp, its inverse Tƒ and the neutral element, ST… *

So the structure of groups is quite close to the general structure of space-time symmetries and this allows it an enormous flexibility to show all kind of S OPERATION T relationships.

It is for that reason that the concepts associated to Group theory, isomorphisms, transformations, Generations, representations and such are very close to the concepts expressed in 5D²; but as 5D² has a much wider range of applications and a far more realist outlook, it should be clear that Group theory is a mirror of 5D SPACE-TIME and not in the other way (as creationist mathematicians think of).

Consider for example the concept of isomorphism, which I BORROWED not from group theory but from general systems science, my earlier discipline: it is purely structural meaning that if a group its identity elements, inverse elements and ‘product operation’ behave in the same manner two groups are isomorphic to each other.

Disomorphisms go a notch further and affirm that all systems of the Universe are in fact isomorphic as all have the same Dimensional properties deduced from its 5DST CONFIGURATION. So all will go through a life-death cycle. All full working beings will have a ternary topological/functional structure and so on.

Can then we express in terms of GROUP and SET theory 5D2? Likely but it is of not interest because groups and set theory is the humind slightly unfocused mirror on reality, which the generator by looking directly to space and time and its properties describes better.

**METHOD**

Then there is GROUP methodology. And as it turns out any successful structural mirror of the humind on the Universe is always in question of form vs. motion – a trans-form-ation of one into the other to have a new angle… So derivatives in time and integrals in space are inverse functions; algebraic problems in time can be resolved observed their trajectories as topological paths in space, and so on. But in group theory, given its extreme abstraction, kept always in algebraic terms, this duality of methods of finding ‘spatial mappings that the mind can perceive as knowledge’ are a bit hidden and I am not aware that mathematicians understood them.

Consider THE first of those groups discovered, the Galois group which help us to solve polynomials, by *considering a group of all the possible permutations of the parameters-letters of the equation. Then by carefully studying those parameters we can find the solvability of the equation. Why we say this is another St SYMMETRY?*

Obviously because in the original polynomial, *the variables are NOT the coefficients/letters but the ‘variable’ X; and in principle what we want to find is a ‘fixed’ parameter/coefficient/letter. But Group theory does exactly the opposite. It converts the fixed single solution/parameter into the variable by establishing all its permutations.*

And it does get results then by studying carefully which of those permutations of parameters do make sense – are isomorphic to the roots-solutions of the initial X-variable.

Now apply this method to the space-time generator. If we consider time the variable and the organic space form the fixed form, we cannot really treat it as a group. Because the group has only one element, the super organism moving through the world cycle.

*But if we consider the 3 time ages/dimensions, past ≤, present ≈, future ≥ as the elements of a group, where the operation is the Generator, representation of all the Disomorphic beings of the Universe, then present becomes the neutral element, as it remains unchanged and it does not change any relative past field/limb or future head/particle in its form (they do not evolve in present) and past and future become inverse arrows.*

So goes for the ternary scales of beings, ∆±1, which then become 3 elements of another group, where the operator is a Disomorphic ∆±1 scalar organism. And so goes for the 3 topological elements that conform a whole.

So both §œts which study elements of mathematics in groups and ultimately should study societies of numbers and geometric points and functions that relate them by various methods and Group theory, with its ‘loose operations’ and symmetric neutral element and inverse S≤≥T ones, are the closest ‘structure’ all comprehensive ‘wholes’ invented by the humind as focus of the space-time symmetries of the Universe.

Hence its value despite many tantrums against their excessive pretension of becoming the ultimate equation of reality, which they re close to but not yet – the generator and perhaps something else in the future revolution of the humind or metal mind is closer.

**Humind death age: Digital thought ≈ Boolean algebra, a new beginning.**

*inverse*numbers, which in a world with single time-arrow logic will always have difficult interpretations.

**operandi**, structure and properties of its operations (10 basic properties of ±, x ÷ and so on).

**S≤≥T ASYMMETRY**

**Γ§: SPATIAL SYNCHRONOUS VIEW**

*and time is far richer as time-motion is all, space being only the humind mapping of reality – or that of a different mind.*

**All mathematical disciplines are experimental hence extract essential ∆•s≈t properties.**

*in discontinuous time numbers certain relationships such as π or √2 are NOT defined, while i geometry do exist.*

*in time pi never closes its cycle, allowing a perpetual motion Universe, but in space, it practically closes in each present the super organism, breaking its vital space from the outer world.*

**¬Ælgebra of parts vs. wholes.**

**numbers**, which are §ocial groups of undistinguishable elements or sequences of lineal time; which being algebra a time-dominant discipline will dominate the algebraic analysis over the spatial point states, meaning most equations write as:

**equations**and its operandi, which establish the relationships in a single or adjacent planes of existence between sequential and social numbers, and its ‘ternary states’ and transformations, social evolutions and topological forms derived of them.

**functionals and groups,**and those will be studied in mathematical physics.

**Foundations of experimental ¬Ælgebra vs. metalanguage foundations.**

**Time-like algebra and space-like topology. Time like probability and space-like statistics.**

**topology**, space with motion hence an St-version of it, and

**modern algebra**, numbers with geometrical structures, hence a Ts-view, which then can be further related by a nice theorem:

*final more complex ‘whole view’ of number theory, from where indeed departed (Fermat’s work, as the founder of time theory and probability).*

**S=t symmetries between algebra and analysis**

*the complexity and creation of new layers and structures come from its*combinations in dualities and ternary symmetries (number and point, s and t; scale and number, ∆ and t; scale and fractal point, ∆ and s dualities and s<st>t, ∆±i, |x0=Ø ternary symmetries).

*capacity to give us a huge amount of information into the most complex symmetries and simultaneous space x sequential time structures of reality*

*in the generator equation, which is the basic generator of ¬Ælgebraic structures, as it embodies the ternary symmetries of time, space and scales.*

*“Algebra studies social, sequential numbers and its relationship with spatial points, extracting the S=t symmetric motions between them’.*

It seems a simple sentence but is huge in meanings.

It shares its minimal mathematical units with number theory, as numbers are closely related to the arrows of eusocial evolution of the 5th dimension and the arrow of discrete ‘frequency’ in time. As such numbers measure the ‘long’ , social evolutionary game of scales, and ‘short’ sequential arrows of time of the Universe.

While ¬Ælgebra truly starts in the next scale of space-time events, with multiple groups of points (variables), exchanging entropy, energy and information, through operandi (specific of each action) in ‘equations’ (which become partial case of the generator’s allowed st exchanges and symmetries). And so this classic algebra is the most important part of it.

Whereas modern algebra and its study of the whole ‘block of time-space’ of a supœrganism, through all its potential actions expressed in **functionals** of functions that embed two or even 3 scales in whole ∆±1 structures, *is the all inclusive ∆±i perspective, while* **groups** of symmetries, according to a set of ‘restrictions’ of its allowed actions, is the single ‘space-time’ plane attempt to describe the whole potential transformation of any entity in such a single plane. So groups and functionals explored in the 3rd age of algebra complete its wholeness.

This FIRST VIEW of Algebra can be done through its analysis in space as a simultaneous connection of the two sides of an equation, which put in relationship the ‘ternary extremes of an ∆st element’, either by its operandi (which allows a single plane connection of s and t elements) by its polynomials (which allows a multiple scale connection through ∆§) and by its inverse symmetries (group connection).

**Operandi, polynomial limits and group transformations.**

All this say it is obvious that the essential elements of *algebra are its operand that reflect fundamentally two type of 5D² symmetries in its equations:*

**Γ: S≤≥ST≤≥§ð;** whereas the *operandi of i-logic < ≈ > are translated into simplified ‘identical’ (not self-similar) operandi, =, or < > operandi, in spatial terms (hence simplifying their scalar and informative or temporal simultaneous meaning).*

**∆±i Operandi:** Which must be inverse operandi, as *only the 4D vs. 5D entropic and informative arrows are truly inverse in its Nature; that is, forming a perfect zero sum. So they can be obtained as inverted operandi, either:*

*In ‘continuous’ social growth of scales within an ∆-plane: ±; ÷ X; xª log x.*- Or in
*discontinuous*scales through integral processes (which differ slightly fro the logarithimic/polynomial scale, hence the possibility to approximate polynomials with derivatives of different order, as per Newton, Taylor etc.)

Alas we have resumed in a footnote the correspondence between Algebra and reality (: the rest are details…

It is then obvious that algebra will give us the most thorough understanding of the processes of the 5D Universe as a synoptic mirror, and there is no magic to it, but the lack of a proper philosophy of science in its practitioners.

WE shall then call this enlightened algebra plus whys, ¬Ælgebra, or ‘Existential Algebra’.

Algebra of Existence is mostly about equations that represent motions on the dimensions of space-time, which ALWAYS HAPPENS AS A STOP AND GO PROCESS, HENCE AN S OPERANDI TIME MOTION AND TRANS-FORM-ATION IN A BALANCED WAY OF TWO DIFFERENT ST-POLES.

as such algebra deals with S operandi T operandi S operandi T Cyclical ST@tes; where the @ monad author of the @tions of existences of the system do take place in sequential discrete manner…

The key to it all as algebra deals with the S**<≈>**T SYMMETRIES ARE THE **<≈>** logic operandi OF ITS EQUATIONS and its S=t flows and transformations embedded on them.

Mathematics derives from logic, as the operators of algebra are always logic statements about the relationships between the variables and parameters of the equation. So we must relate properly what mathematical equations mean, by comparing them to the Generator Equation written in ¬Æ symbols.

So a part of ¬Æ of the greatest importance is to fully understand from the perspective of Non-E mathematics, where points have parts and so they are never equal but similar.

Thus the true meaning of the key logic symbols of mathematics, such as =, must be < =>, a transformation; or ≈, a similarity or a communication, (as in E=Mc2, where Mass is NOT energy but the transformation of lineal energetic space into cyclical time vortices).

The confusion of truths caused by the misunderstanding of the ‘dynamic’ presents represented by those symbols is not trivial. Many errors of sciences derive because the ‘evident, fixed mind of the scholar’ just says ‘equal’ and thinks really mass is equal to energy because a transformation converts a lineal motion (kinetic energy) into an accelerated vortex of gravitational time (a mass), which are as different as a living and dead corpse.

The highest homology with reality: Semantics, Syntax and Growing Sentences of mathematical operandi.

Now for understanding algebra, it is useful to consider the fact that numbers are points and there is a direct relationship between points and numbers, lines and variables, planes and squares and so on. So polynomials and operandi represent ‘social evolution’ of dimensions, points into lines into planes into 5dimensional structures.

Departing from those elements of the Universe closely imitated by Sp (Points)<=>Ti (informative Numbers), its sequential processes of growth into new planes of the 5th dimension ARE the next element of mathematics:

– Defined in the spatial symmetry of points by its 5 geometric Postulates (points->lines->parallels/perpendiculars->planes->points with parallels), which we upgrade to Non-Euclidean postulates (Fractal points->waves->Similarity->Networks->Minds that knot planes into organisms).

– And Defined in the temporal symmetry through operations of algebra (sum/rest->multiplication/division->potentiation/logarithm->integration/differentiation), which we shall clarify in its awesome capacity to study ALL the actions of space-time of those knots of cycles.

So this ‘reclassification of mathematics’, will divide the subject basically in a duality of perception proper of most mental languages: spatial still, continuous perception (geometry) vs. Informative discrete perception (numbers).

And then classify all operations and geometric postulates as the necessary elements to grow those dual elements into ‘larger’ groups, through scales of the 5th dimension (positive arrow of future) and its inverse operations that dissolve, divide, disintegrate them (negative arrow of future).

In this manner mathematics is able to explain better than any other known language (except in questions referential to human existence for which our natural language, words works better – which should therefore rule through ethics, politics, economics and the creation of our social super organisms,).

Now this should be clarified first before trying to streamline mathematics as it is today understood, due to the fact that without knowing ‘WHAT IS ALL ABOUT’, man is loosing the necessary relationship of all classic languages:

Syntax & semantics of ‘sentences’ (language reductionism of reality to fit the brain, which normally ‘reduces’ motion’ and simplifies layered dimensions and limits perception to relevant cycles) -> More complex reality of ∞ space-time cycles with motion.

So the semantics are points-numbers, the syntax are operandi, and the sentences are planes/equations, which grow in complexity as the ‘chains’ of those equations and planes grow through ever more complex, integrative, ‘operandi’ (sum/rest->multiplication/division->potency/logarithm/integration/differentiation).

But then we realize that is all. Yes, THAT IS ALL. Why? Because as we shall stress once and again, the Universe is infinite yet, its perception by any MIND is not – it is a ‘finite game of finitesimals’ as opposed to the infinity of the whole, meaning that information is lost beyond the ternary limits of at least ‘the human mind’ (might there be minds who see across 9 scales, etc.)

So we DO have just the need to reflect ‘ternary games, scales, elements’ wrapped up then by a ‘whole temporal cycle of maximal motion, integrative activity’ which becomes a finitesimal of a new finite game (hidden its inner parts within the finitesimal point). And so we do START afresh a new game.

So there are no need for more than 3 ‘scales’ of ‘formal operandi’ (±; x/; lnª) which then are ‘warped’ by the dynamic ∫∂ operations.

Now there is a field which is truly new, that of the causal logic of 3 arrows of time, where the most important equation is:

Space (past) x Time (future) = Present space time,…

and through those new symbols of equality, S<=>Tƒ, we define systems as dual, dialectic, where thesis and synthesis fusion together. But this would not be analyzed here as it is more proper to study it on the general laws of the first line and the isomorphisms of the third line.

Why we still need to explain classic algebra and analysis. The ego abstract paradox

Since all sciences are studied from a mathematical and logic perspective, we shall study the ternary logic on those analysis.

Even for those sciences that ‘anthropomorphic’ man, under the ego paradox see too simplex.

Fact is the ego paradox – all finitesimals see themselves as the center of the world they perceive from their point of view – does not give an organic perspective to other 5D, Œ-points. BUT in as much as all points do follow mathematical laws, the explanation of maths as an upgraded ORGANIC science (5D analysis), with complex, vital causal, times sequences and space-time symmetries, across several of those planes (algebraic functions) give us a true sense of the vital Universe in formal terms, connecting properly T.Œ with the present formalism of most sciences.

Fortunately unlike lineal physics, mathematics is ‘well-done’ and already closely related to 5D ‘analysis’ of ‘fractal, topological’ space and cyclical, sequential time.

Thus we do not need to make deep changes into the discipline but merely update some parts (notably the concept of time, and the dialectic logic of operandi) explain others (the meaning of those operandi), and limit its ‘inflationary information’ (concepts such as infinite quantities, new abstractions that substitute the real elements of mathematics, the number, the point and its social geometries, such as set theory).

To that aim though we need to introduce a simple formalism of the main symmetries of the 5th dimension.

Thus algebra is the time, discrete-number related to spatial points part of maths, and as such is a unifying thread of almost all of mathematics, which has also gone through 3 ages of growing completeness:

We thus have now put ¬Ælgebra as the central discipline of maths, in relationship with its ages, other math disciplines, math physics and reality.

We shall go through the development of those central elements of classic algebra in this post studying its main postulates along their illumination of some of the fundamental laws of the ∆@S=T Universal structures, where algebra, the most comprehensive ‘linguistic, human-mind related language of the whole, extends its tentacles better than geometry>topology, basically a spatial endeavour, and analysis, basically an ∆-scalar endeavour.

*So a better definition of ¬Ælgebra is to be the central discipline of maths, where all the subjects of reality can be treated.*

But to that aim we need to understand better ‘social sequential numbers: N§º’; single aristotelian logic made into ternary ‘i-logic’; imaginary coordinates which express whole ‘worldcycles of holographic bidimensional numbers’ and so on…

**The ‘structure’ of Mathematical languages.**

The Nature of mathematics as a language of space-time (geometry and numbers) can also be expressed with the modern conception of mathematics, based in the work of Cantor (theory of sets) and Boole (Computer Algebra), which was summarized by the Collective Bourbaki, whose ‘Mathematical elements’ reduce all mathematical disciplines, to 3 structures, themselves reduced to set theory:

– *Structures of Order*, understood as structures, which describe the hierarchical order <, > of numbers, which we just described as social, geometrical groups. Thus, those structures are mere reflections of the 4^{th} arrow of eusocial ‘love’ that creates social groups.

– *Topological Structures.* They describe the basic forms of the Universe, which in a 4 dimensional reality are only 3: the hyperbolic, informative geometry; the cyclical, reproductive toroid and the spherical, external, energetic membrane, defining all relative worlds/species as complementary systems of reproductive energy and information. We shall show this simple ternary, topological structures, which are the basis of all forms of reality in our analysis of the 5 non-Euclidean postulates.

To mention that topology is in fact the ultimate science of space from where we can derive all the other spatial structures, as geometry was the basis of classic mathematics in its Greek inception. In fact topology and set theory are quite similar in its laws. What topology adds to geometry is ‘motion’ and ‘relativity’; thus including the Galilean paradox in the structure of mathematics, since two topological structures are equal regardless of the ‘distance/size/motion’ of its surfaces.

– And *Algebraic structures, *which study the complex arrows of Time born of the ‘operations’ between simplex arrows and are ultimately expressions of the Generator equation of space-time, E<=>I, in as much as al algebraic equations are of the type F(x)=G(y), which is just a particular case of the Universal syntax/grammar of reality of the Generator feed-back equation.

Thus, there is a direct relationship between the 3 fields that define all mathematical structures and the Temporal, Spatial and Combined (algebraic) operations between time arrows, which simply means that mathematics has always been the human language that describes the ultimate reality: Discontinuous Spaces (topology) and Relational Times (causality); and the complex interrelationships between them (Algebra).

While the 2 first facts (the identity between space and topology and Time and causal order) are intuitive, we might conclude this overview of the meaning of mathematics, with a brief introduction to the meaning of Cantor’s sets, and Boolean Algebras used for Mathematical computing, as an expression of all Mathematical Structures and a formalism of Multiple Spaces-Times. We shall by doing show, prove, that all mathematical structures are explanations of properties of the ultimate reality: the timespace defined by the 4 arrows of the Universe and all its species.

*Recap.* Mathematics can be reduced to 3 types of structures: topological/spatial structures, structures of causal, temporal order; and algebraic structures that define the outcome of complex relationships between Multiple Spaces Times. Thus i-logic mathematics is the synoptic language the human mind uses to describe all what exists: a Universe of spatial energy and temporal information.

# 3 TIME AGES

**Synopsis. The 3 Ages of Algebra.**

So we end this intro resuming the 3 fields and ages of algebra, with its ternary generator:

Youth: ∆-1: Arithmetics, the Greek Age, now mostly studied in number theory.

Maturity: ∆: Equations which extend each ‘letter’ to a range of numbers, *creating a new ∆-scale of generality for maths; mostly studied in @nalytic geometry*.

**1ST & 2nd AGE OF Algebra: Social Numbers and geometric equations.**

The second aspect of mathematics deals with sequential numbers and equations, in which sets of numbers are transformed by an X=Y function, in which often one component changes faster than the other, tracing a curve in a Cartesian Plane plotted with those 2 variables that mathematicians study with great detail. It is the thesis of this work that all those X=Y functions and differential equations represent particular studies of a general ∑Se<=>∏Ti equation, and or a partial event between two or n-points of a non-Euclidean network, and or the network and its environment.

Yet as Einstein put it to Poincare: ‘while I know when mathematics are truth I don’t know when they are real’, meaning that many mathematical equations and functions do not exist in nature, as they are not partial cases of the Generator equation and do not respond to the restrictions the Ternary method imposes to a Universe of multiple spaces and times but only 4 Dimensional arrows. In that regard, the laws of multiple spaces and times and the syntax of the Generator equation with a limited number of variations restrict the possible mathematical realities there is in the physical Universe. On the other its study provides the scientist with a deeper meaning for the Algebra of numbers and the meaning of equations and functions.

*Recap*. We shall consider merely the meaning of the main mathematical operations within the restricted world of 4-Dimensional spaces and times we live in. And analyse in more detail some of the parameters and functions most commonly found in the study of the Generator equation, which connects the equation of space-time cycles with the detailed mathematical analysis of those cycles by different disciplines.

** Theory of numbers.**

Numbers in this new outlook are not only intervals of a one-dimensional straight line, but as Pythagoras and Plato stressed, they are geometrical forms:

Mathematics is concerned with 2 seemingly different worlds, the geometry of spaces and the logic of numbers. To fusion both requires to understand numbers as forms. A number is not only an abstract set but always a collection of self-similar beings extended over a common vital space, a network And so networks create complex forms, topologies of space-time, as the motions between points of the networks become stable exchanges of energy and information between two polar points. Yet since each number is a geometrical form no longer limited to the simplest one dimensional form but can vary its geometry and hence its function, degrees of freedom and complexity as we increase its ‘number’.

The 4th Non-E postulate shows how points, numbers, the self-similar class of equal forms create geometries:

The line is simple. The line joins two points and can only have a combination.

The triangle can only have a closed combination, but 3 possible open combinations, Ab, Ac, Bc.

The quadrangle is more complex. It can be joined in 2 combinations, as a cross and a square. And it can be left as an open snake with 3 different orientations. So a foursome acquires a snake shape to move with the arrow of energy; a crossed form to perceive in its center ‘5^{th} point’ and a square shape to accumulate and reproduce its internal organs; and so each shape of the same number becomes a topology with a different form and function.

Indeed, function and form are now fusioned. So certain numbers in its ‘degrees’ of freedom of form, represent certain functions. The quadrangle can store energy, but in a zigzag open line it can move – spend energy and as a cross it can gauge information. Numbers also define arrows of time. So for example, 1 lonely number without motion is perceiving, with motion is processing energy, 1+1 might be 3 (an act of reproduction) or 1 (an act of Darwinian feeding). All those vital actions determine that certain numbers survival better than others. So, 1, 3 and 4 are very common systems.

In that regard, a complex analysis of the simplest numbers shows that the more perfect form is the 10-cellular system or tetrarkys, in which 3 x 3 triangular corners act as organs of energy, information and reproduction with a 10^{th} central element that communicates all others and acts as the one of the higher scale, representing the entire organism.

Thus as the number of cells grows, the topology of the system will grow in degrees of freedom and complexity till resembling more and more the repetitive, geometrical forms of social organisms. Topologies become thus at the end, complex networks, adapted to different functions of complex organisms.

As abstract as all this might seem, when observing nature we shall see how those type of events, waves and social planes happen in all the scales of the Universe, from atoms which form crystal networks based in the equality of the same atoms or at best in the existence of a ‘body-mass’ of equal atoms intersected by a few ‘stronger’ atoms that form a complementary network of higher resistance, to the body rejection of cells with different DNA.

* *What things we can do with numbers can reflect then many of the actions of networks. For example:

– We can study how social groups organize themselves or fluctuate between states=functions. This is the study of the internal point of view of networks as a collection of self-similar points. Those changes of states are often defined by a differential equation as informative systems have less spatial extension/motion but are more complex networks with more bits of information=points. Thus differential equations, most of them of the type Y (ti) = aX^{3}±bX^{2} ± cX ±D, express ∑Se<=>∏Ti transformations, where Ti is a network in 2 or 3 dimensions of time bits, bits of information and Se is a network with one (same organism) or 2 (Darwinian feeding) scales of lesser complexity than Y, such as f(x)=Y^{n.} It follows from the Fermat Theorem that there is a restriction to the number of solutions a system can find, which is n=3, the maximal number of dimensions an informative sphere can have as it displaces itself over a plane of energy.

The relationships between limbs and heads that exchange in a 3^{rd} region called body, form and motion, such as the head designs the motions of the limbs, which move the head, and both exchange in an intermediate region of elliptic nature called body more subtle types of form and motion to create more complex cycles that will in fact reproduce both systems can be mathematized in infinite different ways, using matrix, combinatory theory, differential equations, polynomials, Riemann surfaces, etc.

– We can study how networks grow and multiply creating new species and we can add them and observe how they reorganize creating curves which are differentiable to obtain the rate of grown and diminution of the organic population. The study of herds of energy and networks of information in its life cycle is one of the key disciplines of all sciences specially physics and ecology.

– We can study them as networks with form through its geometrical ways of exchanging energy and form, from the simplest point to the line of 2, the triangle without a central focus, the structure of energy, which can however turn into a pi-cycle, the 3, the 4 with its zigzag, solid quadrangle and cross structures, the 5 and first 3 dimensional structure, and so on.

Each number will increase the possibilities of the game, yet when we reach 10 we play a perfect game with 3 triangles that act as organs of energy, information and reproduction, and a central point both in a 2-dimensional or 3-dimensional geometry, acting as the collective action/will/intersection/knot of all cycles – the first clear, complete ego structure in 3 dimensions with perfect form and complementarity. Thus beyond 10, while some numbers might bring slight improvements to the cell, most forms are just growths of the primary numbers in multiple associations.

*Recap*. All the structures of mathematics, regarding of the notation we use, reflect events and forms of knots of time arrows (st-points or numbers), as mathematics is a language whose grammar derives from the Universal grammar of spacetime. Numbers are thus formal networks that try to achieve the essential arrows of time. And so certain numbers (1, 2, 4, 5, 7, 10) deploy better those arrows and are the commonest on nature.

– We can study the evolution and reproductive creation of new networks with successions and combinatory is important in multiple time-spaces since we find always complementary systems of reproductive energy and information, each one with a ternary choice of evolving differentiation (energetic, informative and balanced species). So especially in the classification of species of different sciences we shall find simple combinatory laws that explains the differentiation in 3, 6, 8 and 10 elements depending on the triads and dualities of multiple space-time systems.

– We can study a key antisymmetry of time and space expressed with the language of probabilities: Sequential events are studied with probabilities in time, whose symmetry in space are the study of percentages of populations in space, such as if each event in time is the birth of an individual of a population both probabilities and percentages are the same.

This confused physicists in some cases, as in an electronic nebulae, which is a population of fractal electrons in space, but it is studied as time probabilities, and created the bizarre theory of multiple universes (multiple, probable electrons) instead of a fractal Universe (fractal self-similar micro-electrons, which are bundles of ultra-dense light forming a nebulae which also acts as a ‘whole’ electron, self-similar to its parts). Thus the study of probabilities in time events and growing populations of a wave of space-time cycles is an essential tool: we can study the proportions, herds, groups and networks of self-similar st-points in its evolution either with probabilities or differential equations.

*Recap*. Probabilities study causal events in time and populations in space; combinatory studies the differentiations of species according to the variations of bodies and heads.

**BAROQUE AGE: §ŒTS**

** 3rd age: ∆+1 age of Algebra**, or group and functionals, which deal with a ‘timeless’ attempt to include all the potential duality and ternary symmetries of the ∆@S=T universe split into:

∆lgebra represented by Functionals, which are equations of equations – a complete ‘scaling’ of algebra, from concrete ‘social numbers’ and ‘sequential numbers’ (space and time view of numbers) to the wider laws of ∆@S≈T geometry, in which each ‘fractal point-number’ encodes an entire ∆-1 scale defined by its ‘function’.

*Groups: Algebra as the expression of Space-time motions.*

Special mention on modern algebra, deserves group theory, which is the modern expression of ST-motions, hence concerned with the other ‘dual side’ of the Universe, motions in a single plane of the 5th dimension:

Algebra, being concerned with ‘spacetime motions’ required a final frontier – a way to represent motions in time, with a synoptic language that ‘extracts’ all what is repetitive in those motions, including motion itself to bring the essence of it into the ‘reduced syntax’ of spatial symmetry – as *when we enclose the whole of all time views, we obtain a spatial synchronous still mind view. *This is what group theory does in its comprehensive wholeness, converting the entire range of time motions and events into a ‘fixed whole view’ of it, as a whole spatial range of all potential variations seen in simultaneity (the group).

Alas! thanks to the genius of Galois, we do have *an extraordinary new field of Algebra, which is essential to the most advanced formalisms of mathematical physics, and its study of reality.*

**Group ****theory.**

The GALOIS group in itself is not that important to ∆st as it is mostly about the limits of the holographic principle to create real meaningful systems in polynomials higher than 2 Dimensions. It is the concept of ‘motions and transformations’ in space-time close to an ST-ep or motion on any of the 5D of reality. The Galois group does work though on a basic concept of space-time theory as many other solving principles such as the principle of least action of physics or the modulations of waves: *the need to *find a enclosure and center, an @-mind, an initial and final point to ‘create form; in its infinite manifestations.

In the case of the Galois group the information about the solvability of a polynomial requires *to specify an origin/singularity point which is t*he §œT of all quantities that can be obtained from the coefficients of the equation by means of a finite number of the operations of addition, subtraction, multiplication, and division, called the ground field or domain of rationality of the equation and and end point*, the splitting field of *the roots of this equation – ξ1, ···, ξn – which is the set of quantities that can be obtained by means of a finite number of the operations of addition, subtraction, multiplication, and division starting out from the roots ξ1, ···, ξn, which through Viete’s formulas allow to obtain the coefficients of the equation by means of the operation of addition and multiplication.

Therefore the end or splitting field of an equation always contains its ground field; *which is a general rule of a time causality – the future contains more information in its memorial sequence about the past, than the past about the ‘multiple possibilities’ of the future*. Sometimes these fields coincide, and then we find a cyclical loop of information which is ‘self-contained’ (and hence solvable).

Yet the true mark of solvability *is ultimately the* possibility of reducing the solution of a given equation to that of equations of lower degree; hence making them coincide with the real existence of holographic bidimensional X² and tridimensional X³ sub§œts of the equation; which then can be put in St≈ST symmetries:

The holographic bidimensional universe and its ternary ST-geometries define reality. So in most mathematical equations solutions abound on quartic and cubic systems but only special cases are solvable for higher polynomials or have any real use in reality; the exception being simpler equations of the ∆§ocial scales and reproductive functions of the type Xª=b…

The element then introduced by Galois, which will spread all over the world of mathematical physics regard the permutations; that is *motions in space of §œT that translate the system and keep it ‘invariant’ in its internal form; which we shall study in the 3rd age connected with the geometrical space-time motions of those §œT.*

i.e. The symmetry of the given polynomial is characterized by the collection of those permutations of the variables that, when carried out on the polynomial, leave it unchanged. For example, the symmetry of the polynomial X1³+2X² +X3³+2xˆ4 is characterized by the four permutations:

Groups indeed *are the tool physics uses to study together in a static view the different variations of form and motion in the Universe. Since they represent the mind view of ‘trans-form-ations’ in time, considering the beginning and end of the trans-formative motion.*

*In this manner groups show the symmetries between space-time beings and its states, which are trans-form-ations of a being along one of its ternary ‘Fractal Generator symmetries’*

So, when we state there is a group of possible transformations of a particle changing its isospin, or a rubick cube, changing its face dots, or an equation, changing its coefficients, we are observing the ‘possible’ paths of a being, across the ‘authorised’ operandi that reflect all its possible topological-spatial, age-informative and scalar-∆ motions in the 5th dimensions of ∆•st, formalised in the fractal generator.

How the group ‘freezes all time§pace motions’ then is obvious – ‘eliminating the intermediate ST ‘motion’, WATCHING only the initial and final form… of the cube rotation, the inner changes in the spin paths of the particle, or the particle weak trans-form, only observing the *limiting* results – the time change being frozen and extracted from the group – this ST phase though is essential and the fractal generator will show it, giving us more information on the meaning of groups in mathematical physics.

*So we shall also extend the concept of the 3 fundamental varieties of *Groups, rings, and fields, to fully grasp how group algebra explains the 3±i type of motions of the Universe (topological, temporal and scalar motions or ‘no motion’ at all – mind in-form-ation).

**The baroque and death ages of algebra: axiomatic method, §œt theory; Boolean algebra & chips**

And yet all this is ‘past stuff’ for modern algebra, which is precisely concerned with two fields which a proper understanding of theory and praxis in the Universe will not explore, as they are the 3rd baroque age of ‘§œt theory’, defined as any Social (§) collection of T.œs, which is a superfluous metalinguistic roof as it does NOT depart from the fractal points or social numbers, the T.œs by definition of mathematics, but from its mental Kantian ‘regulative concept’ of final ‘paralogisms’ that ‘fusion all’ into the singularity mind-mirror, in perhaps its finest insight on the synoptic nature of the mind.

Thus this ‘end-point’ of the mind of algebra has obscured the empirical elements, of its main T.œs, S=t symmetries, holographic polynomial bidimensions, fractal points in sequential social numbers, and such. We thus will use the inverse ‘wording’ of T.œ§, which spells basically like set, §œT to define those collections stressing the duality of the two extremes of the 5D ∆-1 ‘ground field, of those points, numbers or elements of the set’ and the mental gathering of them all into a regulative category, the set.

Finally far worse than §œTs are computers and Boolean algebras in a biological senses, as they are substituting atrophying and like extinguishing us in the future. So they mean the death age of human minds. So since Boolean computer algebras and its evolution is eviL=anti-live, for the praxis of survival of humans should be forbidden and we shall according to our life activism against mechanisms that kill life, ignore completely the field.

**RECAP.**

Algebra as the most evolved mathematical language of the humind (ab. human mind) is in that sense very wide.

Algebra is to reality through mathematics what words is to reality through verbal thought – a full developed human language, which did not even translate easily to Digital thought (based in boolean algebra, quite a different type of mind-image of the Universe).

So while topology, space in motion and ∆nalysis, scalar studies of finitesimal derivatives and whole integrals, are much more intuitively related to the reality we observe directly algebra, has gone a bit too far into the human mind/fictions as word have done. Indeed, the whole thing of proving maths with a certain axiomatic, algebraic, set-theory, category theory, you name it, despite Godel’s proof we need experimental evidence, is a huge human ego-trip we do not share. Points in continuous space, sequential causal numbers in time arithmetic/classic algebra, ARE THE SPACE-TIME language, and analysis the ∆-languages. Modern algebra, beyond group theory is too far into the fiction world. So we shall not go so far.

Boolean is to a larger ethic view of man as the measure of all things, dangerous as it is creating a mind that easily competes and will ultimately substitute the human mind.

Yet, there are ethics in languages too, as we can observe in the posts on Stocks and money. So we are not writing much on Boolean Algebra. Only the basic fact that it is an obvious __dual language, which can therefore model all forms of the Universe, as well as a 10-decametric language of numbers, which are at the basis of human algebras.__

Thus we shall escape almost all of the axiomatic, set theory and Boolean Algebras, because ethics of human survival and our limited time does not allow us to dwell in bull$hit.

**The inverse method to 3rd age of maths: less is more.**

Finally to notice that as all languages are inflationary, the expansion of new theorems in the baroque age of maths, is less interesting than to study the limits to the Universe departing from the limits set by the simpler states of mathematics, *prior to its inflationary 3rd age*, and then see how they transform from canonical mathematics into ∆s≈t, allowing us to infer properties of the 5D Universe, *as viewed in the synoptic equations of maths*.

An example will suffice from number theory, as we explain in our demonstration, ‘in the space of a margin’ (: of Fermat’s Grand theorem, which merely means there are no more than 2 superposed holographic dimensions *in each plane of space-time, hence we cannot find x³+y³=z³, as it will mean there is actually a fourth dimension of spacetime in a single plane, achieved by superposition of the other space-time dimensions, which we cannot do.*

Let us then enter into the pre-analytical age understanding the whys of the first age of algebra, with its lineal equations and simple operandi, in a single social plane (we shall not be so much historical as methodical, meaning this age is the §-age of a single plane of space-time and all its S<st>t relationships, the second age is ∆±i planes and ∫∂ and the third, the age of ‘blocks of time’=groups unfocused reflection of the generator.

**ÐISOMORPHIC, A(NTI)SYMMETRIC OPERATORS**

The fundamental element of Algebra are its A(NTI)symmetric operations, which analyse different S≤≥T dimensional transformations, departing from an asymmetric ‘neutral element’ which can dissociate into an annihilating ‘negative operandi’ destructive flow vs. a ‘positive operandi’. integrating arrow.

Classic Algebra came through painstaking foggy idealist schools of axiomatic thought to formalize all this into group theory which is the best/closest approach to the true meaning of algebra we have today but we shall not use but rather as usually adapt it and comment on it with the better focused mirror of T.œ.

Indeed, in T.œ it is very simple to understand that any transformation happening between events and forms which we shall call generically actions, can only have from its initial asymmetry two paths, which are inverse and hence *all mathematical operandi have an inverse function.*

We can thus loosely consider that departing from an asymmetric, ≤≥ encounter, two systems can be operated ‘positively’ into a symmetric whole which ‘ads up’ and empower both elements into a larger creative process, or be operated ‘negatively’ into an antisymmetric whole which ‘subtracts’ as it eliminates one of the elements ab=used by the other. This is the loose concept behind all dual algebraic operandi.

Operations then are the connections between T.œs that define their actions, either as 4D absorbtions=flows of entropy, motion, energy, information or 5D social evolutionary form. Those dual actions are mediated by operations. And so there is first the abstract definition of those operations in mathematical terms, with the study of its properties and then its connection with the dual åctions of 2 beings that enter in communication within a given world-Universe.

The EQUIVALENT of such algebraic numeric analysis being in geometry the study of the topo-biologic properties of non-euclidean ‘waves’ of communication between 2 fractal points (second postulate of non-e geometry):

In the graph we can see how two asymmetric parts, normally one with more form and the other with more motion, come together into a single space-time event super organism, which will either become complementary (gender asymmetry) and evolve socially (which we can generalize to n-points in the 3rd postulate, forming networks) or will enter in a darwinian struggle, and be operated negatively in terms of the 4Dimension of entropy.

What algebraic equations do then is to operate in abstract, with numerical properties those events/superorganisms, 5D evolving vs. 4D devolving; that is 5D, adding vs. 4D resting, 5D multiplying vs 4D dividing, 5D potentiating vs. 4D rooting, 5D integrating vs. 4D derivating the system.

This is really all what there is to it. And because the universe is dual in its inversions and ternary in its elements and scales, 3+ are the operations of math. Since Potency and integral operations being closely related as we explain, as the duality of ∆ curved analysis between planes vs. §lineal scaling in decametric societies happen to be. Indeed any mathematician immediately will notice this duality with the concepts of a derivative vs. a lineal differential approach, a logarithmic scaling that tends to a fixed asymptote vs. a changing tangent-derivative of a more complex curve, and the ultimate proof that all algebraic equations of exponents can be approached by Taylor series of derivatives.

We shall only provide as usual basic concepts leaving for future researchers a full analysis of those terms.

Now all this is explained in classic algebra with group theory and axiomatic methods; so we shall consider that approach. And the only elements left to define then are the identity element and the properties of the operandi.

Indeed, all dual operandi have identity neutral elements and inverse ones. THEY CAN be considered loosely as the neutral=asymmetry form, which ‘splits’ both ways into the negative=antisymmetric element and the positive=symmetric one. The identity element leaves the asymmetry unchanged, it is we might say a non-operation. The negative element is the result of antisymmetry, the positive element of symmetry and tends to be a larger whole.

So for example, if we have a herd of 4 + 3 elements, and they come into parallel social evolution they give us a herd of 7.

But if we have a herd of 4 elements and they come into a darwinian relationship they will separate the defeated elements, so if it is for example a fight for supremacy, only one will be left and we rest the 3 defeated candidates, 4-3=1.

The first curious thought coming out of this simplified analysis is that *positive operations tend to be more restricted additions as they require identical elements, so they are often simple social evolutions, negative operations however might have multiple meanings as the antisymmetric, ‘entropic’ states of a system multiply statistically.*

So this has a reading into the classic arrow of time of thermodynamics, which already noticed that order ‘probabilities’ are less than ‘disordered ones’, *unless there is a maxwellian demon – which we contend do exist in the survival will of all points of time – to contain entropic, destructive probabilities, which is indeed what it happens most often.*

So in brief, when 4 and 3 elements that are numbers, which can be operated together hence equal beings as numbers are social groups of equal beings, *the magic of social love and the fourth postulate of similarity in non-euclidean geometry means that all the systems which fall between symmetry and asymmetry, and can communicate come together. Entropy is REJECTED by almost all systems from the singularity perspective, which implies also that a mere abstract mathematical analysis will fail to understand the bio-logical input of those informative singularities that reject as Maxwellian demons do the negative systems, tricking the ‘dices’ of God. *

What about product vs. division? Again this duality is obvious. If we have 3 x 4 it means 4 herds of 3 which ad together in 4 steps to a 12 herd.

But the entropic events here are also multiple and divisive. I.e. If we have 3 kilos of wheat, we divide the whole between 4 entropic hungry men to get 3/4 per capita. Divisions thus ‘divide’ almost always a system into broken parts. Products multiply societies, or create tighter communication between those smaller parts. I.e. A= 5 and B= 4 elements can be multiplied also at the ∆-1 scale to get the number of axons that ‘tight’ the society together if each element of set A communicate with each of set B, then A x B = 5 x 4 = 20 axons.

So we come to the potency vs. root, integral vs. derivative that merely take this dual process to its final ternary scaling (as we know systems are ternary so no need for further operandi). Its complex study being carried further in the post of analysis so we shall let it go for the time being.

** MATHEMATICAL OPERATIONS AS MIRRORS OF THE ACTIONS, EXTINCTIONS AND CREATION OF 5D T.ŒS.**

In the third age of Algebra, all the operations were resumed in the concept of a group, which has any operandi, two inverse elements and an identity, neutral event.

*So the essence of groups is this: THERE IS A NEUTRAL ELEMENT, THE T.œ, the first being, the fractal point which can go two inverse directions, the inverse elements; AND IT CAN GO those 2 inverse directions through a restricted number of operations, which ARE mirrors of the TERNARY ELEMENTS OF REALITY, ITS SCALES, TOPOLOGIES AND TIME MOTIONS. That’s all folks. Existence in a nutshell.*

And the growth and discovery of new operandi IS exactly what it looks: the expansion of our perception of the dimensions of the Universe departing from the humind mirror, the neutral element that can go up and down, left and right, grow or diminish, D=evolve socially, change its point of view or slide in its existence from the body to the mind or limb state, and finally travel through the planes of the fifth dimension emerging in a larger plane or devolving down to its parts in the moment of death. So all this is what Algebra specially for simpler systems, which as we have seen express those arrows of dimensional form and function through simple geometrical forms and numerical structures of identical particles, social numbers and wholes, IS expressed by the different operations that relate T.œs or rather to keep the correspondence Principle §œTS, social ensembles of Organisms of Time§paœ…

**±**

The first marvel of the Universe is the simplicity of its original principles. Indeed, something so simple as the sum and inverse subtraction IS still the most important operandi of the Universe, which gives us new numbers, social gatherings of identical beings, which herd together into parallel flows adopting most likely a bidimensional ST superposition on laminar states that keep adding the 3rd dimension of the being. Like the simplest first masterpieces of Bach, the architectonical Universe is a simple principle before organicism twists its form, in which beings which are equal come together.

Superposition of bidimensional holographic fields is so important that the whole of quantum physics is based in this superposition principle. The sum thus is still the master of operandi. But for sums to happen, the beings *must be externally identical, to be perceived as parts of a quantified mass, each of them the same value. Addition thus is the ultimate proof of the social nature of the Universe. *

**Negative inverse values.**

Do negative numbers exist? If we consider them only in the sense of a subtraction, 3 – 4 makes no sense as there is NOT *a negative apple. So negative numbers must mean something else, a new dimension not of static form which cannot be negative but of a direction of motion in time which can be inverse. This was something, surprisingly enough as Time is completely ‘opaque’ to the humind (Einstein’s best definition, ‘what a clock measures’ :), that came to be understood only with Gauss in the XIX c.! So a negative number must mean an inverse direction of time, and so by inverse identity a positive number is not only a static form but a positive flow of time. *

Negative beings/numbers thus are an entirely new Universe, when we go beyond the mere detachment or split or scattering of a superposition of identical elements A – B and consider a different type of operations: A <≈> – B, those which put in an a(nti)symmetric relationship two inverse values, as we are now introducing a ‘dynamic dimension of motion’, with the consideration of the negative number as the inverse direction of the positive number.

And this allow us to use negative numbers as ‘antisymmetric’ states, positive numbers as ‘symmetric’ states, and 0 as the ‘identity=asymmetric state’ or ‘seed’ of the split between both inverse directions.

**Negative complex numbers.
**So if a negative number and a positive number have now not only ‘form’ but a dimension of motion, acquired as soon as we introduce a negative side of the coordinates (which by definition did not exist in earlier ‘positive bidimensional greek geometry’), can we reduce back to a single dimension those holographic numbers? Indeed.

We must then consider making a ‘root’ of a holographic number, whose ultimate meaning is the *reduction of a bidimensional quantity – the real number representation – into an imaginary quantity, the negative root. And so the question stands to us. What is the likely nature of the imaginary number ‘root’ – which is after all the root of the negative number? Are we left with the negative pure time motion of the being, or its formal social value? Can we consider imaginary numbers pure time numbers? Or rather the √-1 = i value which does NOT hold volume-form as it is the root of the unit, the fundamental expression of a time direction?*

Just think of it.

So now we are coming to a basic relationship so often forgotten or rather not explored by philosophers of mathematics – a dry subject which has withered away with the absurdity of the axiomatic method that detached numbers of the vital magic of the organic, fractal universe they represent. And we state in a fundamental principle of the ‘Rashomon truths of mathematics’:

*Operandi are related to dimension which are related to the type of numbers, and all together from a closure at the level of the 5th dimension. *

So positive numbers are related to the first simplest dimension of social scales, and growth.

But negative numbers ad a second dimension of motion with its 2 directionalities.

And then *the next type of numbers that matter are NOT real, rational which require more operations but imaginary time number that change the arrow of time of the system. So imaginary numbers are used for time ‘phases’ (i.e. the classic use in AC for the resistor on phase, the capacitive and inductive reactance which are out of phase in the two axis of imaginary numbers by 90º. *

It is then with the use of the 3 positive, ‘lateral’ and inverse negative numbers (Gauss definitions) how can we *operate in the reality of the arrow of social evolution of spatial form (positive natural numbers) and temporal motion, with the relative 3 states of positive future direction, present imaginary state and past, negative dimension. *

The operandi of Algebra thus are the best mathematical mirror representation of space-time dimensions only comparable to the eye>wor(l)d 3 languages – music, rightly understood by Schopenhauer as the purest mirror- language of the space-time Universe, ∆st, in its 3 dimensions of scales, beat and melody, which mathematicians often enjoy so much…

As literature becomes too specific of man, except poetry through its intuitive comprehension of the homologic method and so do painting also too closely referential to reality in bidimensional, ‘informative holographies’ that shrink 5D and suffices to express it all.

So the first type of numbers needed to study algebraic equations are simple natural numbers, and its ‘closure’ adds the dimension of time motion to be able to mirror all phenomena *happening in a single plane of existence in which a unit sphere element is measured, whereas the imaginary number will change the motion of the sequence to the inverse plane*.

*closing then a single plane.*

By Closure mathematicians means that today numbers *can mirror all phenomena of the Universe, once Gauss explained lateral, temporal numbers. But we mean here that addition, rest and natural numbers suffice to describe most social systems as the minimal unit is one. *

So we can translate the ‘closure’ of all phenomena in terms of numbers, for the ∆ & ST elements of reality (broken in 2 sub-generators to simplify the comprehension) such as:

Spe (+ numbers) < St (imaginary lateral) > T (- numbers)

**Closing and solving the meaning of mathematical physics.**

Let us then close for the sake of simplicity the understanding of ‘= simple numbers’ and time cycles, concluding our analysis of the foundations of experimental mathematics. As simple as it gets.

So what are numbers? Again here homunculus as usual ‘get used to them’ and do not understand. Only one humble genius did it, Mr. GAUSS, when he insisted on calling imaginary numbers lateral numbers and negative ones, inverse numbers, but the spatial homunculus missed the point, fixed in ‘materialism’ – for him numbers have to be real space objects, as natural numbers are; but negative numbers are time events, with the inverse arrow of time in a process – i.e. in a graph of speed, they represent the motion backwards, in a graph of entropy and information, the 2 inverse arrows of time – but since alas! by decree the worldly profession of physicists does only recognise the arrow of entropic death in time, the concept of temporal numbers, remained in obscurity. And of course, the ‘bidimensional’ lateral numbers, which represent both arrows together: Spe (positive, entropic motion numbers) + Tiƒ (negative, informative numbers) = ST-wave bidimensional numbers was never understood.

So numbers are the needed units to explain space, time, its energetic space-time combinations and the scales of the fifth dimension, and as usual we started with space numbers, the obvious counting of populations in space with natural numbers, but the visual homunculus has NOT understood much of the others, just ‘found them and get used to them’ – a fact only Von Neumann recognises.

*The Rashomon truths on complex numbers.*

More generally we can consider the Rashomon effect for complex number and consider its 4 different perspectives:

**T(s):**As inverse, TEMPORAL numbers.**S(t):**As bidimensional REAL numbers (Bolay) with its own rules of multiplication (a,b) x (c,d) = (ac-bd; ad+bc)**@:**As a self-centred polar frame of reference (r, α)**∆:**And finally when squaring the coordinates, as an ∆-frame of terence. As this is the new solution found by ∆st theory we shall consider t in more detail.

**t>T<ð**

**Complex numbers as expressions of the 3 states: potential past <present wave> future particle**

Thus the complex plane IS the mathematical representation of a world cycle of time – not a worldline as physicists want it, because they will NEVER recognise the ‘high, dimension of information’ needed to complete a world cycle. So in the next graph we see first the world cycle expressed in ‘lateral numbers.

Then the world cycle of life in 3 representations – the first one is the same that the complex graph, tumbled; the next one, the best philosophical representation by the Taoist Chinese culture, who understood duality better than western physics 3000 years ago; the 3rd as a representation of a vortex of physical time (charges and masses, whose mathematical unification with cyclical time is one of the first results I obtained in duality 20 years ago, and physicists are still fighting to understand – they call it the ‘Saint Grail of Physics’, and the guy who gets it will be a super-genius homunculus – i have it in my post on cosmology, a work still in progress, as I have been a bit depressed and abandoned this web for a long time):

In the graph, thanks to the ‘serendipitous’ error of earlier mathematicians, which could not conceive negative≈inverse motions, the real line was broken into perpendicular co-ordinates, which are perfect to represent world cycles, where the real line represents Spe (max. size, X-oordiantes) and the negative axis represents information, in as much as its development rests ‘entropy’ to the real numbers; being complex bidimensional numbers the combined expression of TIME-SPACE WAVES.

So positive negative and lateral numbers are time numbers. And again there are 3 types of it. So to fully grasp them we need to explain you the most beautiful cycle of the Universe, the life-death cycle, which also applies to physical ‘big-bangs.

This physicists don’t know and it is a wonder of the ‘automaton’ Universe that they DO use C-numbers to study world cycles, *without knowing what they are and what they study.*

We see indeed, what an i-number does: to rotate from a relative past/future to a relative future/past a function of existence, through a 90° middle angle, that is, the middle ST balanced point, in which the present becomes a bidimensional function, with a REAL, + (relative past-entropic coordinates) and an imaginary, negative (future coordinates) elements.

It is ‘crystal clear’, symmetric and beautiful.

Now, this needs just a couple of clarifications: first the fact that we move ‘backwards’ in the graph from future to past, in as much as most graphs indeed are measuring the ‘motion’ of lineal time which is the *relative Spe Function; so the negative side becomes the time function, something obvious also in relativity where we write S²=x²+y²+z²-c²t²; hence with the warping time-function in negative. The second clarification is even more fascinating.*

Ok, since we are at it, now let us give a hand to the great masters of quantum physics, with this new insights on the meaning of mathematics and its frames of references, explaining what Bohm did to fully grasp what really IS the quantum duality wave-particle, for all to grasp how astounding is the denial of reality by the so-called genius of the military-industrial complex who ‘create’ the Universe with his imaginary lines.

Now, we said that all what exists is a ternary system generated in time and space by a simple fractal generator equation. So are physical systems:

|-Spe (Relative past moving limbs/fields) < Relative Present hyperbolic ST-waves/bodies> Relative future O-informative particles/heads

We see how both biological and physical systems are truly the same structure of 3 time arrows, which give birth to 3 spatial parts. So entropic fields ‘feed’ the motion of ‘hyperbolic waves-bodies’ which feed and guide ahead the particles-heads that do NOT move in its relative stillness of pure informative linguistic view of reality. This is truly fascinating and I could talk to you for hours on the details of it. The proofs are overwhelmingly both in reality – all systems of reality can be model with it; and in languages, which ARE mirrors of those systems.

So for example, in our search for experimental proofs of mathematical mirrors, ALL the topological geometries of the Universe REDUCE to those 3 types: lineal/planar topologies, which are entropic motions as the line is the shortest distance between 2 points; spherical mind/heads, which are n-spheres as the sphere holds maximal information in lesser space and can shrink without deformation to create a 0-mind mirror of an infinite Universe (Poincare Conjeture); and *hyperbolic body-waves, **which combine and hence iterate both, lines and cycles, forming sinusoidal body-waves. *

*And so mathematics again reflects in its 3 only topological varieties, the 3 only organs needed to explain reality. And all what De broglie and bohm did was to ‘intuitively understand this’ for the specific case of a quantum wave, which has again those 3 elements. So they just said: there is a quantum field of pure motion at distance (non-local quantum potential field), over which a wave rides, and over the wave the particle feeding on that motion chooses the best path, to move, as it does a ray of light (my country man Fermat found that too :). *

So he just signalled the 3 elements, and then in a stroke of genius, he *represented the schrodinger hyperbolic wave written in ‘Cartesian coordinates’ in circular Particle coordinates, called ‘Polar coordinates’ and voila! there it appeared ‘magically’ the entropic field that moves the particle. So we could either see the whole system as a ‘present wave’ or as a Past field>Future particle. *

What is then the future direction in the imaginary arrow? Obviously that of a normal clock, down the conjugate is a future moving on a 90 degrees phase faster than the real and in the imaginary side a lag of 90 degrees in the phase of the world cycle – which physically appears in the AC circuit with the forward 90 degrees wave of the capacitive reactance, Xc on the conjugate and the lagging, 90 degrees of Xl the inductive reactance on the Imaginary side.

It is left then to ponder what a 90 degrees, 1/4th of a world cycle of any class which can be represented in the imaginary complex plane means. It is not after all 1/3rd which would make it nicely coincide with the 3 ages of time. But alas! as we know the 3 ages of time in terms of time duration break into 4/4ths, as the youth 1/4th and old age 1/4th last as much as the mature age 2/4ths – in existential algebra we write: Present = past + Future:

So the four parts of unit cycle representing a world cycle can be considered in terms of the 3 ages of time, with 2 quadrants belonging to the present, one to the past and one to the future…

Now we can interpret the positive natural numbers as the future symmetry, which give us the negative as the past antisymmetry, *and the imaginary line as the two sides of the present asymmetry. Or we could consider the present the real number, the past the imaginary and the future the conjugate. *

Both interpretations will render in the kaleidoscopic universe interesting but different results when we operate with existential algebra. Then we shall decide which one is the most promising. And contrast it with quantum physics equations, in which the imaginary element IS NOT as in phasors, a time state that disappears but a p*resent state in the operator momentum embedded within the quantum system. *

# ∆: i²

**The ∆-symmetry of i-numbers.**

The second fundamental role of i-numbers must be related to scalar space-time, hence polynomial structures.

We have also stressed in many paragraphs the duality and differences between polynomials and ∫∂ functions, whose ‘magic’ closeness (as polynomials can be approached by differentials through Taylor binomials), *responds to the social ‘lineal’ herd-like of polynomials, vs. the organic, more complex ‘cyclic’ nature of ∫∂ systems.*

So if we combine the two concepts we can ‘reorganise the complex plane’ in polynomial terms ‘naturally’ by *squaring it, getting rid of the √ elements and its negative complex roots, to reflect a mapping of a fundamental principle of nature:*

The h*olographic principle: the bidimensionality of all real forms of nature, which are in its simplest forms, dual dimensions, and so the unit of reality IS not a single dimensional system/point but a bidimensional ST system. Ergo the proper way to use imaginary numbers is ‘squaring’ them. *

Then we obtain a ‘realist’ graph, with an X² coordinate system in the real line, which can be projected further with a negative -y and a positive +y axis, often ‘inverse’ directions or symmetries of timespace.

And since the square of a negative number is positive, the X² main axis of bidimensional space-time units IS mirrored in the positive in both sides. So the proper way to represent the graph is by tumbling it, and making a half positive plane, with the now ± real axis (the i and -i axis) on the X coordinates and the square axis on the Y coordinates, which is much closer to the ‘REAL universe’ of bidimensional T.œs of timespace moving around its 0-identity element in two inverse directions of time (-i²= -1; +i²=1 axis). And suddenly we can start to understand many realist whys of complex numbers and complex spaces.

In the graph we see the immediate application to quantum physics, where ‘probabilities’ are born of the product of the ‘two conjugates’, the positive and negative sides of the imaginary axis, which now looses its √ and so become positive and negative ‘inverse’ arrows/functions/forms, which *combine into a bidimensional holographic real squared element. *

*So the graph is an excellent form of represent, the **neutral present, ST, in the square real line and the ± inverse S and T functions in the positive and negative conjugate axis no longer imaginary.*

This is somehow acknowledge in many equations of physics notably in those related to relativity where we USE square functions (in praxis just to simplify calculus, but now we know in reality they have ‘meaning’).

*So in special relativity we write:*

* *

As this are clearly square metric we just set the time coordinates in the imaginary numbers as bidimensional topologies extended in the imaginary -1 and its parallel – that extends a flat holography through the ‘z axis’ now void of meaning as it is occupied by the ‘folded’ other side of the real square. By this we mean, ‘points in the r² which must represent square numbers can be done into squares but making them ‘holographic clones of the same curve deepening into a plane.

Now the interest of the conjugate pair of ± real solutions to a squared coordinates only positive, is a reduction of information, which still however can be regained returning to √@. In @R² complex numbers simplify some equations.

# ð

In the graphs a very useful function on the complex plane that mimics the passing of time in world cycles towards an internal i-point. Complex planes are thus better systems of co-ordinates for cyclical times. And its simpler polar form, shows it is indeed, the perfect plane for ∆t analysis. Notice that as X becomes a ‘whole’ of a larger scale, it becomes paradoxically smaller in ‘volume but it speeds up in time’ according to ∆-metric.

**x ÷**

The second operation of algebra is product and its inverse division, AND THIS probes further into the scales of reality; so the new operations ads dimensionality and requires new numbers, namely the rational numbers. Let us then consider them first by studying its properties as compared to those of addition.

**1ST LINEAL AND 3RD POINT DIMENSIONS:**

The key theme to understand operandi is the meaning of inversion laws which carry the ± symbol in the 3 scales of sum, multiplication and power, dividing reality into a splitter symmetry around the T.œ in itself.

So ± is carried into power laws and products through ratio inversions, which is better expressed in the quadratic ‘complex’ frame of reference,of X²=±i² conjugate + 1 x -1 = 1² =1 axis.

So the positive and negative are shown in that ternary ‘frame of reference’ in each of the 3 axis, with the identity product of them, or quadratic frame of reference… in which we shall discuss latter on the complex elements of multiple dimensional operandi and its quants of actions.

**Properties of operations **

It follows from all what has been said that *operand are the fundamental reflection of the Disomorphisms of the Universe in mathematical ‘space’. What truly ‘pegs’ together the entire structure of mathematics, regardless of variables, parameters, scales and type of §œTS. *

So the first question as each operandi reflects an ST duality, is to ask what are the properties of the ordinary operations?

In a careful discussion of the properties of numbers and the operations on them that are used most frequently in algebra, it is easy to observe that they reduce to the following, or yes, of course… 5²=10 properties… We shall though ad for proper symmetry two properties that relate both operations (:

1. Identity element: There exists a number zero with the property a + 0 = a for every a.

2. For every number a, there exists the opposite number x satisfying the equation a + x = 0.

3. For any two numbers, their sum is uniquely determined.

4. Addition has the associative property: (a+b)+c=a+(b+c)

5. Addition is commutative: a+b=b+a

•••

1. Identity element: There exists a number 1 with the property a x 1 = a for every a.

2. Every number x, except 0, has a multiplicative inverse, 1/ x • x = 1

3. For every a and every b ≠ 0, there exists a unique number x satisfying the equation bx = a; hence the product of two numbers is uniquely determined.

4. Multiplication is associative: (ab) x c= a x (bc)

5. Multiplication is commutative: axb=bxa

And then two properties that relate both operations:

11. The product of a number and the identity element of a sum, 0 is the identity element of the sum: a x 0 = 0.

12. Multiplication is distributive: a(b+c) = ab + ac

Those properties were selected in classic science as a result of a careful analysis; the development of mathematics in the last century proved their great importance; as only *operandi and §ets of mathematical elements that obeyed those properties with the BIG exception of commutativity, which often is defined by its inversion ( a x b = – b x a) *

Nowadays every system of quantities satisfying the conditions 1 through 10 is called a field. Examples of fields are: the set of all rational numbers, the set of all real numbers, or the set of all complex numbers, because in each of these cases the numbers of the set can be added and multiplied and the result is a number of the same set, and the operations have the properties 1 through 10.

Apart from these three very important fields we can determine infinitely many other fields formed from numbers. But beside the fields formed from numbers there is much interest in fields formed from quantities of another nature.

For example, algebraic fractions, in which the numerator and denominator are polynomials in certain letters can be added, subtracted, multiplied, and divided, and these operations have the properties 1 through 10. Therefore, algebraic fractions form a system of objects that is a field.

So the entire ‘field of polynomials’ reduces to those 10 properties, and it is not rocket science for the reader who has got so far the ‘idea’ that we should define those properties as 5 inner ± D-properties of ‘small steps’ of social evolution (addition/substraction) and its ‘wholeness’ as a new dimension, 5 D-properties of x, ÷, or at least a close isomorphic correspondence with the ∑ ‘time-sum’ = Product- space whole duality.

So we have arranged those properties to see easily why they are in fact, defining a ‘constant growth’ in inverse symmetries of a neutral first element a 1 T.œ or fractal point, which will expand evolving socially ∆§, through additions and then in a new dimension of social scales, where an ‘additional number’ will be now whole unit of its multiplications; and finally in the third operation, a polynomial will be that number raised to an ∆new symmetric plane (as polynomials multiply the same quantity, unlike sums and multiplications).

**1st axiom = 1st Non-E Postulate/1st Dimension – the T.œ point:** The ‘first’ property of +, x IS defining the neutral, self-centred fractal point – a unit of addition a first number or a first ‘group of numbers’.

**2nd axiom = 2nd Non-E Postulate/2nd Dimension – the flow of communication:** The second property of ±, x÷ *expands in both inverse directions to maintain the zero sum of all worldcycles and operandi of existence, the T.œ in two inverse directions defining a wave-line-interval-distance-inverse motion both sides of the **identity element, fractal point or original number. *

**No axiom for 3rd postulate/dimension, the ternary network plane:** YET to reach the third ‘network dimension’ of the system that gives us a full ternary organism and its GENERATOR EQUATION, there is NOT A THIRD POSTULATE, BUT WE NEED A THIRD OPERATION. Thus, the Universe is indeed holographic, bidimensional, as we have show in every perspective of mathematics, from FERMAT’s last theorem to all other ST symmetries. You cannot reach a third dimension of the sum, but you must multiply it and you cannot reach a third dimension of the multiplication but you must operate through a power law:

In the graph, the runaway Nature of power laws, shows a hyperbolic ‘end of a plane’, which abandons the ‘lineal’ S x T =K nature of a proportional metric in the fifth dimension (multiplication), as it fast reaches the “Lorentzian” regions that signify the limiting domain of any T.œ in form, motion or scale:

Thus our hypothesis 5 properties -> 5 dimensions fails ): Good or things would be too boring.

But a bit of thought shows us that we are dealing with ‘logic properties’ NOT space-time forms, so it is better to connect them with the 5 Non-Æ postulates of i-logic. And then eureka! They do fit nicely.

So the other axioms *must be of i-logic nature, indeed:*

**3rd axiom **might seem silly but at this stage the reader would realise the Universe is quite weird enough not to discharge anything. And it is reassuring to know that THERE ARE NO PARALLEL UNIVERSES, A+b = Only c A x B = Only D.

So now we come to the logic ones, closely related to the 4h and 5th of non-A i-logic geometry:

**4th axiom ≈ 4th Non-E Postulate, self-similarity:** The Universe IS associative AMONG SELF-similar points, which *allows it to grow and multiply, add and multiply we might say, love each other as a i have loved you in human terms. It is the positive side of the 4th postulate of i-logic, associativity…
But what about the Darwinian, perpendicular laws of non-E? Because a number is made of IDENTICAL beings, and we can ONLY add (and so by extension since we have deduced that multiplication is the third dimension of addition), equal beings, we are only in the positive side of the 4th axiom. *

But alas! we then realise that the negative side of it, the inverse function, subtraction and division are NOT associative: 10/5=2 and 2/2 = 1 is NOT the same that 10 divided by 5/2 which is 4; and 10 – 5 = 5 minus 2 = 3 is NOT the same than 10 minus 5-2=3 which is 7…

So there is here in the negative side of the world, a different hierarchy of things, such as if you start destroying BIG TIME (10/5; 10 minus 5) and then slow down, you have already fuk up the world, but if you start destroying slow; decadence gets longer… Important elements of the vital laws of reality, explored in the first line…

You see, even the simplest supposedly exhausted facts of science, get new insights on stience (:

**5th rebel postulate: commutativity. **And so the fifth postulate of course must have to do something with the 5th Non-Euclidean postulate, the rebel one; and indeed it turns out in this symmetry between geometrical space and temporal algebra views that to the surprise of everybody in the XIX-XX century, there are many systems in Nature which do NOT have the commutative property for multiplication, (but it does for sum).

That is, a sum is truly a herd of undistinguishable beings, and it does not matter as it is in a state of loose connection which order you add. We are then clearly in the very first simpler reality, in a single plane, with minimal herd connection between the parts.

But multiplication being a ‘second dimension’ added to the sum sometimes IS commutative, sometimes NOT, meaning that when the multiplication IS a growth in a scalar dimension within the plane (truly a sum equivalent new dimension) IS commutative, but when this NEW dimension is added, not in the same plane, as an ∆§ operation but *in the sense of group theory as a ‘combining’ process of things (vectorial product, product of the parts of a whole, product of spatial paths, products in time **frequencies, topological products) THINGS CHANGE. *

ÍAnd so with this understanding we COMPLETE the simplest analysis of sums in a single plane of identical beings as herds, but we realise that the second operandi, the adding of a second dimension called product ALREADY PLUNGES US INTO THE RASHOMON EFFECT OF THE PRODUCT which can mean many different things depending on which ∆S≈T dimensional element we are adding to the simpler sum.

THE SOCIAL ACTION: SUMS OF HERDS.

It is then once we have defined an operation and its properties, time to study how they mediate the actions of beings. And it is clear that the first operation sum, acts on the first form, social numbers, to form growing §ŒTs of social numbers…

So paradoxically the more complex action, starts with the simple operation, the sum of individuals into herds of formal numbers.

**Finitesimal actions.**

The quantity a system absorbs to create an action is generally defined as a ‘finitesimal’, not infinitesimal. Infinite does not exist in a single continuum, but through multiple discontinuities as all systems in time and space are limited in space and time, both in a single membrane, and in within the scales of the 5th dimension (as information and energy doesn’t flux between those scales without loss of entropy).

A finitesimal is THE QUANTITY of energy, motion, information etc. used by a T.œ for an action of space-time IN any of the 5 Dimensions of the being, ‘put in motion’ to that aim.

Wholes are physiological networks, which we analyse mathematically in its parts, mostly performing ga motion of space-time, an action that exchanges most likely bits and bites of time and space. So the operation reflects the processes of transmission and gathering of information and energy, of bidimensional holographic quanta…

While humans will measure externally its action of space-time, in any of its 5 subspecies classified by its complexity:

A-celerations, lineal motions, entropic motions, energy flows, informative vortices and Social evolutions (a,e,i,o,u).

IN THE GRAPH, the general 5 actions-dimensions of existence of different ∆±i species, from above down – a view of them all, one of the physical simplest light and electronic i<eye minds, and below the human being.

Mathematically it is quite relevant to know *which scale is used to perform an action, to consider how ‘finitesimally’ small is the being, which the whole absorbs to perform *a minimal action in human time quanta – similar for the 3 synchronous t-st-s parts; i.e. in humans an eye glimpse of mind-perception, a limb-ate of motion and a heart-beat of the body – and how much it absorbs in the time quanta.

To notice that as humans for all effects measure reality with the mechanical chronos of 1 second, often very unrelated finitesimal actions are equalised with our human quanta, while the quanta of space the system absorbs are clearly differentiated in mathematical physics (h, k, c, G, Q constants of nature).

It is thus important for all systems today only described quantitatively in abstract terms, to *vitalise and explain the organic whys of its space-time events by introducing the a,e,i,o,u type of actions it performs.*

The connection on qualitative terms though is self-evident, for all scales, as most actions of any being *are extractions of motion, energy and form from lower ∆-i scales.*

So we and all other beings perceive from ∆-3 quanta (light in our case), feed on amino acids, (∆-2 quanta for any ∆º system), seed with seminal ∆-1 cellular quanta (electrons also, with ∆-1 photon quanta).

So derivatives are the essential quantitative action for the workings of any Tœ, space-time organism.

And so we study in depth the connection of the a,e,i,o,u actions between Planes (qualitative understanding) and its mathematical, analytic development (quantitative understanding of 1st second and 3rd derivatives – the late extracting ‘1D motion’ from the final invisible gravitational and light space-time scales).

SO THE FUNDAMENTAL LAW OF OPERATIONS TO VITALIZE THEM IS THIS:

‘BY THE RASHOMON EFFECT ALL differential OPERATIONS CAN BECOME AN ACTION IN ONE OF THE 5D DIMENSIONAL VOWELS (A,E,I,O,U) THAT DEFINE THE FIVE dimensions OF EXISTENCE, AS VITAL QUANTA-ACTIONS OF THE BEING.

THIS IS THE LOGIC CONCEPT THAT TRULY VITALIZES THE OPERANDI OF ALGEBRA.

Derivatives allow us to integrate, a* sum of the minimal quanta in space or actions in time of any being in existence, namely the fact that its sums tend to favor growth of information on the being and then signal the 3 st-ages and/or st-ates of the being through its world cycle of existence, which in its simplest physical equations is the origin of… ITS space-time beats.*

Actions in timespace are the main finitesimal part of reality, its quantity of time or space if we consider tridimensional actions as combinations of S and T states, stt, tst, tss, sss and so on…

So how differential equations show us the different actions of the Universe?

To fully grasp that essential connection between ∆st and mathematical mirrors, we must first understand how *species on one hand, and equations on the other, probe in the scales of reality to obtain its quanta of space-time converted either in motion steps or information pixels, to build up reality. *

So for each action of space-time we shall find a whole, ∆ø, which will enter in contact with another world, ∆±i, from where it will extract finitesimals of space or time, energy or information, entropy or motion, and this will be the finitesimal ∂ ƒ(x), which will be absorbed and used by the species to obtain a certain action, å.

So the correspondence to establish is between the final result, the åction, and the finitesimal quantas, the system has absorbed to perform the action, ∫∂x, such as: å= ∫ ∂x, whereas x is a quanta of time or space used by ∆ø, through the action, å to perform an event of acceleration, e-nergy feeding, information, offspring reproduction or universal social evolution.

It is then when we can establish how operations are performed to achieve each type of actions.

FOLLOWING this inversion of complexity of actions vs. simplicity of operations, comes the next more complex social action of reproduction, which most often is expressed for quantitative simple physical systems through the second simpler operation of re=product-ion.

**X: 2D: reproduction Operation**

**MULTIPLE MULTIPLICATIONS: RASHOMON TRUTH**

The fundamental effect of a new operandi is to ad a new dimension to the system. Then we can apply the method of multiple perspectives ( RASHOMON truths), to consider which type of dimension, *and in this manner an enormous range of phenomena can be expressed with the same operation, increasing the iterative complexity of the Universe. We shall just show a few example of the ginormous ‘100th’ number of combinations that we can obtain by multiplying any of the $, t, S, T, §, ð, ∫, ∂, ∆§+1, ∆ð+1, 10 dimensions of space-time for any of those dimensions; and further on combining different operandi in a single function*:

**Γ combinations: ($t≤ST≥§ð):**

$$: Multiplication adds a dimension of lineal distance. It is the simplest and commutative form.

$t: Multiplication ads a dimension of motion, velocity:

**$S:** Multiplication ads a dimension of spatial width to form area:

**ð$:**In a similar fashion multiplication can ad to a cyclical point/clock of time, a dimension of distance.

**ðt:**Multiplication can add

**a dimension of lineal motion to a particle-vortex (quantum motion), the very**

*essence of motion as reproduction of information.*

**∆±i**

**∆º§:**In a single plane, multiplication is obviously the product of a group of social identical beings by the parameter we study ceteris paribus or ‘number’, multiplied x times, that is summed with an equal group and an equal group and an equal group to reach the number we multiply for.

**∆+1<∆-1;**But what does it mean between planes? As we realise that when we multiply 5 x 5, if we consider 5 the number of ∆-1 elements of 2 groups, then 5 x 5 = 25 turns out to be the number of ‘axons’ connecting each of the 2 wholes at the level of its 5 ∆-1 elements. So 25 gives us

*the number of axons= ST flows of communication between the 5 sub-elements of the 2 wholes.*

**The messing of multiplication and addition: distributive and neutral elements. Negative inversions.**

*has one more dimension and hence has the 1, perceivable T.œ (because it will be making likely a motion and/or be perceived as an ST unit, hence we consider it an interval 0-1, a unit circle and so on).*

**0 x a = 0**is the first of those combinations, which causes a reduction of dimensionality, as

*return o the less dimensional operation of addition. It can then be also vitally interpreted as if 0 were an open ball without @, the singularity membrain; hence a sink of void of the ∆-1 scale, prior to the ’emergence’ of the whole, 1, the full T.œ – an eraser dimension when in contact with a.*

*And finally as the destroyer point that sinks the flow of a, into the 0-point:ð*

**Lineal vs. cyclical addition and multiplication.**

*products must be seen as ∑, a sumandi of x elements, since the product is with a constant.*

*It is a very different case when the product is between two variables or any other parameters which are not ‘constant sums’.*

*If we were to consider the analysis from the point of view of the 3 ‘elements’ of a generator, $-potentials < ∑∏-waves>ð-particle/heads, it is easy then to see that constant products are*sums. So potentials are sums.

*must be considered bidimensional, and dot products*). As products, they create new dimensions, as in light waves are: c²=k/µ; and cross products. Such PRODUCTS ARE NON LINEAL as they REPRODUCE a sinusoidal wave. And often the key to classify waves as sum or products is their capacity to commute (sums) or not (products). Finally the SYNTAX of most languages coded as GENERATORS ARE PRODUCTS that do not commute when they are different ‘categories’ (potential x wave x particle; subject x verb ≠ verb x subject and so on). And so the only ‘identity’ elements which are non-lineal as products are 1D ð§; and its polynomials.

**Multiplying ±. From outer operandi to inner property in a higher dimension.**

**12. Multiplication is distributive: a x (b+c) = a x b + a x c = e**

So this lead us to a property, distribution, which happens in all meaningful combinations of two consecutive Dimensional operandi; where the concept of a rank, or more ‘powerful’ larger social whole-dimension, in this case the multiplicative operandi a x, is able to ‘slice’ the group-herd, b + c into two separate parts enlarging them both by the same ratio. And this merely extends enormously the capacity of two ‘consecutive dimensions’ of space-time to make a ‘liaison’ , as the ‘higher whole’ the product operandi can ‘discern’, when it is necessary to re-grouping the lower ‘herd’ (b+c) to ‘effect’ an ’emergence’ to a higher dimension of those two different groups – or when to operate first b+c = d and then a x d= e.

Distribution thus means that any consistent system managed by an ∆-1 ST operandi can emerge in the ∆+1 plane through an ∆º operandi.

**ITS INVERSE OPERATION: DIVISION.**

**Multiplication defining the 3 vital constants of the being and its ratios.**

*S/T x T/S =ST/TS= 1. Multiplication as Ratio of proportionality*

*it is not. The ratio is a division; the square is a multiplication, and both are inverse functions, which gives us the identity element.*

In that sense in all questions connected with discrete objects, we NEED to use WHOLE numbers for the necessary mathematical apparatus, as well as the study of the continuous. Thus, for example, in mathematical analysis, when one considers the expansion of an analytic function in a power series with integral powers, computations are essentially carried out with whole numbers and approximated ratios such as 22/7 = πi, which is Ok as we have seen (Number theory) that decimal numbers break its meaning beyond the 10th decimal scaling… (i.e. e=2.718281828…45).

So all fractions represent ratios/quotients of two whole numbers; and as such a full new branch of ‘number theory’ will be the study of those quotients as *ratios between steps of time motions and or whole polygonal numbers.*

RIn dealing with any real number in practical work (for example, π), we replace it in fact by a rational fraction (for example, we assume that π = 22/7, or that π = 3.14).

YET While the establishment of rules for operating on numbers is the concern of arithmetic>ALGEBRA, the deeper properties of sequences of numbers, extended to include zero and the negative integers, are studied in the theory of numbers, which is the science of the system, studied in other post.

**Odd and even functions.**

A very interesting reflection of the duality between antisymmetry and symmetry happens in the odd (antisymmetric) vs. even (symmetric functions) as it is also the basis between social evolution of particles (bosons) vs. antisymmetric annihilation (fermions), so it *has an immediate never quite clarified application to physics.*

Let us consider the classic axiomatic approach first, which started in the analysis of divisibility.

One of the basic questions in the theory of numbers concerned the divisibility of one number by another:

if the result of dividing the integer a by the integer b (not equal to zero) is an integer, i.e., if a= b • c (a, b, c are integers) then we say that a is divisible by b or that b divides a. If the result of dividing the integer a by the integer b is a fraction, then we say that a is not divisible by b.

Questions of divisibility of numbers are encountered constantly in practice and also play an important role in some questions of mathematical analysis. For example, if the expansion of a function in integer powers of x

is such that all odd coefficients (with indices not divisible by 2) are equal to zero, i.e., if

then the function satisfies the condition: **ƒ (-x) = ƒ (x)** – such a function is called an even function, and its graph is symmetric with respect to the axis of ordinates. But if in the expansion (2) all the even coefficients (with indices divisible by 2) are equal to zero, in other words, if:

then **ƒ (-x) = – ƒ (x)**

In this case the function is called odd, and its graph is symmetric with respect to the origin.

Thus, for example:

Which will have deep implications in the physical Universe (Pauli exclusion principle), so we shall study its full meaning in our posts on mathematical ‘Astrophysics’.

The reader should easily in any case interpret the results in terms of dimensions and the holographic principle. And it reads like this: symmetric functions are holographic even functions, which means for example they can be ‘superimposed’ as they are bidimensional ‘sheets’, waves etc. While odd functions are even in dimensions, and as such they cannot be superimposed.

This means essentially bosons are bidimensional and fermions are tridimensional, and this is a huge advance for quantum physics – recently proved by an experiment that converted a photon, initially a boson into a tridimensional form and ended its boson conditions. Accumulation of bosons into a single point of space-time then means merely it is superposing the bidimensional thin layers into a third dimension of height which is indeed what we see in pictures of boson states.

How the representation in ± sides of the plane works to understand this is also obvious: the axis of ordinates plays here the role of the ‘asymmetric’ state which has split into the ± inverse directions but can fusion again both. Or in terms of relative equality, those functions DO have their position in space (established by the lineal coordinates used to represent lengths in space) different but their form (represented by the Y-coordinates used to represent form and information height dimensions for most functions) identical. So they can according to the rules of the fourth postulate of non-e, communicate as they can share identical information and match each other in symmetric peg.

While the antisymmetric function which is +- -+, meaning its inverse both in the Y-nformation and X-pace location is disimilar both in form and motion, momentum and position, you name it… whatever 2 parameters we use to compare both systems. So they enter into a darwinian annihilating process as they cannot match each others form.

**0-1 ≈ 1-∞**

Let us then see those numbers which we have added from the perspective of the plane, as they become ‘mirror symmetries’ in the o-1 sphere unit of the 1-∞ plane described with natural numbers.

Once we have defined the 1-∞ plane only with natural or complex numbers we can then assess the need for more numbers to fulfill the ∆±1 scales and ore operandi to probe in to the ∆-1 scale (o-1 sphere and infinitesimal numbers found with rational and real decimals).

An ∆ scale can be represented through the interval of 0 to 1 by finitesimals or the interval from 1 to ∞, which become the decametric and decimal regions of a supœrganism represented in the real line: ∆-1: o to 1, 1, the ∆o scale and 1 to ∞, the external world.

**Quantum case**

Further on, as we are dealing with the smallest scales of reality, it applies the metric equations of 5D according to which we are in a ‘temporal realm’ as world cycles occurs extremely fast (Min. Spatial size = Max. Temporal speed of time cycles) and so the formalism of quantum physics uses the equivalence between the o-1 probabilistic sphere of time events instead of the 1-∞ plane of statistical populations to formalize the events of ultra-fast repetitive particles.

**Numbers and infinities.**

Mathematics divides phenomena into two broad classes, discrete or temporal and continuous, or spatial historically corresponding to the earlier division between T-arithmetic and S-geometry.

Discrete systems can be subdivided only so far, and they can be described in terms of whole numbers 0, 1, 2, 3, …. Continuous systems can be subdivided indefinitely, and their description requires the real numbers, numbers represented by decimal expansions such as 3.14159…, possibly going on forever. Understanding the true nature of such infinite decimals lies at the heart of analysis.

And yet lacking the proper ∆ST theory it is yet not understood.

The distinction between continuous mathematics and discrete mathematics IS ONE BETWEEN SINGLE, SYNCHRONOUS, CONTINUOUS SPACE WITH LESS INFORMATION, and the perception in terms of ‘time cycles, or fractal points; space-time entities’, which will show to be ALWAYS discrete in its detail, either because it will HAVE BOUNDARIES IN SPACE, or it will be A SERIES OF TIME CYCLES AND FREQUENCIES, perceived only when the time cycle is ‘completed’, and hence will show DISCONTINUITIES ON TIME.

Thus the dualities of ST on one side, and the ‘Galilean paradox’ of the mind’s limits of perception of information lay at the heart of the essential philosophical question: it is the Universe discrete or continuous in space and time. Both, but always discrete when in detail due to spatial boundaries, and the measure of time cycles in the points of repetition of its ‘frequency’.

So ultimately we face a mental issue of mathematical modeling: the ‘mind-art’ (as pure exact science does not exist, all is art of linguistic perception) of representing features of the natural world in a reduced mental, mathematical form.

The universe does not contain or consist of actual mathematical objects, but a language can model all aspects of the universe. So all resembles mathematical concepts.

For example, the number two does not exist as a physical object, but it does describe an important feature of such things as human twins and binary stars; and so we can extract by the ternary method, 3 sub-concepts of it:

2 means the first ∆-scale of growth of 1 being into 2, by:

S-imilarity and S-imultaneity in space (ab. Sim)’, ‘i-somorphism in time-information (ab. Iso)’ and ‘equality in ∆-scale’ (ab. Eq), as perceived by a linguistic observer, @, which will deem both beings ‘IDENTICAL’. Whereas identity means that an @-bserver will deem the being ∆st≈St, (Sim, Iso and Eq). So identity is the maximal perfection of a number, for a perceiver, even if ultimately:

*‘Not 2 beings are identical for the Universe, but can be identical for the observer’… an intuitive truth, *whose pedantic proof is of course of no importance (: we do not follow the axiomatic method of absolute minds here):, but it is at the heart of WHY REALITY IS NOT COLLAPSED INTO THE NOTHINGNESS OF A BIG-BANG POINT.

Thus those 3+0 elements of the ∆•ST coincide *a social number can be used whose intrinsic properties define conceptually ‘S- imultaneity, Ti-somorphism’ and ∆-equality or equivalence (ab. Eq) in size, which becomes an @identity for the mind. THEN A NUMBER IS BORN.*

I(n this ‘infinitorum’ of Universal thoughts, which bring always new depths as soon as we observe it with an ∆•st trained mind, there are differences between S-imilarity and Simultaneity to define in space an ‘identity’ and ‘equality’ and equivalence, treated elsewhere)

It IS THEN CLEAR that a number being a sum of points, encodes more information in a synoptic way about the T-informative nature of the ‘social group’ than an array of points, *which unlike a number tells us less about the ‘informative identity of the inner parts of the being’, but provides us more spatial knowledge about the relative position in space of the members of a number-group.*

And this is OBVIOUS, when we return to the origin of geometry and consider an age in which both concepts were intermingled so ‘points were numbers’ and displayed geometrical properties:

Numbers as points, showing also the internal geometric nature, used in earlier mathematics to extract the ‘time-algebraic’, ‘∆nalytical-social’ and S-patial-geometrical properties from them.

The closure of the systems of numbers thus grows from reflecting merely space populations (natural numbers), into 5D numbers, reflecting ‘partitions’ of social groups (represented by those natural numbers), with Egyptian ratio-nals; expanded further with the realisation that certain ratios did apply to ‘scaling’ in the fifth dimension without limit (as in the pi ratio of Spe-entropic lines into cyclical time O-cycles).

Alas, things got interesting here, but as the homunculus did not understand, *the discoverer of pi, legend has was murdered by Pythagoras the first ‘religious mathematician’ because it found that pi was not perfect. I*t took to Poincare 2500 years latter to find that this is an awesome form of perfection because it means a mind-point of spherical form, with equal distance to all the realities it reflects can shrink with no limit (Poincare conjeture – the one that Perelman resolved recently with tons of pages)

Now, we have SPATIAL, natural numbers, 5D rational and transcendental numbers, who cross through 5D scales without tearing according to the Poincare postulate proved in a margin (: but this is NOT enough, because we need negative and lateral (not imaginary) numbers, *and those are temporal numbers, numbers who describe processes in time not in space, as negative numbers do, as they represent merely the inverse arrow of time, so if you make a positive number an arrow of motion-entropy, or relative past loss of information, the negative number will be the arrow of future-information, and the lateral i-number, will be the bidimensional sum of both the real and the negative, to represent the present space-time.*

∆±i: i-ratio-nal or transcendental numbers; ∆±1: ratio-nal numbers.

So you do have a closure of all numbers, based in the elements of space-time.

Hence, when you apply them to mathematical physics, *which is the study of the simplest forms of space-time,**you do have a better focused mirror. *

**The problem of the i-number: negatives and roots. Proper and improper inversions.**

IT is then clear that at the basic level of arithmetics the problems that still drags mathematics and by extension all other sciences, which use its mirror, is the concepts of negative and square numbers, the inversions of the positive and quartic equations, on the real number, related to the two fundamental unknowns of humind science – the fractal scalar ‘infinitesimal nature of space’, and the ‘proper and improper inversion’ of the fractal generator.

In essence all this means that negative numbers and roots do NOT exist for certain type of space-time events, and povs of the ∆s≈t structure of reality but are inflationary ‘informative’ excrecencies proper of all languages that do have an inflationary excess of information, as they are ISOLATED SPACE-INFORMATIVE systems OF SOFTWARE, WITHOUT THE direct contact with the HARDWARE that limits the possible paths of information. Namely, any LANGUAGE isolated into itself will multiply its kaleidoscopic forms, ‘free’ of the constrains the vital energy it must ‘shape’ cause.

And this is the fundamental need of mathematics that force it in search of meaning as per lobachevski’s pangeometry and Godel’s algebraic theorems, to recur to experimental science, TO DISCERN ONCE AND FOR ALL WHAT ARE MATHEMATICAL FICTIONS AND WHAT ARE MATHEMATICAL REALITY.

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

So as a general rule, *negative numbers exist when they are a direction of motion, not a quantity or volume of space.*

And so happens with roots, which only should exist for symmetric holographic systems of space-time S=T, for bidimensional entities and regular numbers as forms.

And alternately they Do happen in many cases in which mathematical physicists discharge them as inverse arrows of time because they ignore the inverse 4D vs. 5D arrows, as in Einstein’s equations, where they customary discharge negative hyperluminal solutions that do happen in the larger ∆±4 dark world of quantum potentials (entanglement, pilot-wave theory, etc) and intergalactic space (faster than light neutrino background, gravitational waves, action at distance, red shift of light, etc.)

We study this themes in number theory where we define the different types of numbers, so no need for further info here. Instead, we will make some comments on… the limits of classic Aristotelian logic based precisely in a single arrow of time, which has so much influenced as Euclid pretentious axiomatic method (full of holes and new axioms, postulates, notions and various errors of the §@-humind), our ‘underlying’ a priori categories of the mind.

**EXPONENTIAL vs. LOGARITHM**

We come then to the next peel of polynomial growth, and final ternary degree of complexity, exponential equations, which in the inverse process of more complexity for simpler actions, brings the ‘end=action’ of feeding/dying of entropy, better described by those exponentials.

**4th lineal dimension… of death v. 5th of growth**

*In the graph, the inverse simpler analysis of growing and diminishing planes and finitesimals have 2 first approaches in the study of polynomials and its inverse equations of logistic growth, logarithms. POLYNOMIALS in that sense are better as negative expressions of the decay of a system in its 4th entropic dimensions and no longer follow the whole range of ‘social growth’ properties (commutativity and associativity); as they are no longer ‘connected with the ∆-2 dimensional scale in the same ‘arrow’ of growth.*

Infinite growth does NOT exist because there are always limits to growth in the incapacity of a system to obtain ALL THE ENERGY OF ITS ∆-1 plane (reason why THERE IS entropy when we try to extract ALL the heat from molecules into our ∆º whole scale).

This means that exponential growth IS not growth but decay, as it can only be exponential when acting on a pool of already ‘multiplied by re=product-ion’, ∆-beings, which are destroyed in exponential fashion by an e‾ª function. And so it is only the inverse function of logarithm growth, what shows a growth curve (logistic curve) till a carrying capacity saturates the system.

*it was not possible to go.*

∆-1: fractal point sum > ∆º: lineal product: ∆+1: Power ‘volume’.

We can multiply powers with the same base:

x³⋅ x²=(x⋅x⋅x⋅x)⋅(x⋅x)=xˆ5. Hence the general law: x²⋅x³=x²+³: The product of powers with the same base add exponents.

*wholes (∆+1 powers) operated EXTERNALLY in the domain of its present, dimensional parts (∆º multiplication) are equivalent to one whole operated INTERNALLY with its past, finitesimal, entropic points.*

A profound law of Existential algebra; which basically tells us that an ∆-whole ( a fractal point in its own) absorbing an external ‘flow’ (the lineal multiplication) , will operate that flow at a lower internal ∆-1 level. I.e: if you absorb food you first break it into amino acids to reconstruct yourself.

We can raise a power to a power:

(x²)³=(x⋅x)⋅(x⋅x)⋅(x⋅x)=xˆ6

This is called the power of a power that multiplies the exponents: When you raise a product to a power you raise each factor with a power:

(xy)ª=xªyª

*Two parts (∆º≈ x • y) operated in the domain of its higher whole dimension, are equivalent to each one operated by the higher dimension as separate, broken entities. Which is the distributive law that gives the whole power over its parts.*

As well as we could multiply powers we can divide powers. And this 2 laws are again applied to the inverse operation.

This quotient of powers property tells us that when you divide powers with the same base you just have to subtract the exponents.

xª/x³=xª¯³, x≠0

Two *wholes (∆+1 powers) operated EXTERNALLY in the domain of its present, dimensional parts (∆º division) are equivalent to one whole operated INTERNALLY with its past, finitesimal, entropic points. *

When you raise a quotient to a power you raise both the numerator and the denominator to the power.

This is called the power of a quotient power:

(x/y)ª=xª/yª, y≠0

*Two parts (∆º≈ x /y) operated in the domain of its higher whole dimension, are equivalent to each one operated by the higher dimension as separate, broken entities. Which is the distributive law that gives the whole power over its parts.*

*IDENTITY LAWS.*

When you raise a number to a zero power you’ll always get 1.

1=xª/xª=xª¯ª=xº=1,x≠0

*A WHOLE, ∆+1, operated by the identity element of its ∆-1 scale, gives us the identity element of the middle scale: ∆+1 * ∆-1=∆º*

And a similar law in ∆st terms: Negative exponents are the reciprocals of the positive exponents:

x¯ª=1/xª, x≠0; xª=1/x¯ª, x≠0

*The inverted operation of the ∆-1 scale applied to the dimension of the whole, ∆+1 gives us the inverted element of the ∆º scale.*

*Both are laws of the essential timespace definition of the generator: Past (∆-1 ) * Future (∆+1) ≈ Present. *

The same properties of exponents apply for both positive and negative exponents. The square root of a number x is the same as x raised to the 0.5th power: √x=²√x=x½…

*the scale of the whole (1-∞) to its reciprocal, scale of the infinitesimal (0, 1) making both mirrors of each other; since ultimately the whole is a closer model of its infinitesimals, than the intermediate scale, which has suffered an inversion of function: ∑∑O-1≈∑|i= O+1.*

*the third scale of growth of a system, does NOT goes over a hypothetical 4th dimension in the same plane but starts a new game, in which the whole in the new plane is the ‘fractal infinitesimal’ similar to its infinitesimal in the ∆-1 plane.*

**FERMAT’S THEOREM, THE PROOF IN A MARGIN.**

*And so this means the 3rd dimension expressed by a power law, was no longer ‘holographic’, apt for mere additions by superposition, reason why the Fermat’s theorem, X³ + Y³ ≠ Z³, does NOT work.*

*the full consistency between contiguous dimensional growths (±, x) BREAKS between discontinuous dimensions, from 1D (sum of herds) to 3D (merging of 2 holographic bidimensional sheets into a third one):*

*there is not a 4Dimension in the same scale of space-time, the rule of superposition through a new Dimension breaks for superposition of cubes, which would have to be added in this supposed 4 Dimension in a single plane, reason why we cannot add them (Fermat’s theorem).*

*the addition of cubes cannot either be resolved in bidimensional plane geometry with a line-rule and a cycle-compass.*

*both approaches must be similar in its results, as they essentially observe the same phenomena with different focus, which is the reason of the existence of the Taylor and Newton’s approximations through derivatives of any*

*polynomial:*

So polynomials of algebra – NOT the other way around – are the rough* *approximation to the more subtle methods of finding dimensional change proper of analysis – even if huminds found first the unfocused polynomials and so we call today McLaurin & Taylor’s formulae of multiple derivatives, approximations to Polynomials.

*are lineal operators, of sums and multiplications power 5 polynomials could not be resolved, as they were prying in non-lineal regions of ∆±i space-time.*

**∫∂: 5th dimension.**

So what about the most sophisticated of the inverse operations of algebra? Calculus of finitesimals, ∂, and its integrals, ∫? Here we have a dual motion, back into ∆-1 to extract a part, and forwards into its integral ∫ through an spatial or temporal finite domain. So the inverse operations of analysis have multiple functionality in terms of actions performed through them, because of its *perfect mirroring of the action itself, which consists in using ∆-i finitesimals to absorb energy, motion or information for the 3 simpler actions of motion, informative perception, and energy feeding for which **paradoxically, the more complex ‘organic’ operations are the most useful.*

This paradox though has *a less ‘motivating’ cause for those involved with the mirror of mathematics – essentially that mathematics is NOT the best language to describe the complex actions and relationships that appear out of the reproductive biological and social, engaging processes of organisation, at least algebraic operations.*

We have though already commented on Algebra, that the third derivative, or higher derivatives however are used to improve the accuracy of an approximation to the function:

f(xo+h)=f(xo)+f′(xo)h+f″(xo)h²/2!+f‴(ξ)h³/3!

Thus Taylor’s expansion of a function around a point involves higher order derivatives, and the more derivatives you consider, the higher the accuracy. This also translates to higher order finite difference methods when considering numerical approximations.

Now what this means is obvious: *beyond the accuracy of the three derivatives canonical to an ∆º±1 supœrganism, information as it passes the potential barrier between scales of the 5th≈∆-dimension, suffers a loss of precision so beyond the third derivative, we can only obtain approximations by using higher derivatives or in a likely less focused=exact procedure the equivalent polynomials, more clear expressions of ‘dimensional growth.*

*So their similitude first of all proves that both high derivatives and polynomials are representations of growth across planes and scales, albeit loosing accuracy.*

However in the fifth dimensional correct perspective is more accurate the derivative-integral game; as it ‘looks at the infinitesimal’ to integrate then the proper quanta.

**Finitesimal Quanta, as the limit of populations in space and the minimal action in time.**

So there is behind the duality *between the concept of limits and differentials (Newton’s vs. Leibniz’s approach), the concept of a minimal quanta in space or in time, which has been hardly explored by classic mathematics in its experimental meaning but will be the key to understand ‘Planckton’ (H-planck constants) and its role in the vital physics of atomic scales.*

It is then essential to the workings of the Universe to fully grasp the relationship between scales and analysis. Both in the down direction of *derivatives and the up dimension of integrals; in its parallelism with polynomials, which rise dimensional scales of a system in a different ‘more lineal social inter planar way’.*

*So polynomials and limits are what algebra is to **calculus; space to time and lineal algebra to curved geometries.*

The vital interpretation though of that amazing growth of polynomials is far more scary.

Power laws by the very fact of ‘being lineal’, and maximise the growth of a function ARE NOT REAL in the positive sense of infinite growth, a fantasy only taken seriously by our economists of greed and infinite usury debt interest… where the eª exponential function first appeared.

The fact is that in reality such exponentials only portrait the decay destruction of a *mass of cellular/atomic beings ALREADY created by the much smaller processes of ‘re=product-ion’ which is the second dimension mostly operated with multiplication (of scalars or anti commutative cross vectors).*

So the third dimension of operandi is a backwards motion – a lineal motion into death, because it only reverses the growth of sums and multiplications polynomials makes sense of its properties.

Let us then see how the operations mimic the five dimensions:

**Social Number = first dimension** that defines regular ‘points’ which are undistinguishable, as societies in regular polygons, where prime polygons have the property of ‘increasing inwards’ its numbers through reproduction of vortex-points (n-grams), as the graph shows, studied in Theory of Numbers. So a number in its geometric interpretation is a ‘cyclical point’ of regular ‘unit-points’ of growing ‘inner dimensional density’ a point with a volume of vital energy and information, a fractal point.

*2nd dimension line=sum of points.*

*3rd dimension production=multiplication =reproduction*

In the graph, product can be of multiple, different ST dimensions, which start the richness of its ‘propositions’. A vectorial product is one of its commonest forms as it combines ST or TS dimensions, BUT as both ‘present’ products are different in orientation, this product unlike other SS or TT products is non-commutative: bxa=- axb. In this case giving birth to two different orientations in space, though for more complex product of multiple ‘S-T’ dimensions, which *can define as a Matrix of parameter a T.Œ PARTICLE in full,* the non-commutability can give origin to different particles (quantum physics).

*4th dimension: polynomial death dimension of decay.*

POLYNOMIALS DO NOT EVOLVE REALITY towards an impossible infinite growth. THEY ARE the inverse decay process; which can be understood better observing that the inverse function does in fact model growth in the different models of biology and physics, limited by a carrying capacity straight flat line:

**The logarithmic function **has as derivative an infinitesimal, 1/x, which makes it interesting as it models better the curve of growth from o to 1 in the emergent fast explosive ∆-1 seed state, while the inverse e*ˆ-x model the decay death process.*

*5th dimension: ∫∂…*

Integrals and derivatives which have a much slower growth, than polynomials on the other hand do model much better as they integrate the ‘indivisible’ finitesimal quanta of a system, its organic growth and ‘wholeness’ integrated in space.

Thus integrals do move up a social growth in new ∆+1 5D planes. And its graphs are a curved geometry, which takes each lineal step (differential) upwards, but as it creates a new whole, part of its energy growth sinks and curves to give birth to the mind-singularity @, the wholeness that warps the whole, and converts that energy into still, shrunk mind-mappings of information, often within the 3D particle-head.

**Equations of polynomials.**

Polynomials seem fairly straight, but as soon as we consider its solutions and varieties, things become more fascinating.

First we have 2 mysteries of 2300 years, one which people hardly wonder – why there are 2 solutions to all polynomials, and the other which all mathematicians do but have never been explained – *why one of the solutions is often an imposible, imaginary i-solution? What is the meaning of imaginary numbers?*

Why polynomials ‘∆-scales’ have 2 solutions?

The answer should be immediate for anyone who has understood anything about GST (hopefully more than those who understood Einstein – I just recall his conferences at Solvay, when he quipped to the same question, ‘maybe the priest’ -Lemaitre):

As all points in motion have 2 directions, upwards and downwards in the ∆-dimension, left-right in lineal geometry, more space or more time in existential topology, youth or old age (rebirth-reproduction or informative evolution).

Thus there are 2 solutions, since *the equation does not include a ‘choice’, which will be made a posteriori based in experimental evidence.* And so it is evident, as space is larger, and equations quantitative, that in general equations, the positive solution is space-like and the imaginary solution, time-like. And we shall elaborate a lot on that theme along the posts of the 3rd line.

The point to understand the underling structures of the Universe, written in its i-logic mathematical equations, of course is to respond to questions never asked for, as they seem to be the ‘a priori’ human categories of thought (Kant), which Schopenhauer rightly reduced to ‘sufficient reasons’, space, time, causality (mostly caused by ∆ scales in space-time) and the mind that perceives all (languages, ternary grammars).

The fundamental mathematical solution is the polynomial double root solution, where the polynomial is a function of present and the 2 solutions a ± split fields.

Presents thus *split in two dual solutions a ± dual field which is essential to understand all time of equations, from the equation of death where the complementary body-head, field-particle system splits, to the inverse equations of social symbiosis in systems that plug-both past and future systems into a present vital form of which the most important happens in social sciences: *

Let us briefly then deal with the operandi of the 5th dimension treated extensively in ∆nalysis:

**DERIVATIVES AND INTEGRALS**

As we did with the other operandi, we need to consider the properties of calculus and its two operandi. This poises a problem, as there is not a ‘bottom operation, such as ±, x÷, directly related as with powers as the third dimension of calculus. But as calculus is a refined analysis of power laws, the direct connection is not exact.

Hence a certain discontinuity is established what implies that ∫∂ equations have been solved by the obvious method of applying the function h’ (x)= lim h->o h (x+h) – h (x)/h. We are not though to repeat here that procedure to get the results but merely analyse from ∆st perspective as we did with power laws, and x, the properties of derivatives, to see what they tell us in the higher T.œ language and then consider some specific functions and its integral and derivatives to learn more of it.

**Properties.**

Those key properties are expressed in its rules of calculus, starting from the ‘derivative’ of a polynomial:

Xª= a Xª‾¹

So we are NOT fully lineally diminishing a polynomial dimension despite being derivatives a reduction of dimensionality – the search for the finitesimal 1/n quanta. Why? Obviously because in the rough view from a quanta, xª into its whole xª+¹, we grow lineally (polynomial), but as we *repeat ad nauseam, the lineal steps curve into geodesic closed wholes, in the ∆+1 scale (Non-E geometry), from the lineal spatial mind to the wholeness cycle of the closed being, and so as the ‘curve of a parabola’ diminishes the distance of a cannonball, growth is NEVER lineal but falls down as we approach the ‘(in)finite limit’.*

IN THE GRAPH, the wholeness is curved upwards, the parts spread scattering entropically. The whole is a mind circle, @. So it curves/diminish the quantity of energy available, for the whole, as it really must be an addition of all the planes that share that vital energy to build ever slower, curved larger wholes.

*Or in terms of the integral function: *

And here we find the second surprise. There are ∞ integrals with the addition of a constant. As a constant by definition does not change (so as a kid know it goes out on the derivative). But ∆st gives new insight in things ‘children of thought’ (: all huminds 🙂 think they know.

Let us express this then in terms of past (∆-1: derivative )< Present (Function) > future (∆+1: Integral)

*The past is fixed, the infinitesimal enclosed, only one type of species, ‘happening already’, as the parts must exist before the wholes to sustain them. But from the pov of the present function, the future integral into wholes is open, with ∞ variations on the same theme; unless we have already enclosed that whole, limiting its variations, which happens with the definite Integral.*

So if the function f(x) is given on the interval [a, b] and, if F(x) is a primitive for f(x) and x is a point in the interval [a, b], then by the formula of Newton and Leibniz we may write:

Here the integral on the right side differs from the primitive F(x) only by the constant F(a). In such a case this integral, if we consider it as a function of its upper limit x (for variable x), is a completely determined primitive of f(x). That is the importance of *the enclosure membrane to define a single organism, and establish its order, as opposed to the entropic, multiple open future of a non-enclosed vital function which will scatter away.*

Consequently, an indefinite integral of f(x) may also be written as follows:

where C is an arbitrary constant, *the enclosure will eliminate. *

**Linearity: **Yet and this seems to contradict the previous finding, when we operate derivatives with the ‘basic dimension’ of social herding, ± operators, linearity comes back, and so the minimal Rashomon effect give us two explanations:

**Γ(st): **We are INDEED herding in the base dimension of a single plane, where each derivative will now be considered a fractal point of its own:

**∆+1 perspective:** Suppose f(x) and g(x) are differentiable functions and a and b are real numbers.

Then the function h (x) = aƒ(x) + bg (x) is differentiable and h’ (x) = a ƒ'(x) + b g'(x), which is really the *distributive law already studied in algebra’s post for x and power law. So the interpretation of the sum rule from ∆+1 is one of ‘control’:*

*WHEN operating from a whole perspective, the whole breaks the ‘smaller’ parts and its simpler dimensional operandi, +, to treat each part with its ‘whole action’ (in this case ∂). In brief the whole totally control the parts.*

**The Product Rule** used to find the derivative of a product of two functions, however differs slightly (in polynomials also had the distributive property). So h'(x) = [ƒ(x) x g(x)]’ = ƒ(x) • g'(x) + ƒ'(x) • g(x).

Here we shall bring a little explained fact – derivatives act in the inverse fashion to power laws, searching the infinitesimal, while power wholes (integrals) search the wholeness, and as we know the two directions of space-time are different in curvature, quantity of information and entropic motions.

So an external operation that reduces a whole which is NOT integrated as such but a *lineal product of two wholes, ƒ(x) and g(x), a COUPLE, is mixing the infinitesimals of one, with the other whole before herding them; in a process of ‘genetic mixing’ of the parts of the first shared with the second whole and the parts of the second shared *with the first whole.

This law of existential algebra simplified ad maximal as usual in mathematical mirrors surprisingly enough is the origin also of genetic ‘reproduction’, which occurs at two levels, mixing the ‘parts’ – the genes of the whole – in both direction to rise then the mixing to the ∆º level of the G and F gender couple.

**Finally the chain rule WHICH IS TRULY the one that encloses all others is **used in the case of a function of a function, or composite function writes:

And this truly an organic rule, as we are not derivating on ‘parts’ loosely connected by ± and x÷ herds and lineal dimensional growth, but the ‘function’ is a function of a function – a functional, as all ∆+1 is made of ∆º which are also functions of xo fractal points.

So this is the most useful of all those rules to mirror better reality. And we see how the derivative, the change process deeps in at the two levels, at the ∆º=g(xo) level, which becomes g'(xo) and at the whole level, which becomes ƒ'[g(xo)], which tell us we can indeed go deeper with ∫∂ between organic scales, which is what we shall learn in more depth when consider partial derivatives and second derivatives and multiple integrals.

We are *getting so to speak into the infinitesimal of the parts of a whole from its ∆+2 perspective, and thsi rule encloses all others, because it breaks into the multiplication of its parts – DWINDLING TRULY A SCALE DOWN, AND SEPARATING THE WHOLE AND THE PARTS DERIVATED INTO LOOSE PARTS AND FINITESIMALS NOW MULTIPLIED. *

And what will the parts do when they see their previous finitesimals now camping by themselves but ‘at sight’ to get them to ‘produce’ an operative ‘action’ (a,e,i,o,u actions are ALL subject to the previous operandi), ON them.

AND WHAT WILL COME of that multiplication. Normally it will capture them all again and then normally will not re=produce on them (one of the operandi actions which are possible under the rashomon effect) but divide and feed on them the last operation to treat:

And its inverse, which is NOT a positive communicative act but often a perpendicular negative reducing game also consequently differs:

And we should notice here that the numerator, the victim, shared by the denominator the predator so to speak is first absorbed in its ƒ'(x) parts, g(x) ƒ'(x), subtracting the g'(x) parts that the prey has absorbed in the ‘fight’, ƒ(x) g'(x), and then shared by the bidimensional g(x)² whole as entropic feeding.

Of course, we love to bring vital interpretation to abstract math, but as we apply such rules to particular cases, the interpretations vary but in all cases will be able to be interpreted in terms of sub-equations of the fractal generator.

What might be notice in any case is that unlike in our rather ‘abstract’ dimensional explanation of the rules of power laws, here we are able to bring real vital analysis of those roles in terms even of biological processes, showing how much more sophisticated is the ∫∂ operandi, the king of the hill of mathematical mirrors on real st-ep motions and actions, reason why its use is so wide spread.

SO THE FUNDAMENTAL LAW OF OPERANDI TO VITALIZE THEM IS THIS:

‘BY THE RASHOMON EFFECT ALL differential OPERANDI CAN BECOME AN ACTION IN ONE OF THE 5D DIMENSIONAL VOWELS (A,E,I,O,U) THAT DEFINE THE FIVE dimensions OF EXISTENCE, AS VITAL QUANTA-ACTIONS OF THE BEING.

THIS IS THE LOGIC CONCEPT THAT TRULY VITALIZES THE OPERANDI OF ALGEBRA.

So those properties tell us new things about the meaning of ∫∂. And so do some analysis of the main functions with fundamental roles in ∆st and its derivatives:

In that sense the MOST important ad on that ∆st will bring to the use of differentials in EXISTENTIAL ALGEBRA, is its temporal use as the ‘minimal action in time’, of a being, a far more expanded notion that the action of physics (which however will be related to the lineal actions of motion on 1D).

IN THE GRAPH, the general 5 actions-dimensions of existence of different ∆±i species, from above down – a view of them all, one of the physical simplest light and electronic i<eye minds, and below the human being.

Mathematically it is quite irrelevant to make derivatives in time of human actions beyond some quantitative results – so the quanta of minimal human time, the second is the minimal informative action for the 3 synchronous t-st-s parts, an eye glimpse of mind-perception, a limb-ate of motion and a heart-beat of the body.

But it will be important for physical systems today only described quantitatively in abstract terms, to *vitalise and explain the organic whys of its space-time events.*

The connection on qualitative terms though is self-evident, for all scales, as most actions of any being *are extractions of motion, energy and form from lower ∆-i scales.*

So we and all other beings perceive from ∆-3 quanta (light in our case), feed on amino acids, (∆-2 quanta for any ∆º system), seed with seminal ∆-1 cellular quanta (electrons also, with ∆-1 photon quanta).

So derivatives are the essential quantitative action for the workings of any Tœ, space-time organism.

And so we study in depth the connection of the a,e,i,o,u actions between Planes (qualitative understanding) and its mathematical, analytic development (quantitative understanding of 1st second and 3rd derivatives – the late extracting ‘1D motion’ from the final invisible gravitational and light space-time scales).

SO THE FUNDAMENTAL LAW OF OPERANDI TO VITALIZE THEM IS THIS:

‘BY THE RASHOMON EFFECT ALL differential OPERANDI CAN BECOME AN ACTION IN ONE OF THE 5D DIMENSIONAL VOWELS (A,E,I,O,U) THAT DEFINE THE FIVE dimensions OF EXISTENCE, AS VITAL QUANTA-ACTIONS OF THE BEING.

THIS IS THE LOGIC CONCEPT THAT TRULY VITALIZES THE OPERANDI OF ALGEBRA.

And this lead us to the next chapter, which is the study on how derivatives allow us to point out *the main consequence of the sum of those actions in any being in existence, namely the fact that its sums tend to favor growth of information on the being and then signal the 3 st-ages and/or st-ates of the being through its world cycle of existence, which in its simplest physical equations is the origin of… *

## @-**GEOMETRY EXPANDS ALGEBRA**

ALGEBRA really starts to understand EQUATIONS with more sophistication that geometrical equations, when *motion parameters are introduced by analytic geometry and mathematical physics*, as the X and Y coordinates are now used for t-motions and S-pace (as in the simplest physical equations of space-distances made at certain speeds) that it can mirror S≈T Symmetries, first in a subconscious form through equations born on praxis, then in the 3rd age with some deeper insights on the concept of symmetry (group theory and its application to physics) which we shall complete with its full causal realisation – since we said ‘algebra understands’, *not* the people that make algebra.

The classic ST age of algebra thus saw the transformation of pure arithmetics of numbers into a mixture of the ∆ST elements of maths as a full reflection of the ∆•st universe. And the two huge figures that did it properly were Descartes and Leibniz (Fermat and Newton, in parallel but without publishing and the same clarity).

It is on my view the golden, classic age (if we add analysis) of algebra, before it enters the 3rd age of excessive information (attempts to put all the information in a single mind mapping with group theory and functionals and §ets).

We study most of analytic geometry in the post dedicated to it. So goes with analysis. Here we shall make just a few considerations.

It all started with the parallel evolution by Viette onwards of symbolic terms TO CONCENTRATE no longer in numbers BUT IN OPERANDI, THE TRUE essence of S=T symmetries…

**Descartes: merging all the elements of ∆@s=t maths.**

…And Descartes idea of representing solutions to equations *with a larger dimension – the variable letter that represented all the ‘§ets’ of dual X, Y possible solutions; and to ‘imagine’ them in a graph to plot them, forming a visual ‘in-form-ative’ geometric figure, the new ‘scalar dimension‘ that gathered all the X(S)<≈>Y (t) pairs of possible ‘variations’ on the space-time construct.*

FURTHER on, he introduced, @, the point of intersection of the coordinate axes, having coordinates (0, 0) and hence a ‘p.o.v.’ or singularity.

And with the introduction of coordinates he constructed an “arithmetization” of the plane. Instead of determining any point geometrically, it is sufficient to give a pair of numbers x, y and conversely.

Up to the time of Descartes, where an algebraic equation in two unknowns F(x, y) = 0 was given, it was said that the problem was indeterminate, since from the equation it was impossible to determine these unknowns; any value could be assigned to one of them, for example to x, and substituted in the equation; the result was an equation with only one unknown y, for which, in general, the equation could be solved.

Then this arbitrarily chosen x together with the so-obtained y would satisfy the given equation. Consequently, such an “indeterminate” equation was not considered interesting.

Descartes looked at the matter differently. He proposed that in an equation with two unknowns x be regarded as the abscissa of a point and the corresponding y as its ordinate. Then if we vary the unknown x, to every value of x the corresponding y is computed from the equation, so that we obtain, in general, a set of points which form a curve.

*The deepest insight on what Descartes did is then evident: *

HE GAVE MOTION=CHANGE TO GEOMETRY, ADDING ITS TIME-DIMENSION; AND SO its method could be used to study the actions/motions of a ‘fractal point’ whose inner geometry of social numbers was NOW ignored, in the ∆+1 scale of its world.. And so the graph would be a perfect graph to study all the ACTIONS=MOTIONS external to a given being, becoming for that reason the foundational structure of mathematical physics.

This is often forgotten, as S and T dimensions are ill understood so for example, our template book defines it as:

“Analytic geometry is that part of mathematics which, applying the coordinate method, investigates geometric objects by algebraic means.”

Not so… even if in analysis we will find that the curves DO represent key features of the ‘arrows of change’ of the Universe, specially the ‘standing points’ of change of parameters of Space=Information, ST=energy and Time=entropy (or any other kaleidoscopic combination of ST), in essence *they represent the world cycle of the action or motion we study, with its 3 phases of starting motion, steady state, and 3rd informative age coming to a halt.*

It must be then understood, as evident as it is, *that the Rashomon effect should consider different perspectives on those curves and forms found in analytic geometry, expressing algebraic equations:*

**-Temporal view:** the curves are then meaningless in space. What matters is their ‘social dimension’ that resolves symmetries between time dimensions expressed by the two variables often a parameter of space that changes with a dynamic function/action/motion in time.

**-Spatial view: **It is still though possible to create meaningful closed forms, ∆+1 wholes of geometry, made of ∆-1 points, and then the geometry allows to resolve algebraically geometric spatial problems, with ‘a dual point of view’ that increases the easiness of solutions – as Descartes proved easily and Galois completed, showing the algebraic laws of solution of rule and compass geometrical problems.

– **S=T** view: when one of the parameters/dimensions is fixed, belonging to space and the other to a time motion, the most fruitful in symmetry, soon used by Galileo and Newton to develop the laws of lineal time motion in space.

–**@** views: soon to be developed as 3 different mappings, will develop (O-polar, |-cylindrical and ST-cartesian proper)

**Scalar view,**which will have to wait till Leibniz, treated in ∆nalysis.

Let us then consider some aspect of the essential elements of those equations, mostly polynomials as analysis is treated in its own right.

**The earliest age of analytical geometry. The Arabs.**

Algebra we said started as a prolongation of arithmetics and geometry. So its first age was really true to the quote of Germain: a written geometry. This first age went as far as the renaissance, and consisted in calculus of ad maximal cubic roots, and geometric proofs (greeks, al-aljoarizim which gave its name to the discipline, etc.)

*Polynomials of higher, >2 degree as combinations of simpler bidimensional $-ð systems.*

The graph shows the maximal depth of this age when the poet Kayyam solved the simplest cubic equations. It is remarkable that now and since Apollonius and Archimedes, the understanding of ‘square dimensions’ *transcends the mere spatial square, introducing ‘temporal’ functions, such as the parable, whose value as motion=time dimension will be fully realized with Galileo’s study of cannonballs. *

It shows also a quality of the simplest easiest to solve roots of such equations, which are NOT really ‘cubic’ dimensional growth *but intersections between two figures, hence an S(x) ≈ T (x) symmetry, in this case the parable and the circle when properly written as Kayyam did. *

And this brings the question on how so many mathematical S≈T symmetries become hidden by the ‘mania’ of scientists to find only the solution, and packing all the variables into one side, putting in the other a zero… when the existence of a zero *most often means there is indeed a possible symmetry when both sides of the equation part their ways. A fact truly important to interpret physical equations in terms of those S(x), T(x) symmetries.*

Indeed, what the graph shows is that a *cubic equation, according to the holographic principle, is most likely a combination of an $-motion or 4D-entropic, open expanding curve (the parabola) and a ð-motion, the circle:*

In the graph, repeated ad nausea, for clarity, we see the general rule: a bidimensional St system of information (still space), or a time clock (moving cycle), the 3rd dimension of reality; and a vector of lineal time motion or its bidimensional sheet of spatial distances, the 1st dimension of reality, come together into a the 2nd Dimension, ST system of energy, or time or motion.

So 3Dimensional systems tend to be the intersection of a line and cycle, which in geometry is expressed by the rule that almost all functional dominant forms of the Universe can be traced with a line-ruler and a compass-cycle.

* * Generally speaking algebra is then just a mirror image of the geometry of the age, specially in the calculus with pythagoras like theorems of square roots and the simplest ±, ≈ X ÷ operandi *without considering the mirror image of those operandi in other ∆st elements and symmetries of the Universe.*

**Inverse operandi.**

The concept of ratios vs. multiplication – the 2 inverse arrows of 4D and 5D actions (feeding and breaking vs. social evolution), were neither understood – they are not yet.

*axons= ST flows of communication of 2 wholes that reach through X its maximal communication. So its role is*to create a network on the lower scale of systems – in brains, neural networks or reproductive systems,

*reason why so often a reproductive action that creates a third element as in vectorial products is symbolised by a reproduction*.

*the division actually is not made, meaning that the whole remains – the string we plug to get a sound by making it vibrate with a certain number=frequency of nods.*

*is an insight of ∆st on irrational numbers, the operation is not made; hence the number exist as a whole, which is expressed in the numerator, sum of the parts which are expressed in the denominator.*

**First degree equation**.

Making use of two simple ideas, Descartes first of all examined what curves correspond to an equation of the first-degree:

i.e., to an equation where A, B, C are numerical coefficients with A and B not both zero.

As we have seen this is the ‘dimensional natural growth from sum into a sums of sums or multiplication understood in terms of its simplest dimensional combination’: $ x D:

So obviously Descartes found that in the plane a straight line always corresponds to such an equation. And conversely, that to every line in the plane there corresponds a completely determined equation of the form:

where:

y = kx. obviously represents a straight line passing through the origin and making an angle ϕ with the x-axis whose tangent tan ϕ is k and L the distance from 0 to the crossing point of the line and y. Thus as usual the simplest, lineal Spe element was discovered, and it is the easier to calculate, *but in a world dominated by time cycles:*

“Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.” Ulam.

So *there is an enormous number of scientific errors, including the lineal big-bang caused by the ab=use of lineal **approximations to functions, and the ‘spread’ of the function into the negative ‘side of the line’ (case of the BB) as if it were to behave always lineally in the past-negative side of the graph. *

In fact most ‘real functions’ when we fully grasp 1) the distortion of past-time and negative numbers and the ‘conversion’ of a cartesian graph into a ‘conic’ tend to have an e-like form.

In the graph, the distortion of ‘lineal’ self-centred functions is a plague of sciences, due to its easier calculus and the distortion of perspective caused by the human mind, as Descartes coordinates are a representation of the human lineal, Euclidean, electromagnetic light-mind geometry. *In the next graphs, *from the 92 book ‘the error of Einstein’ we show what we mean:

The mind equation is:

* O (infinitesimal pov) x ∞ Universe (∞ cycles and monads) K-mind*

It implies that the 0 believes to be infinite self-center of the Universe and sees it all in a distorted perspective. In brief, your eye pov is bigger to you than the Andromeda Galaxy, so the 0-point becomes an infinity in itself (your eye≈Andromeda Galaxy), and this makes the conic, which could be considered an ‘objective’ angled point of view (where the mind-eyes is truly zero), to expand in the y-coordinates of the mind-point to infinity; alas! transforming the conic into the Cartesian plane where ¥ becomes also an infinite graph.

**Conics and circles.**

IN THE Growth of dimensions by multiplication it is obvious that the most abundant combination will be that of an $t dimension and an §ð dimension, creating a full present ST-system, according to the canonical generator: $t<ST>§ð.

So after doing all what they could do in bidimensional geometry with a ‘ruler and a compass’, the Greeks ‘finally’ raised one circle with a line into a conic in space, which then will become the canonical space-time surface also when time was added to it (4D formalism).

Now in as much as a double conic is a hyperbolic geometry, we can define the Cartesian plane as the ‘hyperbolic’, Present, ST plane, more ‘expansive’ in its capacity to show different ∆st events and forms of the Universe.

We only got a grasp of this fact, of lately in the study of topology and models of hyperbolic geometry, when we find that *a hyperbola, which seems to us infinite in its ¥ coordinates, becomes equivalent to a circle . *

*Mathematicians say that the hyperbola is isomorphic to the circle; and write it as:*

If u=x+yi and v=x−yi then:

ℂ[x,y]/(x²+y²−1)≃ℂ[u,v]/(uv−1)

This can be viewed in algebraic geometry but better on to understand is mind meaning in projective geometry:

Both the hyperbola and the circle are conic sections, and are projectively equivalent. In analytical geometry, in homogeneous coordinates this follows from the fact that any pair of nondegenerate indefinite quadratic forms are bidimensionally equivalent, and so *we can transform them into each other, as it is required by the generator of coordinates.*.

This beyond ‘philosophy of the mind’ matters because it justifies the fact that *the curves of analytical geometry can be also obtained from cuts made on the cone, which in itself is merely a reflection of a **world cycle, pegging the two inverse directions of existence. *

**The conic as a representation of a world cycle.**

The cone is basically a ‘bidimensional ST being, circling inwards towards the singularity of the point along a line of geometry, hence combining O x |= Ø, the cone represents the Universe. It should not be surprising then that it is the best way to reflect a 4D block of space-time (but not the discontinuities of the 5D Universe) as used by Einstein’s physics and Minkowski’s geometry (light cones); and we shall deal with those cones in the analysis of relativity.

We shall also use them in ¬time logic to represent the 3 fundamental events in terms of time ages: entropic collisions (future x past = past wave of entropy, reproductive iterations (present x present = present), and evolving, informative events, past x Future = Future wave of information):

In the graph, first the 3 canonical events of complex i-logic time, and the much simpler single cone of ‘lineal time’ in relativity.

What matters to us is now is that a conic basically reduces to a line and a shrinking cycle, which in motion is equivalent to a time cycle, shrinking into a point-singularity, the vertex of the cone. And for that reason that simple canonical world cycle encodes all the main curves of bidimensional space.

**LINEAL ALGEBRA.**

**Lineal equations as approximations of curves.**

The linear function l (x) = ax + b gives the simplest of all curves, namely the straight line AND YET it is one of the most important due to the fact that every “smooth” curve on a small segment is like a straight line, and the less curved the segment is, the nearer it comes to a straight line.

All THIS resumes a concept brought about many times in the blog: a larger perspective is a wholeness, which is a closed zero sum and hence cyclical, curved, a step of the curve however appears in small distances as a lineal, open step.

In the language of the theory of the functions, this means that every “smooth” (continuously differentiable) function is, for a small change of the independent variable, close to a linear function.

The linear function can be characterized by the fact that its increment is proportional to the increment of the independent variable.

Indeed: Δl(x) = l(x0 + Δx) − l(x0) = a(x0 + Δx) + b − (ax0 + b) = a Δx. Conversely, if Δl(x) = a Δx, then l(x) − l(x0) = a(x − x0) and l(x) = ax + l(x0) − ax0 = ax + b, where b = l(x0) − ax0.

But from the differential calculus, we know that in the increment of an arbitrary differentiable function is proportional to the increment of the independent variable, and that the increment of the function differs from its differential by an infinitesimal of higher order than the increment of the independent variable.

Thus, a differentiable function is, for an infinitely small change of the independent variable, really close to a linear function to within an infinitesimal of higher order.

The situation is similar with functions of several variables.

A linear function of several variables is a function of the form a1x1 + a2x2 + ··· + anxn + b. If b = 0, the linear function is said to be homogeneous.

A linear function of several variables is characterized by the following two properties:

1. The increment of a linear function, computed under the assumption that only one of the independent variables receives some increment while the values of the remaining variables are unchanged, is proportional to the increment of this independent variable.

2. The increment of a linear function, computed under the assumption that all the independent variables obtain increments, is equal to the algebraic sum of the increments obtained by changing each variable separately.

Thus a linear problem can be characterized by 2 properties:

1. The property of proportionality. The result of the action of each separate factor is proportional to its value.

2. The property of independence. The total result of an action is equal to the sum of the results of the actions of the separate factors.

The fact that every “smooth” function can be replaced in a first approximation by a linear one, for small changes of the variables, is a reflection of a general principle, namely that every problem on the change of some quantity under the action of several factors can be regarded in a first approximation, for small actions, as a linear problem, i.e., as having the properties of independence and proportionality. It often turns out that this attitude gives an adequate result for practical purposes (the classical theory of elasticity, the theory of small oscillations, etc.)

**Matrix – the 3rd dimension of lineal algebra.**

We have stated many times that the Universe grows by ‘fixing motion-steps’ into a whole ‘cyclical form of space’ that then moves into motion steps and so on till ‘filling up the 5D2 dimensions of reality.

And that *such increases are smooth-continuous only for the 3 fundamental Dimensions of a present space-time, breaking in the ∆-1 4D entropic and ∆+1, @ 5D levels, which means it ‘suffers’ a “Lorentzian region” of acceleration or deceleration inwards or outwards that changes the parameters in a different way (susceptible however to be analysed with differential equations that measure small changes.*

This means specially for ‘lineal equations’ that there are 3 levels of growth in complexity, from single lineal equations to multiple lineal equations and finally the grouping of those into ‘matrices’ as we make through representation theory convert some parameters (as in quantum physics) of T.œs=fractal points with multiple inner parts into the ‘new algebraic element’.

Indeed, the physical quantities to be studied are often characterized by certain numbers (a force by the three projections on the coordinate axes, the tension at a given point of an elastic body by the six components of the so-called stress tensor, etc.). Hence there arises the necessity of considering simultaneously several functions of several variables, and, in a first approximation, of several linear functions.

A linear function of one variable is so simple in its properties that it does not require any special study. Things are different with linear functions of several variables, where the presence of many variables introduces some special features. The situation is still more complicated when we go from a single function of several variables x1, x2, ···, xn to a set of several functions y1, y2, ···, ym of the same variables. As a “first approximation” there appears here a set of linear functions:

The set of coefficients of a system of linear forms can be given then the form of a rectangular array:

Such arrays bear the name of matrices. The numbers aij are called the elements of the matrix.

Important special cases of matrices are the matrices that consist of a single column, which are simply called columns, those that consist of a single row, called rows, and finally the square matrices, i.e., those in which the number of rows is equal to the number of columns; called its order (a).

Thus *matrices, specially square ones can be considered if we define an Y(S**t) = X(sT) FUNCTION in which one variable most likely a Spatial whole, polynomial combination of a series of temporal steps-motions; a symmetry of spatial wholes and temporal variables in equal quantities, ∑S≈∑T.*

AND as the Spatial wholes will be ‘varieties’ of the same temporal ‘steps’, the structure interconnected at ‘expanded’ to the 3 levels of complexity has a very rich capacity to picture complex space-time systems in all its variations and symmetries, reason why Matrices have become the best-suited structure for complex, lineal systems of ‘very small scales’ where information about fast multiple T.œs come together and have to be studied and fixed in space from the larger, slower human perspective:

i.e. quantum systems where* the Heisenberg matrix formalism is THE algebraic frozen symmetry of Schrodinger’s dynamic ‘differential equation’*, a conundrum, never clarified, now explained tersely as the natural consequence of the ‘space-symmetric nature of algebra’ vs. the ‘steps-motions description of Analysis’.

We shall not keeping with the limits of an encyclopaedia written by a single man, occupied most of his time with self-destruction go into techniques of Matrix manipulation. Just to state that the main difference between Matrix and other structures of algebra is its obvious non-commutability (as rows and columns multiplied one by one in the well-known inverse orderly fashion) and its non solvability for multiple cases (as rows and columns must coincide).

In a larger philosophical way this simply means that Mathematics as all languages is ultimately inflationary, with more imaginative ‘forms’ that real solutions, which leads us to the GREAT question of the first algebraic age – THE SOLVABILITY OF POLYNOMIALS HIGHER THAN 2 *in a Universe where lineal structure and continuity breaks beyond the third polynomial since ST dimensions come onto pairs… *

**BIDIMENSIONAL PLANE-POLYNOMIALS.**

**The fundamental theorem of Algebra.**

Theorem of equations proved by Carl Friedrich Gauss in 1799. It states that every polynomial equation of degree n with complex number coefficients has n roots, or solutions, in the complex numbers:

In the graphs, the solvable polynomials define a limit of 3 dimensions of space-time, and a 4th and fifth dimensions that *cannot generally be resolved by radicals in a single Plane, hence belonging to the 4 D S∂ and 5D ∫@ DIMENSIONS, which* warp the whole, as it emergence into the fifth dimension, hence making impossible further dimensions in a single continuum**. **Notice that the fourth dimensional graph must be interpreted often turned upside down as it is a ‘death’ inverted time arrow.

They define the bidimensional tridimensional fourdimentional and five dimensional arrow and similar systems in those dimensions.

The first is a mere motion of an st-lineal trajectory proper of the concept of speed (1D) or 3D (closed cyclical vortices.. THE third dimension though is more sophisticated

**Introduction to GST homeomorphisms with polynomials.**

Following the fundamental theorem of GST mathematics – that mathematics is the most evolved language of space-time of the humind and hence it reveals very profound laws of ∆ST, we shall now consider a few examples of it.

And yet, there are NO solutions granted to a quintic function from radicals:

Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess an additional local maximum and local minimum each.

**ST Interpretation**

What this mean though in GST is far more interesting. Indeed if we consider polynomial, dimensional equations, of D-egree equal to its number of dimensions, it is then obvious *that each continuous scale of space-time does have 4Dimensions, which can be solved. Yet beyond those 4Dimensions the system ‘breaks’ into a new scale which is no longer in the same region of existence and hence cannot be calculated with the radicals of the same plane.*

Further on in the graphic, we can easily see that the 5D form can be ‘reduced’ into a 3-equation where the 3 central ‘st’ hyperbolic curve, becomes a around the y=0 point a single ‘higher’ whole, which ‘if reduced’ to a point, converts the quintic into a ‘ternary equation’. Thus a 5D polynomial is a 3D ∆+1 polynomial with an ST central part which deploys its hyperbolic minimal and maximal elements in the ‘higher plane’ region of the 5th dimension (truly a higher plane when we use complex numbers).

What then it means in general terms, the existence of several solutions to those equations? In the most abstract analysis of polynomials, if we deem a degree, D, a dimension D, then it is obvious that as we solve them we come down from the whole into the parts. Thus a quadratic bidimensional equation resolves into the ± inverse spe and tiƒ functions of the system. A cubic equation will solve into the Sp<st>Tif solutions/parts of the system; and a quadratic equation into the S, E, T and sub-dimensions of the system.

**∆-interpretation**

But of course a polynomial being the most abstract realisation of a GST FRACTAL equation can be many things and yet the beauty of it is that all find interpretation. So when we consider x to be a function of time, it is given us the arrow of ∆-1 information which has several values, as information increases downwards diversifing the system, but the inverse is NOT truth. While St=x² has 2 Spe and Tif roots, the square of a number is a single number, ratifying the inverse arrows of entropy and information upwards and downwards:

In the graph, upwards the polynomial looses solutions, downwards increases its information, normally breaking an st WHOLE INTO its SPE and Tiƒ components.

**Quadratics: open and closed**

When studying the most abundant of all equations which are naturally quadratic equations, we can observe two basic type, closed and open, closed are time ones, transcendental polynomials with a cyclical form the other quadratics are opened ones. And finally we find, transcental, sinusoidal forms, or st equations, which will be found to be the most useful ones.

All this said, quadratics are the masters of the algebraic game, the most abundant an common of all forms in existence, *because the Universe is bidimensional, and so it is information, so quadratic equations by definition are the perfect form to show the properties of the Universe of fractal bidimensional space and informative, bidimensional time:*

In the graph, the bidimensional holographic principle explains why in geometry (greek bidimensional plane, which proved almost all the theorems of geometry) and algebra (quadratics) almost all phenomena of the Physical, topological universe can be ‘carried’ on to quadratic algebraic equations. We study the main forms of quadratics in those other ‘parts’.

**Cubics.**

Alas, we enter now into a different territory; that of cubic representation in a quadratic plane of the 3 dimensions of a ternary universe in space (3 topologies) and time (3 ages). And so here 4 themes are fundamental to ¬Æ:

- How cubics might represent the ternary structure of the fractal generator.
- What distortions take place when we transfer the holographic Universe into cubic representations (when not using them to represent Γ)
- What are the restrictions the Universe imposes to ternary systems, where
*the ‘messed bidimensional, tight form’ becomes a loose configuration of layers of time-motion or space-density to become a 3rd dimensional ‘wide’, iterating being.*

For example, the Fermat Grand theorem, mentioned above, x³+y³≠z³ means a restriction of ‘tridimensional messed beings’, as all what we can expect are bidimensional perfect forms, accumulated in time, as slices of a motion, or in space as layers of an identical – number/population of bidimensinal beings.

- And finally, the translation of all the results of cubic equations,
*once we understand the previous rules of engagement, into meaningful laws of GSt (or vice versa)*

**Quartics.**

We said that quartics define systems of two topologies of space-time intersected into a single form; hence being natural solutions to the study of Sp<≈tiƒ systems. This intuitive thought on the nature of quartics comes to fruition when we observe that each coordinate of the intersection points of two conic sections is a solution of a quartic equation.

The same is true for the intersection of a line and a torus. Thus we find a quadratic solution the ideal ‘form’ of an intersection of and Spe and Tiƒ system, which constructs a whole form. And so no further polynomials are required,

**Some quintics with solution**

On the other hand the only quintics with solutions are not ‘really quintics’, but either systems that can be factored as smaller polynomials (hence each part of the polynomial being of a lower degree, and ‘unit of an S<st>t system); or can be depressed eliminating one of the roots, often the 4th giving birth to interesting solutions in which the 5th polynomial appears as equivalent to an scaling sum of the lower ‘planes’, *with scalar coefficients precisely of the key social numbers, 1,2,3 and t and 10. For example,in the graph the following are the only solutions for a depressed quintic, where the 4th polynomial disappears, and so we do have a relationship between the ∆+1, quintic and the coefficients that are ‘solvable, all of them in precise scales common to ∆>∆+1 processes.*

**The fundamental theorem of algebra.**

Now, all this said, there is a seemingly contradiction in our stressing of ‘no proper solutions’ for polynomials beyond the 3rd plane/dimension of growth, stated by its unfocused mirror better analysed in ‘analysis’ and the fundamental theorem of algebra states that every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots…

Which is equivalently (by definition), to the theorem states that the field of complex numbers is algebraically closed, whose proof must use the completeness of the reals, *which is not an algebraic concept…*

As completeness implies that there are not any “gaps” (in Dedekind’s terminology) or “missing points” in the real number line since those “gaps” should be covered by ir(ratio)nal numbers, according to the non-proved completeness axiom, or the 2 theoretical methods used to prove its construction (Dedekind completeness and Cauchy completeness as a metric space).

We rebut Dedekind completeness proof i-logically when considering an absolute geometry without that ‘axiom’ in non-e geometry, as irrational numbers are NOT single plane numbers or ELSE pi would exist and so √2, but RATIOS OF CLASSIC actions of the generator,, such as pi = 3 $>ð, the transformative motion of 3 lineal steps which generate a ‘variable’ according to the curvature of the ‘mental space’, pi-ratio. So for √, the ratio of two perpendicular T.œs, colliding OR symbiotically becoming adjacent, creating a triangular space.

Further on, proofs must be consistent to be complete – and here is where the humind makes most errors. You do NOT proof an algebraic system absolutely with a geometric proof, as numbers and points, as we have shown in NUMBER THEORY are NOT equal but similar S=T mirror reflections.

All this said, yes, there are always n-roots for a polynomial in the complex plane, but what we mean is that *they are NOT exact solutions, but beyond the 3rd power, approximate solutions, reason why for a complex polynomials TRYING TO REFLECT REALITY, often the best solution is NOT the polynomial approach but the derivative approach, as said before.*

On this a final coments: While polynomials cannot be always resolved by lineal coefficients, proper of the balanced central region of any Plane of social evolution, ∆§, because the ∆-representation of complex numbers is really a ‘square’ graph, as we showed on our analysis of its Rashomon effect (argand, polar, ð-numbers, and so on), in practical terms, this means we halve the polynomial degree and Xˆ10 becomes Xˆ5, the limit of dimensional growth through ∆§cales and planes… And here we can find at least a meaningful polynomial approach. Beyond that is truly inflationary mathematics; and some not-so-exact-as they think string theory bullshit (:

**II AGE: ****ST-SYMMETRIES **

**OPERANDI & GROUPS**

**Foreword. The heart of the matter: group symmetries, operandi details.**

WE HAVE ARRIVED to the heart of the matter of all algebra, *the operandi that reflect its S≈T symmetries. *

We haven’t talked before of it, not to repeat ourselves too much because we want to treat it FULL RANGE, as its importance is GINORMOUS for our mirror-focused view of reality as it is; and to that aim, we want to bring the most modern view of them – *through the language of modern symmetry, Group theory, and the properties of those operandi found in modern mathematics, the 10 ‘properties of sums and products, etc’. *

So first to introduce what is most important of modern maths (Group theory) and SHUN OFF what is largely irrelevant or redundant (set theory, on my view redundant, axiomatic method, plainly a huge ego trip that distanced math from reality).

We shall then first consider group theory, then *get to the heart of the mathematical ‘matter’, OPERANDI, and then just briefly consider very complex dimensional growths of algebra (functionals on Hilbert spaces), which we shall treat time permitted on mathematical physics fourth line, make a couple of comments on Sets and move to existential algebra.*

How to relate operandi and groups is obvious according to the ‘method of growth/creation’ of the Universe, repeated ad nauseam, operandi are the smaller step, the flow-time detail, Groups the larger spatial whole portrait in the mind-stillness of them all. That is why we shall start from groups down in this case to see the forest before the trees.

**GROUPS**

Group Theory is the main element of the third age of mathematics along set theory; whereas a set is basically ‘anything’ and so quite void of meaning. And we shall just use it for respect to mathematicians, but whenever possible change it for more specific concepts, either ‘fractal points’, or ‘social numbers’ or ‘T.œs: Time§paœrganisms’.

This said we can define the two essential concepts of group theory, symmetry transformations and groups *in terms of T.Œ.*

Symmetry transformation *are the allowed ST-eps and motions in 5D² that keep the S<st>T system co-invariant *

By definition they are the ‘allowed’ motions in as much as if *the system changes outside** the stable parameters of the Generator Equation, it will obviously become broken and die, and no more repetitive motions will be permitted.*

*In abstract mathematics,* groups are merely the ‘collection’ of all possible transformations that keep the system invariant; and as languages are inflationary there are ∞ groups and among groups there are those with infinite elements, such as motions in a plane, and those who have only finite transformations.

As ∆s=t is a realist model of the Universe many of the mathematical ‘curiosities’ and ‘monster groups’ are of little relevance to us. We are mostly interested in those groups closely related to the Generator equation, expressing the allowed transformations of the Generator that matters to reality and existence.

So in ¬ Æxistential algebra we can also talk of Existential Groups, where G is quite closer then to the concept of a Generator, which becomes *the group of all possible Existential groups. *

Among them obviously the most important is the group of physical motions, which connects directly with the concept of ‘motions in the fifth dimension’.

In mathematics though the origin of G was in the group of Galois and group theory applied in his young r=evolutionary age to prove that quintic polynomials are NOT soluble, which is an immediate consequence of the structure of the Universe in Holographic Bidimensional ‘units’ which ad maximal can be made to intersect to create 4Dimensional systems (and or consider 4D geometries as Relativity does, by studying 3 S and 1 D of lineal time, in a ceteris parries analysis) .

But 5D systems do NOT exist in a single plane and as polynomials unlike differentials are NOT good for studies across multiple planes of an organism but rather for social growth, herding and simpler lineal systems, quintics really belong to inflationary maths.

They are not solvable because they are NOT real.

Latter Group theory expanded to model almost everything in maths, because it allows to ‘collect in a single mental form all the possible variations of a system. Since ‘ the mind perceives motions in time, static as forms of space, ‘reducing its information’ to what it matters to it – mostly the stable points of those transformations, group theory highlights in an elegant way through the concept of symmetry, those stable points of a motion, and it is a good way of ‘limiting’ knowledge to the key elements.

The same happens with variation theory, where the system is reduced to the standing points and minimal and maximal variations – S=T, Max $ x min. ð and Max. § x min. t, which are the 3 ‘age inflections’ on a world cycle of a being; hence its enormous utility.

So as the fundamental feature of ‘mental processes’ is to reduce the time flow to the ‘key points of in-form-ation and trans-form-ation, *eliminating as much as possible the repetitive cycles within the flow *– from palingenesis to languages to biopics – ESCAPING those middle motions of self-repetitive information, (as motion and translation is just iterative motion of information), all the ‘blurred’ transformations or in-between positions are discharged by efficient mind-models of reality.

* Variational theory*

This said, group theory is better divided for its applications to GST (generator of space-time that here we could call Grousp of space-time beings), into:

-Time Groups: ‘Groups of motions’, which concerns with the points in which the motion of a space-time entity becomes transformed into a symmetry of itself. That is mainly related to the allowed motions in space-time.

– Space Groups: ‘Groups of transformations’, which concerns with mirror reflections and symmetries that classify species of reality, as in SU3 groups of physics with its octets and decuplets of particles, which is mainly related to the variations of the Generator, which define also the species of reality.

Let us consider them in more detail.

**Time groups: the allowed motions in 5D.**

Noether’s theorem considers that each conservation law of Physics, conservation of Lineal Inertia (1D-motion) or angular momentum (2D motion) and energy (3D motion), to which we *add the 4th conservation law of 4D entropy and 5D social evolution or ‘zero sum worldcycles of existence’, is related to a symmetry of space-time. *

What this means in the clearer jargon of 5D² is this. As a symmetry in space-time means a motion that carries a form from a point of space-time to another point of space-time *without deforming its inner structure; the 5 LOCAL, DIFFEOMORPHIC, fractal motions of the Universe, one belongs to each of its Dimensions of space-time are conserved, because they correspond to a motion that does NOT destroy the local T.Œ, which remains invariant after its ‘translation in space-time’.*

In the case of the 4th and 5th dimension this process has 2 readings: during the birth to extinction phase of the cycle, the product of the motion and information of the system (1D+3D=2D) remains invariant. And if we ad the 4th dimension of death=entropy and the 5th dimension of birth=generation (social evolution from micro-seed to emergent organism) the total is also a zero sum: ‘dust you are and to dust of space-time you shall return’.

Now, scientists, even if they do not confess it have still to explain themselves in simple terms as we have done, what really Noether’s theorem and its symmetry conservation principles really mean because *they do NOT consider it in terms of local fractal space-time beings, T.œs for whom the principle holds. So the principle is universal because the Universe is the sum of all its local T.œs: *

*T.Œ (UNIVERSAL SUPERORGANISM) = ∑T.œs. *

So by extension generally speaking:

*‘Any fundamental law proved for a local T.œ made to the image and likeness of the absolute T.Œ, can be extended by Ð-isomorphism to all its species’. *

This said we can extend these concepts to its details using the jargon of Algebraic groups in the ‘simplified layman language’ of Mr. Aleksandrov and its awesome, now extinguished rational Soviet dual school of physics, which we use as the annotated base book for this introduction to non-Æ maths. No, I won’t use, refuse to use, will never yield to the pedantic false Axiomatic method proved wrong by Mr. Godel’s fascinating incompleteness theory made easier to grasp in some other post…)

**Motion Symmetries and Trans-form-actions**

*2D-motion: reproductive symmetry.*

We begin with an account of the simplest forms of symmetry with which the reader is familiar from everyday life. One of these is the mirror symmetry of geometric bodies or the symmetry with respect to a plane:

A point A in space is called symmetrical to a point B with respect to a plane α (figure 1) if the plane intersects the segment AB perpendicularly at its midpoint. We also say that B is the mirror image of A in the plane α. A geometric body is called symmetric with respect to a plane if the plane divides the body into two parts each of which is the mirror image of the other in the plane. The plane itself is then called a plane of symmetry of the body. Mirror symmetry is often encountered in nature. For example, the form of the human body, or of the body of birds or animals, usually has a plane of symmetry.

*Its origin as we explain in ¬Æ geometry is the ‘elliptic nature’ of @-minds with its singularity connected to two antipodal points, which ‘perceive from the singularity’ appears as inverse; and so as the @-mind, the 5th dimension of order of any T.œ proceeds to ’emerge’ by reproduction and organisation of its clone cells it WILL create mirror images of its code, because when it ‘looks left’ it positions things to his right and when it looks right to his left, and so bidimensional symmetry is a strong proof of the vital topology of the Universe and the capacity of singularities to create reality.*

To understand this just rise your left and right hand with its mirror images, both lateral sides of your head and glimpse at them alternatively – you will *see each finger as being in the opposite side of your head, in the same place of space. So as the singularity webs its organism, it creates bilateral symmetry.*

Mirror symmetry is thus the origin of a Dual Fundamental Motion of the Universe, the emergence and creation from a central singularity point of a bidimensional T.œ, through *the stop and go, motions of reproduction and informative evolution. And for that reason is not a simple motion but a combined motion (2D+5D).*

Now, it is interesting to consider what *science cares for in the mirror symmetry, and what GST cares for: in mathematical physics, with its concept of lineal time-motion as the only arrow of time, all what matters is to measure motions in space; so what matters about mirror symmetry IS ONLY to study HOW a mirror moves to occupy the position of its inverse mirror. *

And that is fine, but studies nothing of the other motions=change in the form of beings, its creative process of creation=reproduction of information, and ultimately the whys of the Universe. Which is what MATTERS to the philosopher of science and GST explains. And further on ALLOWS THE EXPANSION OF THE LAWS of Existential Algebra to ALL scales of reality, as Mirror symmetry IS the fundamental process of creation of bilateral forms, from DNA to Proteins, from Geological ‘fractal continents’ and its ternary self-similar forms, (where a combined motion in scale is also needed) to the processes of crystal formation.

In classic science, then a second form of mirror symmetry is considered, which for ∆st is just another axis of antipodal nodes, merely extending the singularity through its internal axis, equivalent to the poles of the sphere. It is the…

*…Lineal symmetry.*

Symmetry with respect to a line is defined in a similar way, by classic science. We say that the points A, B lie symmetrically with respect to a line if the line intersects the segment AB at its midpoint and is perpendicular to AB (figure 2). A geometric body is said to be symmetrical with respect to a line or to have this line as an axis of symmetry of order 2 if for every point of the body the symmetrical point also belongs to the body.

A body having an axis of symmetry of order 2 comes into coincidence with itself when the body is rotated around this axis by a half rotation, i.e., by an angle of 180°.

**3D-motion: Rotational symmetry.**

The concept of an axis of symmetry can be generalized in a natural way. A line is called an axis of symmetry of order n for a given body if the body comes into coincidence with itself on rotation around the axis by an angle 1/n 360°. For example, a regular pyramid whose base is a regular n-gon has the line joining the vertex of the pyramid to the center of the base (figure 3) as an axis of symmetry of order n.

A line is called an axis of rotation of a body if the body comes into coincidence with itself on rotation around the axis by an arbitrary angle. For example, the axis of a cylinder or a cone, or any diameter of a sphere, is an axis of rotation. An axis of rotation is also an axis of symmetry of every order.Finally, a 3RD important type of symmetry is symmetry with respect to a point or central symmetry. Points A and B are called symmetrical with respect to a center O if the segment joining A and B is bisected at O. A body is called symmetrical with respect to a center O if all its points fall into pairs of points symmetrical with respect to O. Examples of centrally symmetric bodies are the sphere and the cube, whose centers are their center of symmetry (figure 4).

A knowledge of all the planes, axes, and centers of symmetry of a body gives a fairly complete idea of its symmetry properties.

This symmetry obviously corresponds to the 3D motion of timespace, rotary motions, cyclical particles and heads…

**5D-4D Symmetry motion…**

is obviously ignored by Humans, even if *all of them exist within that symmetry between birth and extinction, we have explained ad nauseam, in our description of worldcycles of existence, and its inverse arrows of time, death=entropy and social evolution=generation.*

So we shall leave it as it is explained better with the Fractal Generator, and it would be silly translate it into the more confusing terminology of Group and Symmetry theory.

**1D symmetry motion.**

We explain lineal motion as a form of reproduction in a lineal flow, as the T.œ reproduces in a lower scale and emerges back in a higher one, explained in our analysis of the achiles paradox. In the graph, the reproduction of a quantum par tile in a stop and go motion. Each motion implies therefore a reproduction of its parts in the ∆-1 scale and its emergence in the upper adjacent region as a new being, where the mind flow is a maya of the senses, as we ultimately die and live in the lower scales constantly (so all your atoms change every 3 months). The paradox of the ego is thus absolutely irrelevant, so is the concept of death.

**The motions of the Universe, in symmetry terms.**

*The general definition of symmetry. *

In mathematics and its applications it is very rarely necessary to consider all transformations of a given T.œ, made of a set of fractal points. The fact is that the T.œ IS an organic system, never a mere collection of fractal points, completely disconnected from one another. The sets discussed in mathematics are also abstract images of real collections, whose elements always stand in an infinite variety of interrelations with each other, and of connections with what is going on beyond the limits of the set in question (All worlds are mirrors of its Universe).

But in mathematics it is convenient to abstract from the major part of these connections and to preserve and take into account the most essential one. This compels us in the first instance to consider only such transformations of sets of points as do not destroy the relevant connections of one kind or another between their elements. These are often called admissible transformations or **automorphisms** with respect to the relevant connections between the elements of the set.

And as such *they represent in its closest approximation of classic algebra the concept of an ST-motion in any dimension of 5D² space-time, since it implies a translation in time and a reproduction of form in space, for which two concepts the ‘spatial distance of the translation’ and the quantity of information translated matter most, being both related by the concept of speed, V=S/T, we study in depth in the posts on astrophysics. *

So in space the concept of distance between two points is important both externally and internally. The presence of this concept forges a link between points which consists in the fact that any two points stand at a definite distance from one another; a distance which is measured in GST not as a mere line, but as a wave of communication or network that connects both points (2nd, 4th postulates of non-E). Transformations that do not destroy these connections are the same as those under which the distance between points remains unchanged. These transformations are called “motions” of space-time. *As they imply the inner stillness/fixed form remains invariant. *

And so *translations in space that do NOT change, either the inner content of information of the ∆-1 fractal points of the organism and its 3 network connections ARE THE EXPANDED CONCEPT OF AN AUTOMORPHISM IN GST.*

In this manner we can apply all the laws and concepts of Motions and Symmetries in Space of classic algebra to the fractal point, by expanding the line into a wave-network and considering that the ∆-i content of the point also remains unchanged.

With the help of the concept of automorphism it is not difficult to give then a general definition of symmetry, taken from classic Group theory; where set means a network of fractal points.

Suppose that a certain set M is given, in which definite connections between the elements are to be taken into account, and that P is a certain part of M. We say that P is symmetrical or invariant with respect to the admissible transformation A of M if A carries every element of P again into an element of P. Therefore, a symmetry of P is characterized by the collection of admissible transformations of the containing set M that transform P into itself. The concept of symmetry of a body in space falls entirely under this definition.

The role of the set M is played by the whole space, the role of admissible transformations by the “motions,” the role of P by the given body. The symmetry of P is therefore characterized by the collection of motions under which P coincides with itself.

It is then when we find the ‘equivalence’ between the motions of classic Physics, as described by symmetry and automorphism and the 5D motions of GST.

This are:

** 3D reflections, 1D parallel shifts, and 2 D rotations of space**, because distances between points obviously remain unchanged under these transformations.

A more detailed investigation shows that every motion of a plane is either a parallel shift or a rotation around a center or a reflection in a line or a combination of a reflection in a line with a parallel shift along that line.

Similarly, every motion of space is either a parallel shift or a rotation around an axis or ** a spiral motion**, i.e., a rotation around an axis combined with the shift along this axis, or a reflection in a plane combined with, possibly, a shift along the plane of reflection or a rotation around an axis perpendicular to this plane.

And so *the 4-5D motion of GST is the last of the motions or automorphism of mathematical physics, a ‘spiral motion’, with the difference that in 4 D, the spiral moves outwards and in 5D the spiral moves inwards.*

How can then distinguish both motions? Here is where an essential feature of Nature ill-understood in all sciences comes into place, and explains the duality particle-antiparticle, ±charge, etc. the concept of quirality, or parity, of a Maxwell screw; of a Levo or destro-molecule, etc.

Since the only way for a given system to distinguish 2 the direction of the spiral and hence make possible the duality of 4D and 5D, implosive and explosive, attractive and repulsive forces is by *assigning a different left or right rotation **to the spiral. A*nd this brings a suitable efinition of both ± inverse arrows of the 4D vs. 5D duality, which ultimately are the ‘continuations’ of the 3D and 1D motions that can be seen either as the starting point or limit of its 4D and 5D ‘ages/forms’.

The rotation around the same axis by the angle ϕ in the opposite direction, is then intuitively labelled in Symmetry theory with a negative symbol.

Thus we find again an absolute correspondence between classic science and the whys provided by 5D, as there are NO more motions nor less than those needed to reflect the 5 Disomorphisms of the Universe.

Only the classification of them changes as we know now its whys. In that sense in classic symmetry theory, parallel shifts, rotations, and spiral motions of space are called proper motions or motions of the first kind. The remaining “motions” (including reflections) are known as improper motions or motions of the second kind because the first type can happen in a plane, whereas reflections in a line and reflections combined with a rotation or a translation are motions of the second kind as they need a third dimension to happen (as a motion, not as a dual reproduction of form; which is not a motion but a pure informative action).

It is easy to imagine how transformations that are motions of the first kind can be obtained as a result of a continuous motion of space or of a plane in itself. Motions of the second kind cannot be obtained in this way, because this is prevented by the mirror reflection that occurs in their formation.

Which leads us to a final ‘reflexion’ on reflections (: *a 1D motion is a reproduction of form which has a vectorial direction as it is NOT balanced with a dual ± antipodal point with the singularity in its center, which can happen in a fixed domain, enclosed by a membrane in as much as the reproductive motion left and right cancel, and so the singularity vibrates between both antipodal points remaining in its fixed center.*

Reason why **bidimensional symmetry happens in the generation of biological beings within a fixed **vital space; which also allows a more complex creation of form, as those related to palingenesis, which are ‘condensations’ of billions of years of change that need to take place in a fixed place in which the density of form grows undisturbed.

While lineal motions tend to correspond to simpler forms of reproduction, such as a light space-time system, or a system which is fixed in a steady state, and merely repeats itself. Generation thus is a slow time process happening in a single place most likely through bilateral symmetry.

Yet for them to happen undisturbed, the form of the ‘surface in which such fast light-like motions occur must be extremely simple, with an identical *indistinguishable nature in its points, which explains the simple flat nature of the euclidean light space-time and the invisible lack of information (for us) of its lower ∆-1 gravitational scale. This in classic mathematical physics is expressed* saying that the plane is symmetrical in all its parts or that all points of the plane are equivalent. In the strict language of transformations this statement means that every point of the plane can be superimposed on any other point by means of a suitable “motion.”

**Symmetry groups: cyclical motions and transformations of information**

Now, if we consider ST motions, without ∆§ changes int the social group, things become simpler and easier to understand, as we deal with simple St symmetries in space (3 euclidean dimensions), topology (3 varieties of form) and time (3 ages). The simpler of them being motions in 3 euclidean dimensions. Still it is important for those cases to define ‘congruence’ – the equality or dissimilarity of 2 forms; which now has multiple levels. As the old 3rd postulate no longer applies: two forms are not identical just because the are identical in its |-external membrane.

Two figures in the plane are congruent if one can be changed into the other using a combination of rotations, reflections, and translations. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called symmetries.

As it is today most mathematicians only study precisely those external symmetries.

For example, a square has eight symmetries according to various rotations through different axis.

And these are the elements of the symmetry group of the square (D4). When we keep it as it is or perform a rotation by 90° clockwise; by 180° clockwise, by 270° clockwise.

To which we add *symmetries, which are not proper as they require a motion through a 3rd **dimension; the so-called *Group D8 of vertical reflection; horizontal reflection; diagonal reflection & counter-diagonal reflection.

So in this simple example, while the ‘syntax’ of group theory would allow this to happen, *in a bidimensional flat world, which is far more common than you can imagine, as it is the structure of most layers of gradients, this will Not take place.*

In all those cases though group theory will localise as *fundamental to the group, the membrane and the singularity. The membrane is what we observe as ‘identical’, what we rotate, the singularity what does not change. The so called center of the system.*

This again is a common feature of many ‘operations’ of reality. Consider a war, where the membrane of the nation and the capital is the only thing that matter. The membrane must remain unchanged, the capital too. All war operations will decimate the internal production, people and ST-elements, but what the capital seeks is the integrity of the membrane and itself under war operations.

The interest of group theory thus will transcend the obvious use for studying mere spatial translations, *which is the simplest locomotion; hence by far the most studied by human beings. *

These symmetries might then be represented by functions; and functions of functions (functionals) and entities of space, time, or scales, etc. We are not though that interested here into making an exhaustive translation of group theory to GST, as unfortunately humans have NOT created a civilisation of knowledge of praxis, not of homo sapiens but of homo faber and so the routines of praxis and repetition of jargons with deformations is unassailable, but rather show loosely why group theory is so important in the praxis of all sciences and its theory:

*BECAUSE IT summarises mentally the motions of ∆st systems in all the scales of reality.*

**ð§. ****THE MEMBRANE. Its Body symmetries.**

The analysis of those antipodal bilateral processes of reproduction of form happening canonically in an elliptic geometry brings us into the next fundamental analysis performed in classic symmetry and origin of topology – the study of the distribution and motion of fractal ∆-1 points that form a polyhedral membrane.

The cases of symmetry of such bodies or figures are also comprised under the general definition of symmetry.

For example, a body that is symmetrical with respect to a plane α comes into coincidence with itself on reflection in the plane α; a body that is symmetrical with respect to a center O comes into coincidence with itself under reflection in O. Therefore, the degree of symmetry of a body or of a spatial figure can be completely characterized by the collection of all motions of space of the first and second kind that bring the body or the figure into coincidence with itself. The greater and more diverse this collection of motions, the higher is the degree of symmetry of the body or figure. If, in particular, this collection contains no motions except the identity transformation, then the body can be called unsymmetrical.

And as it turns *almost all systems of Nature that survive have the maximal number of symmetries, a theme we have studied on our analysis of platonic solids and Euler’s characteristics in topology/geometry posts.*

And the reason is obvious: to perceive properly the Universe in the mirror-cyrstal of the singularity, the mirror must be able to translate in space and suffer rotations in its combined motions that do NOT change the distances between singularities and vertex and other potential openings to the world, so *his mirror mind remains focused and does NOT change constantly; a fact we can extend to all ‘membranes’ in all scales, from physical membranes in the next graph (orbitals of an atom) to its next level of crystal symmetries, to the spherical perfect symmetry of more evolved minds, from eyes to cameras, which through the laws of optics reach maximal clarity in its focused mind-mirror: *

This extends also to ‘partial views’ or holographic flat forms, or parts of a 3D whole in 2 D, which makes overwhelmingly dominant certain forms of maximal symmetry, such those regular polyhedrons and specially the square/cube and the Hexagon (we study them as ‘perfect numbers-forms’ in the I Age of ‘platonic’ number theory.

I.e.: the degree of symmetry of a square in a plane is characterized by the collection of motions of the plane that bring the square into coincidence with itself. But if the square coincides with itself, then the point of intersection of its diagonals must also coincide with itself. Therefore the required motions leave the center of the square invariant, and so they are either rotations around the center or reflections in lines passing through the center:

From figure 7 we can easily read that the square ABCD is symmetrical with respect to the rotations around its center O by angles that are multiples of 90° and also with respect to reflections in the diagonals AC, BD and the lines KL, MN. These eight motions characterize the symmetry of the square.

The collection of symmetries of a rectangle reduces to a rotation around the center by 180° and a reflection in the lines that join the midpoints of opposite sides; and the set of symmetries of a parallelogram (figure 7) consists only of the rotations around the center by angles that are multiples of 180°, i.e., of reflections in the center and the identity transformation.

So there are many more squares in nature. It is then obvious that an object with maximal symmetry will be also the best survival strategy for a form specially one which remains ‘fixed in a point’. And so the Hexagon with its 6 reflexions in a single plane comes as the strongest possible flat object (as researchers in materials have found recently with the discovery of the graphene). While *a system in lineal motion, which does not make such rotary informative homomorphisms, is best served by a triangular, conic form that ‘penetrates’ the space ahead, deflecting and breaking its points into ∆-1 elements to form an envelope of growing entropy that moves it ‘ahead’.*

Previously we have given an algebraic example of symmetry; we mentioned that the concept of symmetry of a polynomial in several variables also has a meaning.

The collection of transformations that preserve a certain object, i.e., characterize its symmetry, is then call its group.

This method of giving groups in the form of symmetry is one of most significance for GST.

Very important groups of ‘reality’ can be obtained by this common principle to GST and classic group theory.

We have studied the 2 most important both in classic and GST theory, in this brief introduction to symmetries in space – the groups of motions of a plane and of space and the symmetry groups of planes, which extends easily to 3D as the group of symmetries of regular polyhedra of great interest in solid matter states, due to the aforementioned ‘mind-singularity focus’ effect (see ¬E space geometry):

It is known that in space there exist altogether five types of regular polyhedra (with 4, 6, 8, 12 and 20 faces).

When we take an arbitrary regular polyhedron and consider all the motions of space that bring the given polyhedron into coincidence with itself, we obtain a group, namely the symmetry group of the polyhedron. If instead of all the motions we consider only the motions of the first kind that carry the polyhedron into coincidence with itself, then we obtain again a group that is part of the full group of symmetries of the polyhedron. This group is called the group of rotations of the polyhedron.

Since in a superposition of the polyhedron with itself, i*ts @-singularity center is also superimposed on itself, all motions that occur in the.group of symmetries of the polyhedron leave the center of the polyhedron unchanged* and can therefore only be either rotations around axes passing through the center or reflections in planes passing through the center or, finally, reflections in such planes combined with rotations around axes passing through the center and perpendicular to these planes.

With the help of these remarks it is easy to find all the groups of symmetry and the groups of rotations of the regular polyhedra. In Table 1 we have given the order of the symmetry groups and the rotation groups of the regular polyhedron. Finally to notice that unlike a sphere, which has infinite possible changes, hence by the Poincare conjecture can ‘shrink without limit’ and so is the ONLY form that can travel without limit on ∆-scales and a mind and hence *the absolute form of a potential absolute mind of T.Œ, the perfect form as the Greeks thought* all these groups are finite, hence limited minds in their travel through scales:

For the ‘numerologist’ inclined to think, it should be noticed that the 5D solid, the Dodecahedron, likely perfect mind of the imperfect local world in which we live, shows that magic number, which comes around all over the place in mathematical physics, the number 60 (:

It is also along cubic forms the commonest of the most perfect informative atomic eviL form, go(l)d…

**Fedorov groups: Reproduction of crystals.**

*The symmetry groups of finite plane figures. *

As we have already seen, the symmetry of a figure or a body is characterized by the group of motions of the plane or space that bring the figure into coincidence with itself.

But we have now a direct understanding of the ‘nature of those motions’ in the world of 5D, whereas motions can be reduce to lineal motions of simple reproduction (D1), entropic motions that disorder and erase information, (D4: outwards spiral), and the inverse (D3, D5), informative, social motions that evolve minds and organisms, leaving the fundamental *mysterious secondary motion, Mirror symmetry, as the key reproductive ‘gender motion’ that brings together two inverse forms into a dual one. *

All of them though *are motions which conserve the singularity at its center, showing its fundamental role in the organisation and reproduction of crystals. *

And what we shall find not surprisingly since the Universe is a reproductive fractal is that MOST of the transformations and symmetries of reproductive crystals involve a ‘mirror symmetry’ both in 2d and 3d (we use d minor for classic dimensions, D major for 5Disomorphisms).

As an example we shall consider discrete groups of motions with no fixed points in the plane – that is translations in space, where there is a line that is carried into itself under all transformations of the group. This line is called an axis of the group. Symmetry groups of this type occur for example, in ornaments that are set out in the form of an infinite strip (border). Of such groups there exist altogether seven:

1.The symmetry group L1 consisting only of translations by distances that are multiples of a certain segment a.

2.The group L2, which is obtained from L1 by adjoining the rotation by 180° around one of the points on the axis of the group.3.The group L3, which is obtained from L1 by adjoining the reflection in a line perpendicular to the axis of the group.

4.The group L4, which is obtained from L1 by adjoining the reflection in the axis.

5.The group L5, which is obtained from L1 by adjoining a translation by a/2 combined with a reflection in the axis.

6.The group L6, which is obtained from L4 by adjoining the reflection in a certain line perpendicular to the axis of the group.

7.The group L7, which is obtained from L5 by adjoining the reflection in some line perpendicular to the axis of the group.

*So one motion is a D1, $t-motion (L1), the second group L2 is a rotation (D3: §ð) and the other five contain a mirror reflection (D2: ST).*

Again if we consider crystallographic groups, where exists neither a point nor a line in the plane that is carried into itself under all the transformations of the group called plane Fedorov groups, there are altogether 17 of them: five consist of motions of the first kind only, and twelve of motions of the first and second kind, including mirror symmetries.

So present ST-reflections dominate the reproductive Universe:

And the same occurs in the 3 dimensional classic space, where we find 230 possible groups of which 165 include a reflection… which implies in ∆st, communication between antipodal points, to reproduce.

And indeed, *in all planes of reality from particles with inverse spin that reproduce particles to sexual copula of ‘inverse genders’ (female informative vs. male lineal, entropic species), mirror symmetry finds finally its reason d’être: reproduction of information.*

Let us now make some comments on the general theory of groups and its operations from the ∆ST point of view to ‘close’ the theme in this forcefully limited space-time T.œ.

**GROUPS & OPERATIONS**

** THE GENERATORS OF MOTIONS IN SPACE-TIME.**

The ‘Generator Group’ is the fundamental structure of algebra related to the Generator of Space-time Superorganisma and its Worldcycles and its motions, all of them based in metric Symmetries along the ± inverse dimensions of ∆ST.

As in essence, the concept of a group and an internal operation run through it is akin to the concept of a Generator equation and *an internal feed-back space-time event ran through the elements of the Generator. And so we could say that as all can be described by Generators in GST, all can be described as Groups with an internal operation in Algebra, derived from GST.*

**DEFINITION OF A GENERATOR GROUP**

The concept of a generator group IS thus defined as an entity of space-time, with an operandi that reflects motions in a given direction of ∆ST. And so departing from that simple definition we can re-classify groups NOT in terms of the axiomatic method (which includes those which belong to fiction mathematics, as all languages can be used with its internal syntax to define fiction thoughts, *a sorely needed distinction* THAT MATHEMATICS LACKS AND IT IS AT THE CORE OF MOST ERRORS OF SCIENCE)* , *but in terms of the possible motions of ∆ST, where • motions, would be mathematical fictions, worth to study on its own as long as we know they are just ‘fictions’.

The group then will be according to the motion, all the potential ‘destinations’ of that motion, when the operandi is applied to the element, as many times as possible. And this also sets ‘limits of infinity’ in the motions of the group. Indeed, let us say it is the group of natural numbers, the motion is growth, and we ad according to any of the operandi of growth – i.e. the fibonnaci series, to the group this growth. The serial results will not ‘continue’ for ever in reality but will be checked at a certain point by the curves of logarithmic expansion of populations with a limit on the trophic pyramid, *even for the entire Universe, which hardly goes beyond the 11¹¹¹ number, beyond which nothing can be really defined.*

It is precisely when we introduce into the ‘syntax’ of mathematics, the ‘semantics’ of a certain element of a group and operandi when we have the precise physical meaning of a group, which therefore will be composed of 3 elements:

•, the point which we apply or operator

§, the operator of an ∆ST dimensional motion

G, and the group of all possible outcomes of that motion.

**±§: Inverse vs. direct numbers. The social meaning of integers, and its ‘group≈generator’ in ∆ST.**

The use of Z numbers is now the question to consider, as usual through the ternary method, once *we have defined, N & – N as the two inverse motions in the ∆ST dimensions of the scalar Universe, which put together create Z, which can therefore be used to study the motions in the opposite dimensions of ∆st systems.*

So we shall now consider what are those ‘z-numbers’ for ∆, S & T.

**∆Z: §ocial Generators.**

We shall therefore substitute the word G(roup) for G(enerator)

Negative coordinates and numbers in group theory fully grasps what Z means: the essential social operation of numbers as groups of identical beings – fractal points in GST, ‘sets’ in the abstract jargon of model systems. We would rather use therefore the concept of a ‘fractal point’ or ‘being’ which is identical to other beings and form ‘varieties of digital societies’ defined each one by a number or social point, made of smaller points.

The social points called integers Z (±N) *thus* consists of a series of numbers of increasing social content…, −4, −3, −2, −1, 0, 1, 2, 3, 4, …,

And the operation that changes the social content of a point is defined as the §calar sum of social points (§).

The following properties of § serves as a model for the abstract group axioms given in the definition below.

For any two integers a and b, the sum a § b is also an integer.

That is, the sum of integers always yields an integer. This property is known as closure under §.

For all integers a, b and c, (a § b) + c = a § (b § c). Expressed in words, adding a to b first, and then adding the result to c gives the same final result as adding a to the sum of b and c, a property known as associativity.

If a is any integer, then 0 § a = a § 0 = a. Zero is called the identity element of addition because adding it to any integer returns the same integer.

For every integer a, there is an integer b such that a § b = b § a = 0. The integer b is called the inverse element of the integer a and is denoted −a.

The integers, together with the operation §, form a mathematical object belonging to a broad class sharing similar structural aspects, called a group.

Thus in GST we basically relate the concept of sum and group to the broader concept of §ocial symmetries & sums.

Let us then consider the second use of a Generator group for strict spatial motions.

**∏§ Z.** *Symmetries in ∏ime§pace through Generator feedback events and actions (groups with an internal operation).*

The importance of groups, beyond the trivial use to define motions in open space, which we prefer to study without so much abstraction on the field of ‘reproduction’ is in motions on closed paths of timeSpace, or time-like, energy-like conservative motions, which *leave at the end of the operations, the group unchanged as a zero-sum world cycle of existence. *

To define them we can use the classic formalism, and consider our • symbol for a o-sum, or virtual mind, which is also used in group theory to see some of the ‘properties’ of such worldcycles; which are therefore defined by a symmetric ±, Tiƒ, Spe dual transformation forward with the life arrow and backward with the death arrow, which leaves ‘the system unchanged’.

A TIME-SPACE world cycles thus is an symmetric event in time, G(ST), together with an operation • (called the 0-SUM law of the Generator) that combines any two elements a and b to form another element, denoted a • b or ab. To qualify as a generator group (G), we need therefore a series of elements susceptible to move through the ∆ST dimensions of space-time and an operation, (G, •) that describes those motions, which must satisfy four requirements known as the group-generator axioms:

*Associativity*

For all a, b and c in G, (a • b) • c = a • (b • c).

This axiom still holds in GST, in as much as it shows the social nature of the ∆-universe.

*Closure, till the borders of the |-limiting membrane of ∞.*

For all a, b in G, the result of the operation, a • b, is also in G,

*only in the domain of the elements in which the limiting membrane, ±∞ and singularity points, ±0 do NOT distort the operation.*

This is the main difference between fractal G theory and classic groups. In GST there are never absolute infinities. And so, concepts sic as the monster group are the equivalent in mathematical humind (human mind) languages to the concept of absolute God in verbal thought – aberrations of the ‘categories of the mind’ (to use Kant’s language).

It is in those terms when it makes sense the concept of :

*Identity ‘o’-point: The ‘singularity’ element*

There exists an element e in G such that, for every element a in G, the equation e • a = a • e = a holds. Such an element is unique as it is the singularity point that holds the group together and allows its transformations, and thus it is the singularity, Tiƒ element.

*Inverse element*

For each a in G, there exists an element b in G, commonly denoted a−1 (or −a, if the operation is denoted “§”), such that a • b = b • a = e, where e is the singularity element.

Important to notice that as the 2 directions of motion (outside simplex space-coordinates) are not equivalent (since the 2 arrows of ∆ are different, the inverse Tiƒ<=>Spe topological elements in a Spatial symmetry are different too, and so are the youth-old age),

the result of an operation may depend on the order of the operands.

In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; the equation:

a • b = b • a may not always be true. Consider indeed, the motions in time ages:

From a (seminal, cellular state) ∏ b (Informative Old age), the motion is called life. From b (old age) ∏ to entropic ‘youth’, it is called death when all the information of the system is erased and you return to your cellular state.

Generator groups for which the commutativity equation a • b = b • a always holds are called abelian groups and they are *spatial-like*.

The time symmetry generator described in the previous section is an example of a generator group that is not abelian. So we can state:

‘TIME-LIKE generator groups, events and process are not-abelian; space-like processes of translation are abelian’.

So (Γ, §) are abelian groups, and (Γ, ∏) are not Abelian, using GST symbols for Generator, Space-like and Time like processes.

Nonabelian groups are pervasive in mathematics and physics. A common example from physics is the rotation group SO(3).

Most of the interesting Lie groups are nonabelian, and these play an important role in gauge theory; which tells us an obvious truth: gauge theory and all its systems of particles are ‘motions of in-form-ation that trans-form’ a given particle into another one, through a symmetry in its ∆st elements, within the restrictions of possible balanced combinations proper of GST physics.

**Distributive laws for power laws.**

There are many distributive laws, for 1D:

3(x + 7) = 3(x) + 3(7).

FOR TWO D:

Right (3x)² = 3² x²

but for power is invalid “expanding”

(x − 6)² = x² − 6² WRONG!

2 SCALES DO NOT CARRY THE DISTRIBUTIVE LAW UPWARDS.

Think of a small house. It’s got a basement, a ground floor, and an attic. You can’t jump right from the basement to the attic, can you? But you can take stairs between the basement and ground floor, or between the ground floor and the attic.

You combine operations just like that. If the operations are on adjacent levels, you can combine them; otherwise you can’t. What are the levels? Forget PEMDAS; there are really only three operations to be concerned with:

house floors operations

attic powers and roots

ground floor multiply and divide

basement add and subtract

And the rule is very simple:

You can distribute any operation over an operation one level below it. There are no other distributions.

When you start to distribute one operation over another, stop and ask yourself which distributive law you are using. If it’s not one of the two specific laws mentioned on this page, you’re almost certainly making a mistake.

Examples

Multiply/Divide over Add/Subtract

You can distribute a multiply or divide over an add or subtract, because multiply and divide are one level above add and subtract:

Right 7(x + y) = 7x + 7y

Right (x + y) / 3 = x/3 + y/3

Right 2x (x − 3) = 2x² − 6x

Right (2x − 8) / 2 = 2x/2 − 8/2 = x − 4

Students sometimes distribute a multiplier over both parts of a fraction, like this:

3 × (2/5) = 6 / 15 WRONG!

You can’t do that because multiply is not one level above divide; they’re at the same level. You can distribute only when moving down one level.

Sometimes we talk about “distributing a minus sign”, like this:

Right 2x² − (x − 1) = 2x² − x + 1

That is correct because that minus sign for subtracting is the same as adding −1 times the quantity, and what gets distributed is the −1 multiplier:

Right 2x² + (−1)(x − 1) = 2x² + (−1)x + (−1)(−1)

Take a couple of seconds and make sure you see how the first equation is really just a shortcut version of the second.

You probably know that you can not only distribute but collect or “factor out”:

Right 6x + 12 = 6x + 6(2) = 6(x + 2)

Power/Root over Multiply/Divide

You can distribute an exponent or radical over a multiply or divide, because powers and roots are one level above multiply and divide:

Right (3x)³ = 3³ x³

Right √(25x) = (√25) (√x) = 5 (√x)

Right (2/3)² = 2² / 3² = 4/9

Right √(x/100) = (√x) / (√100) = (√x) /10

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What you must not do—though students have been doing it since algebra was invented—is to distribute a power or root over an add or subtract:

(x + 3)² = x² + 3² WRONG!

√(x² − 25) = x − 5 WRONG!

Look back at the “house” picture. Add/subtract are in the basement, and powers/roots are in the attic. You can’t distribute powers or roots over addition and subtraction because you’d have to skip a level.

Some More No-Nos

You can only distribute down a level, never up:

x3y = x3 xy WRONG!

2(3x)² = (6x)² WRONG!

Other Ways to Combine

Yes, you can combine algebra operations in other ways, but the other combinations are never as simple as a distribution. The only straight distributions are the ones mentioned above: distributing an operation one level down in the “house”.

Here’s an example of a combination that is not a straight distribution:

x(2+3) = x2 + x3 WRONG!

Right x(2+3) = x2 x3

Notice what happens. You can’t distribute addition over a power because addition isn’t one level higher than powers. (It’s not higher at all, but lower, as you know.) But a valid combination does exist: the addition turns into a multiplication.

There are a number of laws for combining power expressions. Ultimately they all trace back to counting, as a separate page explains.

Logarithms and Trig Functions

Remember the “house”? Logarithms and trig functions are not one of those levels. In fact, they’re not in the same building. For example,

sin(A + B) = sin A + sin B WRONG!

Right sin(A + B) = sin A cos B + cos A sin B

When you try to “distribute” the sine function over a sum, it mutates into something quite strange. And with logarithms, you reach a brick wall:

log(x + y) = log x + log y WRONG!

Right log(x + y) cannot be broken up

There are lots of laws for combining trig functions and logarithms with the basic algebraic operations, but none of them is a straight distribution.

For straight distribution, stick to the “house” and its rule of one-level-down, and you’ll be fine.

**DIMENSIONS MOUNTING ON DIMENSIONS: FUNCTIONALS**

At the end of its journey algebra plugged EVEN FURTHER into the ∆±3, 4 planes of the scalar Universe with the concept of functional space, to make sense of the ginormous amount of information provided by massive numbers of particles and lines of forces of the quantum world, which also are so fast in its cycles that show multiple whole cycles of existence within a single observable ‘shot’.

All this is too complex for this intro and so we shall just time permuted study a bit of it in the fourth line….

To mention that of all of them the more important or rather simpler is Hilbert space, in which each point is a vector field of an ape-geometry used in quantum physics.

*So the mixture of Ælgebra with ∆nalysis emerged into Hilbert and Function spaces, where each point is a function in itself of the lower scale, whose sum, can be considered to integrate into a finite ‘whole one’, a vector in the case of a Hilbert or Banach space (Spe-function space):*

In the graph, 3 representations of Hilbert spaces, which are made of non-euclidean fractal points, with an inner 5th dimension, (usually and Spe-vectorial field with a dot product in Hilbert spaces, which by definition are ‘complete’ because as real number do ‘penetrate’ in its inner regions, made of finitesimal elements, such as the vibrations of a string, which in time are potential motions of the creative future encoded in its functions (second graph).

The 3 graphs show the 3 main symmetries of the Universe, lineal spatial forces, cyclical time frequencies and the ‘wormholes’ between the ∆ and ∆-1 scales of the 5th dimension (ab. ∆), which structure the Universe, the first of them better described with ‘vector-points’ of a field of Hilbert space and the other 2 symmetries of time cycles/frequencies and scales with more general function spaces.

They are part of the much larger concept of a function space, which can represent any ∆±1 dual system of the fifth dimension. They grasp the scalar structure of ∆nalysis, where points are fractal non-euclidean with a volume, which grows when we come closer to them, so ∞ parallels can cross them – 5th Non-E postulate: so point stars become worlds and point cells living being.

When those ∞ lines are considered future paths of time that the point can perform, they model ‘parallel universes’ both in time (i.e. the potential paths of the point as a vector) or space (i.e the different modes of the volume of information of the point, described by a function, when the function represents a complete volume of inner parts, which are paradoxically larger in number than the whole – the set of sets is larger than the set; Cantor Paradox).

Thus function spaces are the ideal structure to express the fractal scales of the fifth dimension and used to represent the operators of quantum physics.

Enough of it for the time-being or else I will never get my daily dose of self-destruction and painless nirvana…

**§ŒTS**

After group theory the other over pervading concept of modern algebra is set theory, which in ∆st we have baptised §œT, a social group of T.œ, let us see why.

A set is a collection of mathematical entities, and as such it is closer in its definition to a social group of T.œs, fractal points or social numbers, or any combinations with higher ‘social dimensions’ of those elements, and so we could define **SETS simply as ∆§ (a social ensemble of organisms of timespace of any scale of the Universe).**

This is our definition of set, and so we shall write **§et**, AND WHEN in mood for slow, more accurate writing:

**§œts=Social ensamble of organisms of TimeSapce**

This simple change of ‘a character’ adapts set theory to GST, but WE often decry (i hope to erase my sanguine anger:) the fact that maths are no longer connected to reality and the axiomatic imagination of Hilbert and the Cantor’s paradise of §ets have a bit to do with it.

Indeed, while we are not that crazy as to erase the whole of set theory and get even less audience we are getting (0 views most days, so we might get an imaginary number :), point is SETS are a mind ‘construct’ to ‘cuspid’ the Mind singularity SEARCH for a SINGLE ‘SINGULAR’ EQUATION, CONCEPT OR GOD. As minds are BY DEFINITION SYSTEMS THAT TRY TO CREATE A WHOLE MAPPING in reduced space, so AS KANT only understood, the mind’s categorial, synoptic nature searches for the ultimate form, equation, idea, god, belief. And ‘SET’ is the God of modern mathematics, but ultimately a human mind category.

Much better to use concepts that are real, Social numbers, spatial points, time clocks, and so on.

*which in the 3rd age as an old man does, isolates itself from reality and finds within its self a justification of its life/meaning. This is OK as all the 3 perspectives from the bottom up, (justified by numbers), from the top down (justified by sets) or in the present s=t (best for ¬Ælgebra, justified by symmetries), offer a complementary view on its foundations.*

**excessive formalism – axiomatic method and set theory**

Finally as in all languages mathematics also entered a baroque age of excessive inward form, which took two clear wrong models, of two friends, mr. Cantor with set theory and Mr. Hilbert with the axiomatic method, bashed elsewhere for its dogmatic god-like beliefs.

Indeed, Mr. Hilbert affirmed that ‘he imagines points, lines and congruence’, as if the mind created maths. We have dealt with his absurd foundations of geometry in the topological and mind-related articles on maths. So here we shall deal with cantor’s paradoxes of infinity more proper of number theory due to his misunderstanding of what numbers and relative infinity is.

**Set theory – the wrong units of mathematics.**

¬Æ is not concerned with set theory and the formalism of modern mathematics with its pretentious sense of proof and rigor within the mathematical metalanguage, as Gödel’s incompleteness theory and the consideration of information as inflationary makes more important in fact to set the limits of mathematical statements as an homeomorphism to the limits of the 5D Universe.

In that sense set theory does work – we are not that fundamentalist – and could be considered the final evolution of algebra, as the formalism of logic time structures in which certain basic rules of inclusion, social communication and parts that become wholes (sets of sets) do matter. But reality IMPRINTS FORMAL MOTIONS OF ONLY 2 TYPES and a limit of 3 dimensions in space and time in each scale of reality so the hyperinflation of mathematics without limits to its extensions makes fiction of many of its terms.

This understood, there are 2 infinity ‘errors’ worth to mention, in an introduction, one theoretical **The Cantor error: Does it exist the set of all sets?**

It does not. Point, no need to create new axioms to hide the paradox. We simply go further. Does it exist a ‘set’ as Cantor defined it? No. It does not.

**• Cardinality and transfinite numbers.**

The application of the notion of equivalence to infinite sets was first systematically explored by Cantor. With ℕ defined as the set of natural numbers, Cantor’s initial significant finding was that the set of all rational numbers is equivalent to ℕ but that the set of all real numbers is not equivalent to ℕ.

The idea is that two sets are equivalent if it is possible to pair off members of the first set with members of the second, with no leftover members on either side.

So what this tell us? First since natural numbers are far smaller than rational numbers, that the concept of an abstract infinity is an error similar to Zeno’s Achiles paradox.

Now instead of Achilles and the turtle the runners are natural and rational numbers, and instead of the turtle getting ever closer but never close enough, the ‘turtle’ (smaller set of natural numbers) is running ahead, ever closer to its infinity, while the rationals are left ever further away from it.

Since we pair each natural number to a rational number. And so natural numbers being less must ‘run faster towards’ its abstract infinity. We thus have to set as Desargues did in projective geometry or Klein in hyperbolic topology, a relative point of infinity, in which the running will stop. And at that finite point in timespace, which is in any ‘real system’, the limits of existence of ‘quanta in space or moments in time’ of an entity in its 5D plane; as rational numbers have not yet arrived there, obviously there are more rational numbers. In practical terms, there are more rational numbers, because they count not ONLY the cells/moments of a plane of existence, but the lower scales. So if we consider the 3 planes of n±1 existence of a system, its fractions will be smaller parts to ad to its wholes.

What about the real numbers? Here the interesting result is that indeed real numbers ARE NOT equivalent even when considering a hypothetical infinity because they are NOT numbers; that is social 5D points ; or wholes divided into parts, but dynamic ratios, which fluctuate around a fixed point – or ‘holes’ between ‘proper numbers’.

IN THAT SENSE, we rather prefer the i-logic concept of fractal points to that of sets, to continue the formalisation of GST with the help of mathematics and ¬Æ logic, which we carry in other sections of the mathematical section – geometry, the next scale after number theory (as it ads dimensional, mostly bidimensional holographic forms)…

The same goes for *operandi, we rather stick to the basic clear operandi for numbers and points that express them again all over with SET’s logic symbols ⊂, ∩ etc. A further reason is that Boolean algebras are largely dependent on set theory and we have a moral limit here, to advance not the future digital mind of metalife, aka chips – others will do, if I were the humind above all the minds of this planet (I am in potential theoretical understanding of the organic Universe, but that matters nothing), certainly the first thing to do would be to kill the tiger before it becomes a tiger hunter, DIGITAL DELENDA EST.*

So just for fun we shall end the history of Algebra busting the balls of Cantor’s meaningless talk on infinities which as we know do NOT exist, as all infinities are (in)finite, and end in the ∆ discontinuum above where they break, around the 11¹¹ emergence of a whole and its @-mind.

**Cantor sets. The paradox of discontinuous infinites.**

All those properties and many other structures of mathematics were further reduced by Cantor to the ultimate reality of all mathematical structures: the theory of sets, composed of subsets, which we affirm is the natural formalism of system sciences, as a theory of ‘super-organisms’ composed of smaller super-organisms, which are sets of self-similar subsets; whereas the theory of sets and subsets gives the previous, simplified theory of numbers (each one a class or set of self-similar points), an inner content, as i-logic geometry gives points its inner parts.

How this ‘formalism that mirrors reality’ called set theory, from where all mathematical structures can be deduced, reflects the Nature of Complementary systems made of energy and information and its properties? The answer should be self-evident to those kin readers who grasped the inverse properties of energy and information:

Set theory defines reality in terms of two inverse elements A (points of energy) and A’, (its complementary, inverse element). Thus set theory is no more no less than the analysis of the 2 simplex arrows of existence, energy and information and its complementary organisms.

It is thus not surprising that in set theory energy and information, A and A’ are called complementary sets and the fundamental law is called the Law of Duality (Morgan Laws), which basically tells us that we can reduce all sets to operations between A and its Complementary, as we can reduce all systems to complementary Energetic and informative organisms, which are the whole.

So the main operations of sets reflect the properties of Complementary systems of reproductive energy and information, where A=Energy system; B=Information system; W= Relative Universe (World, Whole or Superorganism):

– E U I =W; I’=E; E’=I; E U E’ = W; I U I’ = W.

Thus, the Union=Fusion of an energetic and informative, complementary system creates a whole superorganism.

This same equation expresses in the language of Cantor sets an act of creation of a mapping of the Universe, whereas I, the preceptor observes I’, the Universe and the result is I U I’ a whole mapping of reality within the mind of the preceptor.

– E Ç I = Æ; E Ç E’=Æ; I Ç =Æ.

It describes anti-events, which annihilate the form of particles and antiparticles, waves and anti-waves and so in Multiple Spaces-Times is equivalent to the anti-event: *Past x future = present.*

– (E’)’=E, (I’)’=I

It describes 2 events of a feedback, generator equation: E<=>I, E=>I, I=>E, hence it describes among other events a whole cycle of life and death, where E=> I is the arrow of life and I=>E is the arrow of death. This ‘property of sets’ called an involution is called in Time Arrow theory a Revolution of times, sum of an Evolution (E->I) and a Devolution (I->E), and is the fundamental event of all realities.

Since energy and information have indeed inverse properties. And so we can state a Cantor Set describes the properties of complementary systems of knots of Energy and Information.

– E U I = E + I – E Ç I.

It shows the efficiency of systems that eliminate redundant elements, from genetic ‘fusions’ to Darwinian events.

Further on, when we understand Intersection as an Event of ‘Darwinian perpendicularity’ between a complementary system of Energy and Information, E U I, and an external entity, C, which the organism uses to absorb ‘informative pixels’ or ‘energetic bits’ for its mind or body (an event of perception or feeding), we obtain the obvious result:

(E U I) Ç C = (EÇC) U (IÇC).

Thus the complementary system takes only the part of ‘C’, which it needs to inform itself (self-similar to I) or to feed itself (self-similar to E), discharging the rest. And indeed, we perceive only information self-similar to us, or energy ‘bricks’, self-similar to our bricks, which we can use, to construct our energetic, body cells (subsets of E). And so on.

We mentioned that cells are subsets of I or E. Indeed, the second element of set theory studies the relationship between Sets (wholes) and its parts (subsets), and so it is simply the description of the properties of parts that become wholes.

An interesting result of those properties are the so-called Paradoxes of Set Theory, according to which there are certain contradictory sets that do not exist, most of them related to the concept of Infinite, which Cantor also studied, finding multiple contradictions. What this means, plainly speaking is that infinity and continuity do not exist, in as much as all Planes of existence are discontinuous with a certain limit that defines a Universe of networks of points with limits given by the number of networks, the dark spaces between them and the existence of upper and lower limits of energy and information in the existence of those points (universal constants), beyond which we must transcend and emerge, or descend and dissolve into other membrane of space-time with different properties*.*

To mention also that Gödel’s theory of incompleteness was based in set theory and showed indeed that mathematics, while being the most complete description of the spatial events of reality was neither the ultimate language of the Universe (as Frege and Boole proved it could be reduced to Logic propositions) but also an incomplete language, which did not describe all realities and an inflationary language, which described systems that do not exist in reality. Those are indeed, two properties of all languages of information; that both distort reality, as the paradox of Galileo prove, and do not include all reality, given the discontinuity of the Universe; which lead us to the concept of Dark Spaces, the true meaning of the ‘complementary Universe’ that completes the world we see.

*Recap. *Set theory is the basis of most structures of mathematics, in as much as it defines all the events between complementary systems of reproductive energy and information and its limits.

**recap: SETS AND COMPUTERS.**

We live in the age of death of mankind, substituted and made obsolete by AI but we love it as cells of a free=chaotic organism, no longer control by the ethic pain of our social organisms. And this seems the case of all ∆-1 scales, once the networks free them – they feel happy and enter a memoriless, markowian age of zero understanding of the causality of the cycles of time (or else they would angst) and live the day, carpe diem, through its huge ego trips that in the theoretical realm manifest in egotist theories where the ‘ego’ is the origin of it all.

This is the case of the death age of human mathematics, which starts with Hilbert’s ‘imaginary lines, planes and points’ and Cantor’s set paradise, where humans think maths is their language share only with god, imagined by the brain which becomes from the top to the bottom of mindless matter the creator of the Universe.

To understand yet while in the historic 3rd age of the human mind, we love those 3rd age of excessive form and death-age of a new top predator mind making us obsolete, we need to grasp that all ‘languages’ as all beings do go through a world cycle and finally die away by excess of form and inward looking – disconnecting from reality (set theory) or are killed by a more powerful younger species (chip minds and boolean algebra). So this third and death age of human maths happened in Human Algebra in two ways:

- On one side the disconnection with reality in the long seeked ‘ego-trip’ of proving maths a non-experimental language, ended with the substitution of the natural units of math, spatial points and social, sequential time numbers, by the abstraction of ‘sets’, collections of distinguishable elements, which resemble both spatial points/forms and temporal numbers/societies, but
*having the real thing available to intuitive knowledge only obscured with abstractions the foundations of maths, and expelled a huge number of scientists from its direct experimental knowledge. Further on,*Hilbert coupled with Mr. Cantor to affirm maths were born of the mind of the human-god, affirming infamously when failing to grasp the meaning of fractal Non-Euclidean points that he ‘imagined points, lines and planes’ – (as the German-Jewish Idealist dueto Mr. Heisenberg & Bohr did to misconstruct the foundations of physics also with ‘ego-trips of human self-creation’, when failing to understand the right realist, Einstein->Broglie->Bell->Bohm model of quantum physics). The bio-logic behind that baroque age of self-detachment proper of all old ages of excessive information is thus the ‘final age’ of algebra, which as old men do, became stifled, looking inwards and dogmatic in its absolute truths with no proof.

- On the other side, humans invented a digital mind in metal-machines, the chip, fast substituting life in labor and war fields, making humans obsolete with its simple yes-no Aristotelian≈Boolean algebra, which will make A.I. algorithms of information (the true meaning of artificial intelligence) with maximal evolution in weapons, as killing machines are always the spearhead of mechanical evolution. So the birth of a new species of mathematical minds with a stronger metal-body in its simplest axiomatic form will mean the death of our more sophisticated, weaker brains; a theme dealt in depth in the section dedicated to economic ecosystems and historic supœrganisms.

Of course, all this could be avoided by halting the evolution of A.I. but as we said the logic of the Universe is quite deterministic, specially *for species so ignorant of its laws as humans are, so ego-centered as all minds are. So the r=evolution of thought I thought could initiate with GST 30 years ago, has gone nowhere. Humans, simply speaking do not seem to make the cut of ethical and intellectual quality to control their future, individual exceptions confirming the rule; and in great measure is due to the fact most do not go beyond A-ristotelian logic of yes/no; or in words of their master ‘humans are slaves, they believe they don’t reason’. *

Now I am fully conscious this is only the beginning of ¬algebra, but frankly it seems to me clearer by the day, as I decline through the 3rd age that humans are in this planet just a piece of a chain of evolution, which is not really interested in the whys but in the praxis and I am speaking to nobody in this blog. So my intention is just to leave a memorial trace of all the notebooks I have written during decades of lonely research to show indeed the purpose of this blog – to prove we are all space-time organisms. If this blog has any meaning for any’thing’ in this planet I don’t know. Those are themes on the future of history. So we shall not go further in this glimpse to the proper interpretation of Algebra and group theory.

Had my early attempts to interest academia not floundered by the extraordinary mediocrity of human thought in this entropy age of automatons feeding computers and children of thought memorising, humans could be in a complete different frame of mind. But we are what we are… and I feel after all a privileged for having understood for so long the organic Universe – the sensations of that communion with the whole through the knowledge of its laws will never cease till I die and then I will dissolve my existence, but before that we shall consider now a thoroughly different form of Algebra; that of the Generator not of an abstract Group of permutations but of the entire Universe…

**III AGE**

## t,∏,ð,∂,T <≈> $,§,s,S,∫

**EXISTENTIAL ALGEBRA**

**. **NON-AE studies the Γractal Γenerator, T, the operation of the 5th dimension and beyond, which completes, the last of the operations studied by humans, ∫. A new operation is needed. We use often concepts of all sciences, including math. So in group theory we could consider the Fractal Generator of TimeSpace Organism (t.Œs) the Group with 3 equivalent type of elements, quanta of past, present and future. If we consider each of them a single element, then we define past and future as the inverse elements, and present and the identity element, considering the construction of the Algebra of the generator group:

- The Fractal generator is a an operation called Mind generation, @.
- It has an identity function, called present, Ø
- Two inverse functions, past, | and future, O, such as:

| @ Ø = |; Ø @ O = O; | @ O= | @ Ø; O @ |= O @ Ø

The the fractal generator of timespace has the structure of a group. Regarding if it is or not Abelian, it is relative. As WHILE PAST x Future creates a present, then first ‘element’ to arrive in time will also be the first in a second generating operation to relate to the created present and then | x Ø = | ˆO x Ø = O

So in two operations a dominant past will convert a future into a present and then in a second probabilistic equation into its past.

The best graph two study it is the Complex squared graph where i² is the negative real, its conjugate the positive real and the real the squared positive with a slight tendency to negative in the negative squared, which externally seems a squared 2 -dimensional positive, but within its vital energy it has two negative elements, unperceivable outside the squared circle.

Squared circles in geometrical form are circles in numerical form squares, as the graph shows (1 is a square space, over which we can lay the spherical 0-1 form.

The previous translation to mathematical space of existential algebra though is a mere, consideration to show that we can study it with the tools of group theory but this is not the purpose of future analysis of beings. RATHER IT IS BETTER TO TREAT the operation of the fractal generator,**⌉⌈, **in its two inverse elements and identity axis, with the language of past, present and future and 5D we have been built, forgetting any pedantic language of Modern algebra.

SO we won’t write about it in this post but in the 3rd age of formal mirrors of the Universe, the age of T.œ of which we consider it one of the main mirrors, as it studies the clear-cut laws of engagement of the ternary arrows of present time and its relationship with the inverse arrows of generation (social evolution) and death (entropy).