:

*In the graph, above different time vortices, bellow the reproductive fibonacci spiral and its algebraic fundamental element, the golden ratio, studied in Number theory.*

**Abstract:** The spiral is the simplest representation of a world cycle. As such it is a profound form of the 5D universe of relational space-time beings, reason why we have taking it away from the general post on non-E geometries, as it deserves its own deep thoughts.

Archimedes departs from his book ‘on spirals’ to complete an analysis of dynamic pi numbers. As a spiral is nothing but a pi cycle, whose closure has gone ‘a bit wilder’ on the lineal opening. So in terms of Spacetime holographic fields, it is a simplified bidimensional form of a conic, which also combine a cycle and a line but in 3 dimensions. And it bears witness to the simplex fundamental holographic principles – that SS, ST, TT variations are the units of reality, that all curves that exist can be decomposed as a sum of conics.

“If a straight line one extremity of which remains fixed is made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line is revolving, a point moves at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane.” *On spirals*

We find then 2 merits on Archimedes’ definition of the spiral, 1) to be a combination of a lineal and cyclical geometry and 2) to have motion, not just still geometry; making it the fundamental simplest ST bidimensional representation of a worldcycle. So Greek geometry generated a ternary geometric mirror in different dimensions of a worldcycle of existience making truth that the simplest ‘young’ mirrors of reality are often the most essential:

1D musical strings > 2 D Spirals > 3 D conics.

The fractal point on the spiral as a living organism.

What matters on that definition to mirror a worldcycle is that it is the combination of a lineal moving point within a larger cyclical world. If we take any point of any potential spiral cycle within a fixed spiral structure, the entity within the spiral is moving inwards thinking it exists in a lineal path Universe, as we all think, but because the super organism in which it exists, the curved spiral, is moving in cycles, the fractal point in fact is turning cyclically – he lives subjectively as we all do his lineal time, but the world objectively is ‘wearing it down’ the informative 3^{rd} age or central region of the spiral.

Then each spiral will be a variation on the same ‘mental phase spacetime’ mirror of the worldcycle. And we can apply the symmetries of GST to classify them. In practice though as in so many cases *the fundamental ¡logic symmetry that diversifies spirals *is the gender mirro symmetry, so S=T Archimedean spirals are ‘female’ balanced even spirals and logarithimic spirals are S<T>S informative & entropic. So we reduce our analysis to them.

** CLASSIC, S=T, REPRODUCTIVE, ARCHIMEDEAN ‘STEADY STATE’ STABLE SPIRAL. **

In the Archimedean spiral’ the internal point-being follows a cyclical path at a fixed rate.

Thus the Archimedean spiral is the steady state present spiral and as such the commonest in total space-time, as it is clearly a more stable configuration than an accelerated log-spiral of the form S>T. It does NOT have an inner motion that implodes it but basically is the ‘natural distribution’ of the internal vital energy-space of a spherical form.

The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance (arithmetic progression). In contrast, in a logarithmic 3^{rd} age accelerated spiral these distances, as well as the distances of the intersection points measured from the origin, form a geometric progression. And this reveals its ‘vital age’ & possible multiple functions, in its relative S, T, ∆, @ ‘survival tasks even though its primary design is to reproduce a point or communicate ‘2 points’:

If it is symbiotic as part of a supœrganism it might represent both the inner vital cycles of a flow of energy and information or its surface membrane in 3D projected into a 2D Archimedean spirals as a bidimensional phase space of a spherical form:

In fact we have two solutions/locations for the spiral that trace two dual paths often of communication between two similar beings. So the Archimedean spiral can create by cyclical reproduction in geodesic curve paths, a 3rd temporal dimension of height to construct spherical membranes.

The Reproductive Fermat spiral is a type of Archimedean spiral even more apt for reproductive purposes. In a single plane, it can act as a reproductive parallel communication for a 2-particle system:

In the graph the relationship between the communicative spiral and the shape of its external membrane, as the spiral is the common form in which a vital space of cellular elements is established within it.

Then the spiral is a communicative dance between two similar forms, merging the two elements at the end of the ‘life of the spiral’, as in black hole’s merging..

As a reproductive spiral, it shows the fact that all ‘reproductive’ actions are the structural merge of ‘dual elements’, we call gender. So it has 2 branches – more in a Fermat spiral whose purpose is the maximal packing of its reproductive forms in the space it fills, reason why is so pervading in plants and other highly reproductive systems of nature.

The reproductive Fibonacci spiral achieves this with its reproductive golden ratio, studied in Number theory.

Spirals as fixed bodies of space are thus Archimedean spirals. And we can further apply the fractal principle to subdivide them:

The normal Archimedean spiral occurs when *c* = 1. Other spirals falling into this group include the hyperbolic spiral (*c* = -1), Fermat’s spiral (*c* = 2), and the lituus (*c* = -2). Virtually all steady state dynamic spirals in space, as part of super organisms are Archimedean ones; such as the Parker spiral of the solar wind, or the pattern made by a Catherine’s wheel) are Archimedean; where the motion is conserved, as part of the vital energy-body of the system. Since the distance between its two cycles is fixed.

**S«T.** But in pentalogic all forms have multiple tasks, so as the spiral models a cycle=dimotion’s recurrent frequency it might be an E-ntropic feeding dimotion that brings the mouth to a certain point; and then bringing down the body, as in earlier cephalopods.

Time is a curved hence with at least two dimensions, besides its young age length: 3^{rd} age curvature and reproductive 2^{nd} age frequency. Time cycles thus break reality into an inner and outer ‘vital space’, and add a singularity point or focus of its motion. Because pi is not exact, steady state clocks are less common than vortex spiral or fluctuating ±π mouths. So once it swallows those spirals bring ‘home’ to the singularity the vital energy extracted by the external membrane by several methods.

One is shown in the graph, a scroll compressor: a motion of both arms compresses and moves towards the center the flow coming fro the eternal region of the being.

Notice in the graph above also the difference between the black holes in a communicative spiral. When the black hole in an event of feeding on energy transformed into its ‘quark forms’ as an accelerator vortex, similar to those processes studied on Earth’s accelerators, which leads us to its second fundamental form.

**ACCELERATED INFORMATIVE+ENTROPIC LOGARITHIMIC SPIRALS S<T>S.**

**Pentalogic: S<T>S **The St-logarithmic spiral is informative for its central knot and entropic for the point that moves inwards, *accelerating* through the log spiral, which in physics is a vortex=force source of accelerated timespace.

**It is also the ∆-SCALAR **spiral as its curves are self-similar in diminishing scales. , Bernoulli called it Spira mirabilis, “the marvelous spiral”. because he was fascinated by such unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve, a property known as self-similarity.

**S=T:** As a result of this unique property, the spira mirabilis has evolved in Nature, appearing in certain growing forms such as compartments in a nautilus shells and sunflower heads, that will store the reproduced cells of the system, in its *inverse arrow of creation of a larger whole starting with the simplest cellular reproduced units*, in exponential Fibonacci growth series*.*

**T>S:** It also appears as ‘informative spirals’. Which accelerate and diminish the size of a form, as it comes to its perceptive point.

**¬S«T:** Finally from the perspective of the bite of energy or bit of information, which ‘goes through’ the tunnel of the spiral is a killing machine, now observed not as a reproductive cell system but a trapping channel for bits and bites accelerated towards the central stomach or eye of the spiral. Where the ∆-1 point will die in an entropic/informative split – when the perceiver-predator will split the system in itsits ST>S parts take the information or energy of its body, depending on the event and let it die.

So we can see easily how the logarithmic spiral allows events in ∆-scale (generation), S-entropy functions (feeding) and T-informative and reproductive functions (perceiving, storing cellular/atomic network forms).

**G(¬∆@st): **Which leads to the use of the log spiral as a model for the 3±¡ ‘ages’ of life of the micro ∆-1 entity; as stars in a galaxy, which is a ‘farming trap’ for the central black hole to let the stars grow on dust to feed it eons latter.

Let us see some of those pentalogic perspecti es.

**ð: Perceptive vortex log Spiral **

The logarithmic spiral is an accelerated, hence perceptive spiral.

It can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.

If the point that moves inwards *accelerates* we have a vortex of accelerated time. It is then a logarithmic spiral; an attractive form and *it escapes the simplification of mental geometry to become the commonest real worldcycle*.

** Spirals in physical systems. Charges, masses, eddies & galaxies**

In te graph time is curved, and breaks timespace into closed conserved paths, time clocks are infinite as Leibniz and Einstein understood when he said ‘I seem to be the only physicist that thinks there are infinite time clocks in the Universe’.

All those physical scalar spirals (ab.∆±¡) are topologically similar, differentiated only by the ‘speed or frequency’ at which they close their ‘time§pace clocks’ , according to 5D simple metric rule for all ‘families of time§pace clocks’: Size in space x Time frequency =constant. So LOG spirals are all pervading in nature, as they represent the dimotion of accelerated time or main arrow of future that increases the information of a system. As ‘accelerated timespace vortices’ they are the informative 3^{rd} age of all physical systems in its ∆±¡ quantum, thermodynamic and cosmological scales – an attractive vortex of space-time, which is by expansion of the Principle of equivalence between acceleration and mass, to all scales the physical definition of an Active Magnitude source of a force… NOT a solid static particle, a ‘tiresome’ error of idealist huminds that keeps coming since Pythagoras. As all what exist is a motion in timespace. Still space forms are a Maya of the senses. In the graph, accelerated vortices of timespace in physical systems, in different scales of the fifth dimension: charges, masses and thermodynamic eddies become then the main clocks of timespace that carry with different speeds according to 5D metric (S x T=K), the information of microscopic quantum charge worlds, human-size thermodynamic scales and cosmological gravitational scales. Since E=hƒ + E=mcc->M=ƒ(k); mass is a frequency of accelerated inwards space-time in its 3^{rd} age. So are charges and eddies, which will finally in its central point, become a ‘conic’ and ascend and/or descend lineally the axis in its entropic death.

Thus spirals respond to the fundamental property of time cycles: to have an arrow of future increase of information that diminishes its spatial size according to 5D metric: S x T = K, this accelerating inwards, which makes vortices of physical time (masses, ∆+3, charges, ∆-3), definitively the time clocks of both physical scales. For that reason time-space spirals, its subspecies and transformations are one of the fundamental space-time events of the Universe.

**Logarithmic spirals in nature classified by pentalogic function.**

In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follows some examples and reasons in terms of ∆STœ events:

Å(e): Feeding: The approach of a hawk to its prey.

Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral’s pitch.

__Å(i): The approach of an insect to a light source.__

They are used to having the light source at a constant angle to their flight path. Usually the sun (or moon for nocturnal species) is the only light source and flying that way will result in a practically straight line.

__Å (i):__ The nerves of the cornea:

(this is, corneal nerves of the sub epithelial layer terminate near superficial epithelial layer of the cornea in a logarithmic spiral pattern).

__Å (æ):__ The bands of tropical cyclones, such as hurricanes.

__Å (e: growth and reproduction)__: Many biological structures including the shells of mollusks.

In these cases, the reason may be construction from expanding similar shapes, as shown for polygonal figures in the accompanying graphic.

**Ideal Spirals as representation of worldcycles.**

Our planet-star lives in fact as a fractal point of a spiral, which acts as our Island-Universe in the longest of our worldcycles – that of the planet we exist within. Our own galaxy, the Milky Way, has several spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees. So the stars go through the spiral in its space-time world cycle of existence – from a 30 years old graph (: hence primitive digital form, profound human thought):

How long is the life of a ∆-1 points, which has ‘fallen’ inside any of the attractive vortices of a spiral organism?

**Mathematically** it means that starting at an external point π, of entrance in the spiral, and moving inward along the spiral, one can circle the origin an unbounded number of times without reaching it; yet, the total distance covered on this path is finite; that is, the limit as θ goes toward ∞ is finite. The total distance covered is r cos ϕ, where r is the straight-line distance from Pi to the origin.

So as it turns out, the number of cycles a being can turn about the spiral (frequency cycles) is infinite if time space were continuous, thinner and thinner but the real length of the life-motion or world cycle of the spiral (length to the center) IS finite, and moreover it diminishes in objective time, as the last phase is shorter and faster in frequency a deep fact about the duality of objective and subjective measure of time existence.

Since an old man has a shorter life, falling fast in decay in its last ‘life cycles’, which if we add the inverse ‘balance’ of its inner subjective time which is slower than the objective faster decay, makes the 3^{rd} life even shorter in subjective time, as a kid has much faster mental cycles, so his days are longer in experience. That is, at 50 in total real time actions we are much older; almost all our life ‘bits’ of action-information have concluded.

The scalar duality of the larger spiral organism and the smaller part shows also the scales of time: the short ‘frequency moments’, bits and bites of space-time actions which are Dominant in information, as the motion is inwards and will add to the whole world cycle of a being, causing its 3^{rd} informative age.

If we postulate a further generalization of the spiral as the ideal worldcycle for a whole range or phyla of species, it shows that all beings will live and die a finite, similar quantity of time-actions, making all lives absolutely relative. So 3^{rd} age points within log-spirals shrink in size, increase frequency and shorten its lifespan, completing its 3 ages: S<T+S=T+S>T.

**Spirals as entropic killing machines. **

Each inward closed path ads a bit of informative frequency every time the being repeats their cycle, aging it.

How spirals end the existence of one fractal space-time T.Œ? As all worldcycles with a reversal of timespace.

The life cycle is slow moving to the center: T>S, an action of information that implies to reduce the dimensions of reproductive space-width for those of cyclical motion, till the relative ∆-1 FRACTAL POINT looses ALL its energy and becomes, if the center of the spiral is an eye, a bit of information, if a mouth, a bite of energy, dying.

All time cycles tend inwards in a 3rd age of warping and in-form-ation till the flow of motion ‘stops’ in the still singularity, ejected perpendicularly in an explosion of death-entropy. It is the last time quanta of the clock when motion becomes a 5th dimensional still image before dying into a lineal 4D entropic flow.

So in its death age, the reversal of time happens, as in magnetic fields coming out of masses, dark energy fields coming out of galactic black wholes, when the log spiral uncoils first into Archimedean spiral as we ad a 3rd dimension of height information, that converts its shrinking revolution into an elongated receding motion, for an entity living within its revolution that becomes devolution, as the world cycle keeps accelerating but now acceleration is perceived as an ascension and elongation in height that erases its in-form-ation.

**Spirals as informative eyes.**

Humans express all those bio-topo-logic functions of the spiral with a highly efficient synoptic algebraic mirror image of 2 numbers for the key lineal & cyclical, parameters that construct the spiral: |xO=Ø.

Its simplest=most real form is in polar coordinates, which means spiral are temporal self-centered systems, in which the simplest ‘perception’ is that of the central point of view, making truth a self-evident GST theorem: the simplest mathematical formulation of a space-time event/system, in one of the 3 relative canonical coordinates, St (polar, spherical informative-head/particle), ST-Cartesian(hyperbolic, iterative-wave/body) and Ts-Cylindrical (lineal field/limb), defines the main organic function of the system, as S-Form=T-function.

So if an spiral equation is simpler in polar coordinates it is an informative space-time, form/function,

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**DYNAMIC SPIRALS AS PERFECT SPACE-TIME SUPŒRGANISMS OF 3 STATES.**

The logarithmic curve can be written as: r = a x e^{bθ}

And then depending on the value of b it will *transform either into a circle, or a line. The *derivative of r (θ) is proportional to the parameter b, which controls how “tightly” and in which direction the spiral spirals.

In the extreme case that b=0 (ϕ=π/2) the spiral becomes a circle of radius a. Conversely, in the limit that b approaches infinity the spiral tends toward a straight half-line

So a spiral can be considered an informative ‘state’ of a full organism, which can convert itself into the other two states as a present wave and uncoil as a lineal motion. Again this is a canonical law of vital, i-logic geometry: a system can be converted between the 3 functions/form as systems are ‘modular’ and its functions are constantly changing between the actions better performed by limbs (entropic function), bodies (reproductive functions) and particle-heads (logic, informative functions).

Such transformations are the staple food of existence and development, being the spiral and the tree, then 2 fundamental ST combinations of S & T elements – but the spiral is the commonest dynamic form that allow change of states with ease; while the tree is the commonest ‘fixed’ still simultaneous system of O-| elements:

IN THE GRAPH, all spirals are ‘potential fields’ or sinks and sources (+ inward life vs. – outward entropic death duality); which explain the organic, functional and mathematical description of spirals as bio-topo-logic beings.

So the vital spiral *is *in its vital mathematical description the most efficient organic form=function: A galaxy coiling its flat irregular young form a worm coiling to sleep in its 1/3^{rd} still informative SS-inwards state (completed with its feeding, entropic state, and wave-like motion) are in spiral states; which is ultimately an extremal case of the upper and lower limit of pi that leaves always an opening, dynamic mouth to the spiral allowing its simpler beat of existence, closing inwards (T-State) and outwards (S-tate), never ending the perfect pi cycle.

So most vital living spirals can uncoil to become lineal forms, or can close to become spherical circles, managed by its central head (1D: informative state) that becomes its future ilogic forwards head (2D/5D: locomotion and feeding-hunting). So the spiral can be considered an Ø-intermediate present system, perceived from the perspective of its dominant central point of view in polar coordinates (r, θ).

**RECAP. Varieties of time spirals in pentalogic dimotion.**

Spirals represent the 2 complex dimotions of time-space in a classic S=T female, S<T>S male symmetry:

S=T: Reproductive, communicative Spirals are Archimedean or Fermat’s based in the Fibonacci’s golden ratio constant in which several branches of the spiral allocate with maximal efficiency the reproduction of new ‘infinitesimal parts’ of the whole.

S<T>S: Entropic + Perceptive spirals, (as those of time vortices, charges and masses) where a flow of ∆-1 points falls into the perceptive or digestive central point, allowing both a potential and vortex description of them.

Based in those rules that vitalize the mathematics of Spirals as essential elements of a fractal point in its dimotions of perception feeding and reproduction, we can interpret better the maths of spirals in vital terms and define multiple organic beings as entities, which in its coiled position become organic spirals, often when resting – processing its internal systems of energy and information.

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