**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.

**FOREWORD**** **

**SPIRALS AS REPRESENTATION OF THE WORLDCYCLE.**

Archimedes defines the spiral as *the combination of a lineal and cyclical motion – make it hence one of the fundamental simple $ð combinations of the Universe, along the cone*:

“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*

Often today the purely symbolic expression of mathematics finds such findings. What matters on that definition is that it is the combination of a lineal and cyclical motion: 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 circle, is moving in cycles, he in fact is turning cyclically. *

This is the Archimedean ‘steady state’ spiral’, in which the being follows the cyclical path at a fixed rate.

Yet if the point that moves inwards *accelerates* we have a vortex of accelerated time. It is then a logarithmic spiral; an attractive form.

The arms of spiral galaxies. 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:

Time is a curved hence with at least two dimensions, besides length: curvature and 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, but 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

If *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’.*

So the next step of our inquire from the absolute simplicity of ‘eidos’, the first idea or form of the platonic world – the closed path of a time clock, must be to order the multiplicity of those timespace clocks, and for that we need a new dimension of timespace and a metric equation that allow us to travel through it – its mathematical proof. Alas, that equation exists and it is astoundingly simple, and will allow us to order all the time clocks and enclosed ‘vital spaces=energies’ within them.

As they order in ‘scales’ of spatial size, whose similar forms and properties open a new astounding wealth of laws of science hardly explored till this blog, as we are all timespace beings, who share certain common properties because we share the 5 Dimensions of space-time. Laws that we shall call Dimensional Isomorphisms or ‘Ðisomorphisms’ (ab. Ð).

All those scales (ab.∆±¡) are topologically similar, differentiated only by the ‘speed or frequency’ at which they close their ‘time§pace clocks’ (closed paths that ad a bit of informative frequency every time they repeat their cycle, according to a simple metric rule for most ‘families of time§pace clocks’: Size in space x Time frequency =constant.

As such we have an organic, fractal Universe with a 5th dimensional metric, Space Size x ðime frequency, which allows travel through it, opening a full new formalism to all sciences, which have not studied properly the information of those time clocks of different frequency and size as ‘an ecosystem with its own rule of laws, the laws of the fifth dimension of ‘informative speed’, as the herzian speed of time cycles are ruled by 5D metric.

In the graph, once we understand the WHYS of the spiral which present science CANNOT provide, it is all good i-logic, mathematical, me(n)talphysical fun. As we *know now what is the meaning of those equations in its deepest sense.*

Now, all this which is the visual form and bio-topo-logic functions of the spiral, humans express with its limited but highly efficient mirror image in mathematics, of the key parameters of the spiral. Let us consider them briefly (the bulk of specialised studies goes to 2nd and 3rd lines).

To start with, we use polar coordinates, which *means spiral are temporal self-centred systems, in which the simplest ‘perception’ is that of the central point of view, making truth a self-evident theorem of GST: the simplest mathematical formulation of a space-time event/system, in one of the 3 relative canonical coordinates, T(polar, informative-head/particle), Cartesian(hyperbolic, iterative-wave/body) and cylindrical (field/limb:entropic, lineal), defines the function and form of the system.*

So if an equation is simpler in polar coordinates it will be an informative space-time, form/function, such as a spiral.

MOST of those spirals will be log-spirals as they shrink in size.

## SPIRALS AS FIXED BODIES OF SPACE.

Sometimes the term *Archimedean spiral* is used for the more general group of spirals

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 static spirals appearing in nature are logarithmic spirals, not Archimedean ones; where the MOTION IS INTERNAL, THROUGH A WORLDCYCLE OF TIME.

Many dynamic spirals (such as the Parker spiral of the solar wind, or the pattern made by a Catherine’s wheel) are Archimedean. WHERE THE MOTION is external, as part of the vital energy-body of the system.

Spirals in space, as part of super organisms, tend to be more of the type of Archimedean form, where the distance between two cycles is fixed.

But can a spiral convert itself into the other two points of view. 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 – the spiral, the commonest dynamic form to allow both other states with ease, the tree, the commonest simultaneous system of O-| elements:

But of course there are infinite many more little details to those spirals now ‘vitalised’ beyond its mathematical abstraction by its organic functions.

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

As it turns out, the number of cycles a being can turn about the spiral (frequency cycles) is infinite but the length of the life-motion or world cycle of the spiral (length to the center) IS finite, a deep fact about existence: you can have all short of ‘frequency moments’, bits and bites of space-time actions in the world cycle of a being, but ultimately all beings will live and die in a finite, self-similar quantity of time-span. This is of course a key me(n)talphysical postulate, which can be derived from many perspective of GST.

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.

**Pi-spirals.**

What brings us another huge discovery on irrational never-found numbers such as pi, since the space between the upper and lower limit of pi leaves always an opening, dynamic mouth to the spiral that allows its simpler beat of existence, closing inwards (T-State) and outwards (S-tate) its never found perfect pi cycle:

So of the many consequences and detailed conclusions obtained of the study of the spiral equation with ∆stœ i-logic topology, we just will bring the key ‘trans-forming event’, as spirals can uncoil to become lineal forms, or can close to become spherical circles. 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).

**SPIRALS IN NATURE. CHARGES, MASSES AND EDDIES.**

Why LOG spirals are all pervading in nature. The answer is that they represent the dimotion of accelerated time or main arrow of future that increases the information of a system. As such spirals are ‘accelerated vortices’ of time, the informative state of most quantum and cosmological perfect systems of nature and the best possible representation of the time cycle.

*The spiral ultimately responds 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.*

Symmetries of space-time require changes along the 4 parameters, ∆STœ of any system, which have an overall final change in the topology of the being, which will change from S to T state or from ∆ to ∆±1 often the form of an spiral event.

As we know there are 3 fundamental arrows and 7 combined motions of dynamic space-time. How they are represented in real space-time? Though topological changes in the forms that define such transitions between topological forms, ages in space, scales of existence and languages and actions of the mind. In the deepest knowledge of GST systems one can in that sense recognise *an organ or function in space by its static topological form and its most complex event as a time-space being ‘moving along one of the 7 canonical motions of its function of existence’, by the specific form and inner parameters of the system.*

*Let us then* consider a classic example: spirals.

*How spirals show the existence of one definite space-time event? Exactly as we have said.*

Consider the 2 commonest events of space-time: S>T an action of information that implies to reduce the dimensions of reproductive space-width for those of cyclical motion, till the relative ∆-1 bite of energy looses its e-nervy and becomes a bit of information (genetic linguistics often hides informative code on the meaning of wor(l)ds).

**VARIETIES OF ****TIME ****SPIRALS**

SPIRALS then once we understand to REPRESENT THE COMPLEX DIMOTION OF TIME, CAN BE DIVIDED roughly by the DIMOTION they represent into:

- Reproductive, communicative Spirals, 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.
- Perceptive spirals, (as those of time vortices, charges and masses) where a single flow falls into the perceptive central point, BOTH as a potential and vortex description of them:

IN THE GRAPH, the duality of time and space= moving and simultaneous= vortex of attraction and organic structure of still communication defines the dual description of all spirals of nature as ‘potential fields’ or sinks and sources (+ inward life vs. – outward entropic death duality); which complete the functional and mathematical description of spirals.

Based in those rules that vitalize the mathematics of Spirals as essential elements of a fractal point in its dimotions of perception and communication and reproduction, we can interpret better the maths of spirals.

**ð: Perceptive vortex**** log Spiral**

The logarithmic curve can be written as: r = a e ˆbθ

And then depending on the value of b it will *transform either into a circle, or a line. Indeed, t*he 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

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.

Yet the log spiral can be considered an Archimedean spiral if we ad a 3rd dimension of height information, by converting its shrinking revolution into a receding motion, for an entity living within it.

It doers then represent a world cycle accelerated but perceived as an ascension in height.

Moreover such 3 Dimensional view of the log spiral can be considered a curve along a cone of space-time, the proper representation of 5D worldline:

**∆-spirals: Logarithmic **

So we do have the logarithmic spiral which represents from the Œmind point of view the absorption of such bit of energy into a trans-formed form of in-form-ation:

Logarithmic spirals are self-similar spiral curves which often appears in nature, extensively investigated by Bernoulli, who called it Spira mirabilis, “the marvelous spiral”. Why? Because he was fascinated by one of its unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve, a property known as self-similarity. Possibly as a result of this unique property, the spira mirabilis has evolved in nature, appearing in certain growing forms such as nautilus shells and sunflower heads. As it reproduces the exponential, reproductive growth of a Fibonacci series, which *let us remember was already defined in terms of vital mathematics as a series of reproductive events of a couple of ‘rabbits’.*

However it also appears as ‘informative spirals’. Which accelerate and diminish the size of a form, as it comes to its perceptive point.

And finally it can be seen from the perspective of the bite of energy or bit of information, which ‘goes through’ the tunnel of the spiral (now observed not in its reproductive generation but its final informative perception), as the 3 ‘ages’ of life of the micro ∆-1 entity digested or perceived by the central stomach or eye of the spiral.

And so, the multiple functions of the logarithmic spiral give us a longer ‘space-time beat’ of such spirals, which are:

- Generated in its ‘first motion’ of time, as a series of ‘cells’ (compartments in a nautilus, seeds in a sunflower), which create a structure that will store:
- The reproduced cells of the system.
- Or will guide backwards, flows of energy or information accelerated towards the central stomach/eye perceiver (which for the ∆-1 point means in both cases an event of final death, and entropic/informative split – what the perceived-feeder will take when the system splits its ST>T parts and die, the information or the energy of its body, depends on the event).

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).

It is for that reason that the spiral is so common:

**Logarithmic spirals in nature**

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 subepithelial 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.

**§: Reproductive** **Archimedean** **spiral**

THE Archimedean spiral is then the steady state present spiral and as such the commonest. It does NOT have a motion that implodes it but basically is the ‘natural distribution’ of the internal vital energy-space of a spherical form. A stable spiral.

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 spiral these distances, as well as the distances of the intersection points measured from the origin, form a geometric progression.

And this also reveals interesting questions about the ‘vital functions of arithmetic and geometric symmetries/transformations)…

Again the number of possible functions of such spiral by the ‘ternary method’ (all systems do have by definition multiple functions, in its relative S, T, ∆, œ ‘survival tasks) are multiple.

We are though in a fundamental ‘time event’ of recurrent frequency of a cycle, often of feeding nature that repeats at a certain point and reaches clearly a more stable configuration than a hierarchical event of the form S>T. As such archimedean spirals as time cycles are bidimensional and spherical in forms. In fact you have two solutions/locations for the spiral that trace two dual paths often of communication between two similar beings. The archimedean spiral also appears in the creation of a 3rd temporal dimension of height and the construction of spherical membranes, showing what is its 2 fundamental events for n-particle systems:

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.

It might though be that the spiral is truly a communicative dance between two self-similar forms, and then we have multiple cases of merging of the two elements at the end of the ‘life of the spiral’, as in the much publicized black hole’s merging:

Notice in the graph the difference between the black holes in a communicative spiral, above.

And 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.

How one of those spirals bring ‘home’ to the singularity the vital energy extracted by the external membrane? There are 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.

**The Reproductive Fermat spiral… is a type of Archimedean spiral even more apt for reproductive purposes.**

finally is the reproductive spiral, and this obviously shows in the fact that as all ‘reproductive’ actions it is the structural merge of ‘dual elements’ (most often what we call gender’. So it does have more than a branch, and its purpose is the maximal packing of the reproductive forms in the space it fills, reason why is so pervading in plants and other highly reproductive systems of nature:

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