*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).

**∆-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.

Worldcycle:

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:

Classic mathematical descriptions of the log Spiral

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.

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).

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).

So 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. 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:

**T-Spirals: Archimedean.**

The logarithmic spiral 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.

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:

(to be cont’ed)