everything is finished; infinite is only
the multiplicity of manifestations

The achieved solution of the golden spiral

Here we are at the epilogue of this meticulous journey, which, if on the one hand began by contrasting the more widespread and superficial expositions of the spiral phenomenon, after the first clarifications of the relationships that link them to the primary statement of the golden circles, soon ended up having to confront with the most various natural manifestations that ac­com­pa­ny it with all evidence.

If you have reached this page from an external link, you should know at least the introduction to the first console, designed and set up to allow you to un­der­stand and draw real golden spirals, with extreme ease and precision directly online. From the integrated whiteboard it is possible to access a page of the same independent of the text; which then evolved into a more ad­vanced one, in which it was possible to load and process spirals cor­re­spond­ing to one's own images, with a specially updated help section, to then a­chieve an ultimate stage, including the special feature discovered and ex­plained on this page.

What was immediately translated into a not inconsiderable effort aimed at verifying effective correspondences with living organisms, entailed in the first place a recognition of the diversities that regulate these geometric ex­pres­sions in the various kingdoms, first of all with a distinction between an­i­mals and plants, for then move on to astrophysical aspects, barely touched upon for now.
An attention that may appear exaggerated for the first experiments re­peat­ed using the console built ad hoc, especially for that Nautilus now made the bearing symbol of the golden spiral – which is not the only well­-known and praised one, but it is the one I discovered and developed at the term of the second stage of research – it subtly introduced and grad­u­al­ly confirmed with various tests a series of reflections, which in the specific case only hy­poth­e­sized a normal adaptation possibly justified by growth and en­vi­ron­men­tal effects.

In fact, this 144° spiral proved to solve various cases quite well with a min­i­mum of approximation aimed at focusing a sort of classification for certain marine creatures.
But it couldn't be enough. The subsequent installation of a gallery of images and tests carried out and to be carried out, led me to a new impact with the plant world, decisive for understanding a much more complex and profound problem, where the spirals in play repeated themselves by rotating in quan­tities originating from a single center, and involving serious difficulty in trac­ing them, given an evident geometric irregularity in those marvels of na­ture, which nevertheless flaunted it so much.
Inevitable compromises, such as the need to establish a starting radius suf­fi­cient to compensate for the anomalies of the early stages of development, were no longer enough, in the face of curves that followed a spiral tra­jec­to­ry only up to a certain point, only to then fold back on themselves.

This is the plant that cornered me when I attempted the spiral: a perfect va­ri­e­ty of aloe polyphylla, perhaps the most representative in its development of a spiral structure perfectly in the form of a pentagonal star, which as proof of what I have introduced regarding this polygonal base, well justifies the almost mi­rac­u­lous revitalizing and curative properties in multiple directions, re­nowned everywhere since ancient civilizations.
It is very loosely named "golden-ratio-aloe-plant", even if it is impossible to make it correspond to that golden spiral – which, however, seems the only one known – if not with voluptuous fan­tasy.

aloe - pentagonal-star spirals
To tell the truth, what is most striking, in attempting a spiral winding that corresponds to its volutes, is having to ascertain that the spiral trend is not at all consistent with a presumable geometric model, since not only the initial curve central is more open than in the medium growth, but which even joins in a circle in the external part of the plant, rather like a curve that could be defined as 'containment'. So much so that we couldn't even call it a spiral.

I didn't have to trouble too much to reach a goal, which in addition to providing a plausible metaphysical ex­pla­na­tion, could also be the cul­mi­na­tion, polar achievement of all this research, started on a semblance of an irregular spiral, to discover only some of the mysteries that ac­com­pa­ny it.
Let's look at both sides of the issue.
1. Any defined geometric spiral is always equal to itself, it has no meas­ur­a­ble beginning or end; it has no centre, if not its virtual axis, nor pe­riphery. It is pure abstraction, more so than polygons are.
2. In creation, the situation is reversed, since everything that has a be­gin­ning has an end: each of its manifestations has a starting point or phase, and a completion phase, which is not given by a mere ge­o­met­ric interruption but by a a process that we could consider con­clu­sive, like the vital confirmation of an envelopment that has achieved its precise formation.
The illuminating side is that this event responds to a principle of nat­u­ral balance which sees two opposing tendencies: centrifugal and cen­trip­e­tal interacting right from the start. Expansion and contraction are active in different proportions along the entire journey, so that at the very mo­ment in which life takes shape and expands, the opposing force is trig­gered which entails containment, I could say as a grav­i­ta­tion­al fac­tor, which always concludes by circumscribing reality of the object, or of the subject.
Mathematically it could be argued that each point of a spiral responds only to the simple equation that determines it; but in the reality there is a law which prevents every moment of this potential vector, u­su­al­ly in­ter­preted as single, from continuing its trajectory along the tan­gent; and it is this law that nature and the spiral itself indicate to us.

If we observe one of the five developments of the succulent plant with a clinical eye, we are perfectly aware of this effect, which ac­com­pa­ni­es the creature throughout its existence, regulating its growth within a bound­a­ry given by an evident balance of both phys­i­cal and geometric components.
Naturally the plant species, capable of reacting to climatic factors within hours, is much more sensitive to this process than the animal one, es­pe­cial­ly if this has a shell.

It is a sort of anti-spiral that maintains the plant, not only as gen­er­a­tor of the spiral itself, as it is in geometry, but preventing its essential bi­o­log­i­cal state from expanding to infinity; a force that causes the spiral to fold back on itself case by case – makes it vital – turning the initial mo­men­tum at high φ ratio; in a practically circular shape, which covers the external circumference by one fifth, overlapping the other four.
I call it anti-spiral to distinguish it from a current definition of counter-spiral, commonly attributed to a reverse rotation, clockwise or anti-clockwise, even if it is a relative definition since the default rotation direction, if it exists, de­pends solely on the spatial point of observation; the degrees orientation itself is just a convention, which varies from one programming language to another.
However, the anti-spiral does not reverse the direction of the spiral, but slows down its dynamics, up to overturning its expansion effect into its opposite.

In fact, this consideration leads to the fundamental premise of focusing on the effective spiral, as the geometric expression of two complementary a­gents: the golden ratio in the crescendo of the curve, against the central pivot of the π.

In short, it is a matter of conceiving the spiral not so much as an indefinite ge­o­met­ric trait beyond an equation, but as a tangible space-time factor, ex­pres­sion of a constant relationship between past and future, precisely for this rea­son based on Φ, and therefore virtually governed from the π which is the root and the barycentre.
The manifestation and maximum synthesis, therefore, of the two forces al­ways at play, constituting the universal principle of creation.

And here is understood even more the implicit charm of the Golden Spiral, since better than any other it conveys the sense and the mystery of the in­ces­sant vital mo­­men­tum.

the developments
The best way to investigate and verify as far as possible this first intuition was evidently to intervene on the console, already set up with reliable re­sults, in­te­grat­ing a new function, which would make this vision applicable by ex­per­i­menting with suitable reduction parameters.
Aloe polyphylla- spirale da 36°xΦ
with the anti-spiral active at 270°
The first question that arises con­cerns the meth­ods of in­ter­ven­tion. Since it re­lates to ma­te­ri­al growth top­ics, I will rely on in­cre­men­tal ad­di­tion­/sub­trac­tion in­stead of mul­ti­pli­ca­tion or power.
A complement factor, mul­ti­ply­ing by the de­gree coun­ter (which in my for­mu­la­tion ex­tends from 1 to any dig­it, with clear suc­cess) would be add­ed to what would be the Ra­di­us jump to R × φ, thus length­en­ing its time and re­duc­ing its ex­pan­sion more and more as the vo­lut­es ex­tend. In the fig­ure be­low, the black pro­file re­p­re­sents the curve with a jump of the ra­di­us of 36° × Φ and e­qual start­ing ra­di­us, left to it­self.
If we wish a sym­bol­ic stroke, 36 e­quals 360 ­/ (2 × 5), a sub­di­vi­sion of 5 for the ex­pan­sion and 5 an­ti­thetical.

However, each plant may require different or adapted parameters; I have tried a similar one in the console gallery, but it was only a first draft, which also rotates in the opposite direction.
It must be kept in mind in cases like these that in the center the inner side of each volute covers the outer side of the next one, distorting its apparent initial curvature; and that with almost never perfectly orthogonal shots, the same spiral drawn on a first angle will not adapt as well to the others.
And then natural deformities intervene. In this close symmetrical syn­chronicity, guaranteed by the thickness of the more or less fat leaves, one or the other will easily end up overflowing.
Finally, a certain three-dimensionality on a flat photo can reduce a wider spiral to vision.

I determined this parameter starting from a value within 360°, in order to es­tab­lish a reduction factor proportional to the angular coefficient (1~359 o phi_leap – %) set by the user.
As for a homeopathic treatment, it is necessary to micronize that pa­ram­e­ter to the point of making it applicable for a realistic effect, therefore divided by φ ×106 from which phi_leap += ( degree × complement ); where phi_leap is precisely the starting golden ratio on 360° (in this case field 8), or a spiral start button; ratio to which the increment ctx.dif­fer­en­tial / φ ×106 will be added at each degree, moreover mul­ti­plied by the num­ber of degrees, therefore ratio in increasing relation to the increase of the curve.
Obviously, this is only a first attempt.

So in this case the spi­ral will wind a­round the pe­rim­e­ter of the plant, even turn­ing in­side if it were to con­tin­ue; un­til sys­tem­at­ically re­joining its cen­ter as an ex­treme con­se­quence. The figure to the side pro­vides an ex­am­ple, based on the same pa­ram­e­ters of the a­bove plant, but start­ing from ra­di­us 10 for great­er vis­i­bil­i­ty, and ob­vi­ous­ly more mod­er­ate re­wind­ing than the in­i­tial ex­pan­sion, but com­plete­ly reg­u­lar, marked by a gold­en cri­te­ri­on (col­or­ing is just a graph­ic ef­fect that ac­com­pa­ni­es the view).

Whatever way you want to interpret this exposition, it is impossible to ap­ply to a similar plant – albeit of such a prestigious symmetry – any spiral meet­ing the usual criterion, called the golden; you can entitle its name on the web, but nothing more than that.

the pragmatic tweaks
To implement this new experimental possibility, I had to make some chang­es to what has already been done, optimizing spaces and icons to keep eve­ry­thing compact in the dashboard.
By now it is clear that the first seven golden modes implemented on the con­sole, while affirming a precise theory based on the golden circles, do not ex­haust the range of possibilities, since the Divine Proportion is present at every level of existence.
Nonetheless the eighth customizable can satisfy any case, as well as sim­u­late the previous ones, even in addition with variations in color and thick­ness; so as not to unnecessarily extend the keyboard with a myriad of cas­es that may arise, but above all to guarantee an effective and necessary tool for authentic research, therefore, this last chance was just missing, most likely sup­ple­menting a more realistic conception of spirals.
To make room for it, I replaced the keys for the numerical lists with the re­spec­tive icons:[full#] #, [module#] $ e [clear] con /
The % icon makes its appearance with the reduction parameter input field next to it, which in essence will increasingly lengthen the jump of the Radius to its value × φ, thus increasing the curvature.
Turning it off, the new length can make the curve unexecutable.
Warning: When activated it may require you to double or triple the length of the spiral, as the density of the stitches can make the design sort of like a black hole around the center.
It is here that we really need to linger, to extrapolate an unexpected passage.
When one thinks or describes a golden spiral, by an almost physiological law one conceives it in the widest of views from an initial minimum to a max­i­mum imaginable expansion, but with extremes always contained in the pe­riph­er­al to which to delegate it, and above all in our perceptive potential and therefore of mental simulation.
In reality, we could not imagine a winding in the center so dense and con­cen­trat­ed as to exceed figures that we are able to represent only nu­mer­i­cal­ly, but not graphically.
It must be said that the application of zoom can virtually penetrate this ge­o­met­ric space beyond all limits, provided that this has a reason to be, precisely because the spiral will always be equal to itself, at least in a three­-di­men­sion­al system, whatever portion we reproduce.
But here we are at a point that leads back to the unsolved mystery of black holes, better defined as "spinning black holes", which I mentioned in 2010 on a page of my solution work of the Śrī Chakra yantra.

“Astronomers Found a Black Hole Rotating So Fast, It Could Be Spinning Space Itself”

They are generally accompanied by eddies of increasingly tight spirals, a perennial intrinsic expression of the golden property – Φ inverse outlined here, until they appear as concentric circles.
This can be observed quite well by extending an anti-spiral to the limits of video resolution and noting its pace.
golden spiral free, and compressed by anti-spiral up to 8000px

Should it even derive from a stellar influence, this is the curve that ac­com­pa­ni­es a type of aloe, whose anti-spiral soon concentrates by su­per­im­pos­ing itself in the pixels of the screen.
Note that the graphic projection [turquoise] is that of a single golden spiral, with a ratio φ of (360°/5)/3, where (360°/5) = 72° modulates each of the 5 se­ries of leaves or bracts, which wrap around the whole plant; but which fits into the plan of the five arms only by reducing its ratio to φ to 1/3.
After that (according to our approach) rather simultaneously an anti-spiral of 3/4 of a circle (270°) intervenes, whose effect can be observed continuing in red, up to the central densification.
It is evident how, unlike the momentum of the initial opening, the spiral twists with ever more regular and concentric volutes.
Nonetheless in this case of a living structure which, as I said, particularly ex­alts the potential of pentagonal symmetry, the glob­al spirals must be five, and their expected effect manifests itself with greater evidence, although it gives rise to an even more saturated centre, which, however, in turn introduces the raison d'être of the rotating black hole.
They can be seen by tapping the image, and clicking activates a suitable zoom.
Sure that the greater detail of a PDF would have retraced the central effect of the images above better, I proceeded to create it, not without obtaining re­newed evidence: in fact as can be clearly seen with a zoom to the limit [click for PDF], at a certain point the bending ends, to outline an emptiness inside, which perhaps we will not be able to in­ter­pret as space, but only as an un­ex­plored dynamic field.

A free zone between two opposing and complementary fields of force, which stabilizes as a creative and destructive gap at the same time, of emanation and reabsorption.

There are many simulations, few certainties; perhaps this paradigm of the anti­-spiral will be able to contribute to understanding its dynamics, pres­enting itself to my perception with a far greater range than that of the as­sumed grav­i­ta­tion­al principle.

These graphic jumps naturally vary according to the ratios of each button, and it will take a lot of study to focus on the appropriate parameters and for­mu­las, this can only be an initial approach, published in the bud to start the project.
Its value must be expressed in degrees, the higher it will be, the greater the reductive effect, up to the circular value compared to 360 (I will have to return to this concept); and for consistency also that of the pre-existing 8 option is now required in degrees. Alas some images already posted show the pre­vi­ous formulation in tenths of a circle in the data field, and if I don't reconstruct them it is also to keep the various phases of the work distinct.

The Gallery combined with the console will be able to serve to accentuate cu­ri­os­i­ty, but also to show some convincing results, albeit only sketchy, as an invitation to improve them directly.

I therefore increasingly hope that the tool I have created can help unravel the fascinating skein.


perfect solution of the Golden Spirals

back to: Golden Section and false spirals