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, 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.
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 R_{adius} 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×10^{6} 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 ×10^{6}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.
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.
Turning it off, the new length can make the curve unexecutable.
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.