Beyond the borders, or behind the borders?
the Quintessence of the golden spiral
and the golden section in the pentagon
I do not think there are valid reasons, apart from the lack of information, for associating the classical spiral, based on the quaternary, with the pentagon; however, if an indisputable harmonic principle also affirms its harmony with the geometric construct of 5, all that remains is to take note of it with the main two examples that do not appear forced, here originating from the same golden system of the pentagon, obtained by me and demonstrated in the dedicated page (in the paragraph following the link to the present page).
It can be seen from the application of two by two of the 4 concentric golden circles, which amplifies the pentagon by turning it upside down, to reach the starting inclination every four.
In fact, the spirals, or rather the spiral, start from the center of the polygon to systematically meet:  with one of its angulations only the vertices in successive expansion,  with the other one only the sides by tangentiality, as is obvious given the common denominator: two golden circles that give specularity to the pentagon every 180°.
For its part, the suitably rotated five-step spiral reaches in the polygons in scale φ² to every fourth golden circle, both as tangent to the fourth following side on the five of the concave pentagon, and the fourth vertex of the convex. I am not looking for further possible combinations.
All that has a beginningIt is here meaningful to recall that the first factor which potentially distinguishes the circle from the spiral is that while the circle can assume an indeterminate number of measurements of the radius and the circumference, giving rise to an infinite quantity of circles different from each other, the spiral is only one, not subject to any type of enlargement or differentiation, except in the apparent visual angle of a circumscribed section of it.
has an end, but not the spiral
By tapping on the image below, a purely illustrative overview opens up which, although accurate, contains inaccuracies due to a freehand improvisation in which one can get lost, overflowing with surprising correspondences – all verifiable in ratio Φ – that stimulate the investigation, where a more attentive study could reserve other wonders.
February 2023 – the final disclosureBut the best was yet to come. Meditating on the starry triangle of the pentagon after ascertaining the properties of the spiraloid (which followed this page - link at the bottom) led me to understand how decisive its influence is in the natural world, and as an enlightenment, the certainty that it, and not the universally practiced classical spiral, was responsible for the development of several biological forms and whatnot.
The console I had set up to trace any kind of golden ratio spiral was there, ready to perform that task and give me the confirmation response, and that was it.
The confirmation is unambiguous; to better understand, we will examine the question by historically taking a step back.
As all schools teach, the most immediate way to geometrically draw the golden section starts from a square, centering an arc in the middle of the base with a radius at a vertex on the opposite side.
Thanks to the Pythagorean theorem, its meeting point with the extension of the base marks the length
By then building a square on a long side, space is given to a new overall golden rectangle rotated by 90°, a repeatable procedure to the point of inspiring, as usual, that connection of arches which, by simulating the spiral pattern, give the idea, welcome albeit deceptive, of that commonly acclaimed as the golden spiral.
The real problem that follows is not so much the geometric error to which almost everyone indulges, but the fact that the growth of spiral shells and much more has been claimed to be attributed to said graph, and we continue to claim it more and more up to the 'aberration of use of the Fibonacci numerical series (which he is certainly not responsible for this), insisting on superimposing images and graphs that never coincide, nor could they; and not only because said spiral deprived of the rectangular frames appears visibly ugly, flattened and irregular, but because it is too large for certain molluscs, on which it is superimposed as if it were their home. BR>I think I can share the underlying frustration deriving from such discrepancies, even if for the most part we tend to ignore them, given an instinctive - albeit unsatisfied – longing for harmony; and I am very happy to finally be able to offer everyone the real solution, resulting from a committed and by no means light effort.
At this point, however, the uncomfortable but effective premise must be underlined, that having been able to discover the golden section by means of the Pythagorean theorem ended up trapping our thinking in that rectangle, and in its suggestive and easy development, as if was the only, or the most representative application of the spiral.
But I had overlooked the most important, perhaps the most hidden... but no less convincing for this.
The double meaning of the pentagon, engine of vital and existential growth introduced in the update of the section dedicated to polygons, in addition to the golden spiral based on the convex cadence of 5 golden circles, deserved to be revealed in its halved rhythm of 2.5, equivalent to 144°, which is the angle between the vertices of the pentagon concave.
In my first console console it did not appear, but by applying the parameter of 144° with 0.4 to the [@] button I obtained a full-page layout with a very interesting result.
In deed, the curve is immediately more contained than the classic spiral, and so much more suited to most of the examples, that it can be considered the very realistic spiral; and so I proceeded to integrate it.
I could only try to apply it to my first tests, memories of the past, and first of all, the shell of which I only keep the door provided me with its detached rigor the most welcome confirmation.
Although I wouldn't have started but from an old archived image, it is evident that nobody is perfect, neither the molluscs nor the photographic shots for our purpose; nor do the vertical cut points exactly respect the symmetrical half of the shell, which can have a decisive influence on the terminal side.
The life-size image of the mollusk door at the time I obtained from a scanner, to be certain of the natural size and without perspective distortions, and here the spiral is clearly drawn by the mollusk itself, not by a graphic intervention.Although visibly comparable to that of the Nautilus, which follows its curvature up to the point where growth stops, its functional requirements are very different.
It should be noted that the unwinding of the shell's spiral stops at its point of tangency with the external oval ring, the moment when its protective conformation starts, which appears to be marked by the transversal crack in the bone probably due to cooking of the mollusk.
In fact, I had picked it up along with other smaller ones scattered on the ground, among the ashes of a barbecue.
I fixed this option too, updating the Help with various steps and utilities; but here I leave the work done before.
Even a starfish, abandoned on the shore by a storm surge, demonstrates in his own way how the spiral can be considered the engine of vital growth, at least in the early stages of evolution. This special creature, an extraordinary being that makes scholars fall in love with its abilities – and who knows why, it makes me think of the magnetic atom discovered by Pier Luigi Ighina – I could have adopted it as an emblem of pentagon qualities in biology; but I'm late, I'll let others do it.
Its finished structure extends horizontally and flat five arms directed in radial star symmetry, hence the scientific name 'sea star'; therefore with a pentagonal figure perfectly defined and maintained by the central body, without any link with a spiral formation in its life when stationary or in motion.
Nonetheless, at the moment of its end this image shows that it wants to rise from the ground on which it rests, rising precisely on its spiral-wrapped arms (some look broken at the tips), perhaps thus expressing the supreme yearning for elevation to its vibratory source, where a creature of any other species would have slumped inert; and this in a movement that is very similar to the curve I'm talking about.
After all, even the chakras of the human body are spiral, each of its own vortex; perhaps it would be appropriate to study the number of petals, or rays, to enrich the range of golden frequencies.
This shell also visibly adapts to the 144° spiral, but it is difficult for me to render its expansion consistent since it develops a lot on the three axes and any photo is limited by perspective, to the point that it would be almost impossible even to dissect it, unlike the Nautilus which tends to maintain a certain flatness.
March 2023To dissolve any doubts and help in the research, I will produce a comparison of some reconstructions of images already seen or new, under the true golden spiral headed to the triangle of pentagonal derivation; evidence that should convince once and for all to abandon erratic presentations.
So let's go back to Nautilus, starting with a classic, at least for me, having adopted it as wallpaper for accessing my system since windows8; perhaps one of the many, or rare, signs of destiny.
To instead increase the thickness of the curve by repeating the clicks, set Rot to 0.
The first result, not without a slight compromise, was this one on the side.
First considering the divergence of the profile at the exit, I immediately wanted to compare it with another Nautilus, obviously very similar, and immediately with the same parameters, the result almost turned upside down, as the spiral ends inside the shell .
Trying to highlight the differences between the two, I superimposed them, rotating the 2nd one until the stage of maturation of the mollusc matched. Touching the previous image, we notice that certain difference in the stages of development, which makes the differences even minimal in the spiral trend.
There the spiral appears darker up to the exact overlap.
Early cellular development and the reductive decay of aging may not be equally responsive from start to finish; and the various stages of growth encounter environmental stresses as well as structural pressures of various kinds.
It must be taken into account that swellings or initial structural deformations, even slight and not significant for the subject, can lead to increasingly consistent deviations with an increasing radius of a spiral, even more than in any trajectory. Natural differences and more or less perceptible disproportions mean that from a geometric point of view, the initial phase of development cannot be represented except with a curve starting from a minimum radius, in these images presumably around 20 pixels; and in any case the same organic tissue could not start with a radius =1.
And here is indeed a clear improvement precisely on the 'Nautilus windows8' which, in addition to seeing the spiral collimate on the final stretch in which it previously reentered, justifies a slight settlement of the center at x=136 for a pitch of 0.44, even closer to 0.4 of the stellar spiral, which reasserts itself as the dominant in place of the rectangular 0.25, the only one generally proposed and adopted without actual reason.
Someone among the more serious observers had already remarked it, but only to fall back on a concept of logarithmics, not sufficiently aware the extent of the Divine Proportion in the existent. Indeed, it is the golden ratio that rules the energetic expression of the creative Intelligence.
The definition of logarithmic after all does nothing but produce undifferentiated spirals of any type, that is, without a motivated classification; fitting one to an organism proves next to nothing.
Various Nautilus samples I've examined hover around 0.44, and you can probably find out better.
On the side, the minor difference between the two spirals in comparison; the one with parameter 0.4 (144°) and the lesser one, at 0.44… (now processed at 0.44945).
In fact, there is an aspect to take into account: the sectional cut of the mollusk will hardly be performed at the precise vertical center between the two sides, given the difficulty and the non-necessity of a similar accuracy for illustrative purposes which do not deal with exact geometry but marine biology. Any excess on the edge of the photographed half will be discarded, in order instead to reproduce the flat one, slightly smaller than the total extension.
This could partly explain the tendency of the various photos I tested to be inside a 144° spiral, bringing the parameter to about 160°.
As proof of this, a test on a photo of the living mollusk adapts very well to the spiral in question, of 144°, indeed it seems to fit tightly.However, the effort employed in achieving a convincing verification, has stimulated a new observation: despite the possible approximations and diversities, the trend of various Nautilus models tested reaffirms a stable proportion that cannot be neglected: the average value tested oscillates around 0.4495, which can be seen as a constant, the result of 161.8°, a φ ×*100 /360.
At the conclusion of this study, it can however be deduced that the spiral development options of these and many other molluscs gravitate around this pentagonal frequency of φ and to its double, rather than the 4 quarters made of square and rectangle; nor can one think of counting on a logarithmic spiral adapted to each case, in order to be able to enunciate a law of nature.
Maybe we should set up a special golden frequency?
In this case the formula would be nothing less than:
R =In truth, I immediately re-applied it, optimizing the layout parameters, with a result (figure to the side) that leaves no doubts. It can also be noted that the first stage of development is suited to an even double spiral of the 'classical' one (in red), to then quickly normalize.
But the game is by no means exhausted; here we are dealing with an X-ray sample of nautilus. At the first test I certainly didn't have to struggle, the result gratifies the intended purpose; although a probable slight horizontal tilt may have compressed the right and left sides relative to the top and bottom.
In a posthumous comparison the photo seems not so regular; but since the work done renders a certain idea quite well, I leave it as it is.
What's interesting is how the 90° spiral (red - quadripartite structure) followed by the 144° (purple - pentagonal structure) prelude a mutual compensation around the 36° of our diagram.
I mean that settling in growth of the mollusk, certainly kept constant from 130° onwards. Done.
It is the curve section of the shell on the upper side of the double arrow, which more or less delimits a semicircle in which the red sector with respect to the purple could have amplitudes in the golden ratio between them; but I won't go so far; there is enough to verify the consistency of the golden attribute connected to the concave pentagon.
For the pace of Fibonacci enthusiasts because, whatever the most correct parameter to apply, the spiral that defines the Nautilus has absolutely nothing to do with the pseudo spiral attributed to the mathematician by the general public.
I want to stress once again that the Fibonacci number series is but a worldly surrogate for the golden ratio, that it will never reach.
The first merit of it, in addition to breeding plans, is unintentionally to introduce in a concrete way to some properties of the golden section, which is a transcendent entity, all those who did not know it, or who have not yet focused it properly.
Those who want to try their hand at the console will soon discover that there are many ways to face, rotate, size a spiral on a figure, as well as center it, since it is easier to adjust on the external profile than on the center of figures that are not large enough; if tweaking the beam could surprise you with an excess of contractions~expansions, now I solved it (see help); and in the end the compromises can be credited for various solutions of the same case, to be considered satisfactory.
Other natural examples, as proof of what I have mastered since the beginning of the presentation of the golden circles, can be found in the various passages of the discussion.
The secret of the 'spiraloid'
the most admired Golden Spiral
As in the past, I have to ask the reader, especially the scholar, to bear in mind that all the content of these pages was greatly developed and completed in the course of a research, of which I was barely aware of the initial idea, but certainly not the finish line. It has matured part by part, for which technical and conceptual additions and updates have been the subject of continuous enrichment (or at least I hope so), but in a not entirely preordained way and without repetitions.
I gave precedence to the concepts written straight away at the expense of only the initial systematic drafting, both intended for the reader and notes for myself, potentially still useful and, as has already happened to me, the result was an essay that I would have to rewrite from scratch.
But time flies, and I've spent a lot of it, at any time of the day and some nights... After all, if I decided to draft the treatise on the π at the best I could – like perhaps I should have done – the latter work would probably never have taken shape and content; which is why I prefer to have made both.