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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.
Let's even say that it took us on the road, but also channeling us into a one-way street; in fact the method derived from the rectangle, while a­rous­ing our enthusiasm, could not in itself be sufficient to represent all forms of evolutionary harmony, not covering the complete golden range.
Then tracing the method on the triangle has only led to scenic nonsense, devoid of parallels in nature, and in any case geometrically even more un­found­ed. Evidently it was not the right way, and yet this study of mine has shown that a grain of truth may have prompted such general insistence.

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.

In fact, I had picked it up a­long with oth­er small­er ones scat­tered on the ground, a­mong the ash­es of a bar­be­cue.
Ob­vi­ous­ly, the whole oval has noth­ing to do with the spi­ral ex­cept for its glob­al ex­pan­sion, aimed at en­clos­ing the mol­lusc in its spi­ral-shaped shell.
To re­spect the anti­-clock­wise di­rec­tion of the spi­ral want­ed by the ja­vas­cript, I had to re­flect it hor­i­zon­tal­ly.

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.

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.
If we carefully ob­serve any cen­tral area, we can al­ready no­tice by it­self a con­sid­er­a­ble dis­crep­an­cy be­tween the pro­por­tions a­chieved in growth, and the gold­en ra­tio with re­spect to the hy­po­thet­i­cal cen­tre.
Furthermore, the two green arrows are equal and indicate the same amplitude at a distance of 90°.
The same difference between the dimensions and frequency of the first five compartments, and those of a now regular development, teaches that to remedy the initial settling stage it will be necessary to adjust the starting radius of the spiral.

It is assumed that these exist between one individual and another of any species; but as far as spiral growth is concerned, the purpose of the discussion being the correct attribution of the type of spiral, I wanted to highlight the cen­tral issue by comparing them with a third party.
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.

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.

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.

Maybe we should set up a spe­cial gold­en fre­quen­cy?
In this case the for­mu­la would be noth­ing less than:

R = φ ^{±P/(φ×100)}

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.