Beyond the borders, or behind the borders?

the Quintessence of the golden spiral

Golden spiral for five golden circles
and the golden section in the pentagon
  • The fifth surrounds splendidly, however you turn it, the configured pen­ta­gon, the polygonal structure that best represents the golden pro­por­tion; and squares and rectangles no longer appear. If the four-walled spiral re­-proposes the rhythm of life, this one seems to express a cry of joy­!
    Golden spirals of the step of
    four / five golden circles
    tuned to 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: [1] with one of its angulations only the vertices in suc­ces­sive expansion, [2] 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 beginning
    has an end, but not the spiral
    It 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.
    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.

    The momentum of this spiral is in itself truly and unexpectedly driving.

      February 2023 – the final disclosure
    But 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 (5 +1)/2 in ratio φ with the side, and from that point that rectangle of height equal to the square is erected, well known as the golden one, having the adjacent sides in the aforementioned proportion to each other.
    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.
    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.

    As far as I'm concerned, I almost reproach myself for not having thought of it before, involved as I was in defining the types of spirals based on the traditional modules of the quarter and third of a circumference, at the end of which I also placed the one that started this page .
    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.
    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.

    I fixed this op­tion too, up­dat­ing the Help with var­i­ous steps and u­tilities; but here I leave the work done be­fore.

    It is equally interesting to note that since this has the function of 'door' which closes and protects the mollusk such as the Astrea Rugosa in its spi­ral and deep shell, probably with a golden development of 3/4 as the fourth shell displayed in the console gallery, its growth with ratio φ at 144° inside the shell it denotes the identical spiral-shaped development, also trans­ver­sal, of the shell itself.
    Basically, the golden ratio (not generically logarithmic) appears to in­ter­vene in every modality and direction in which nature evolves.

    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.
    Sea Star at the last resort

    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 2023
    To 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.
    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.

    1st phase of Nautilus Radius: 9, spiral 1/4
    It is clear that the first phase of con­for­ma­tion in the de­vel­op­ment of a spi­ral or­gan­ism man­i­fests it­self in a mode of ex­pan­sion much wid­er than the av­er­age one which will take shape in nor­mal growth; cor­re­sponds to the gold­en curve 2 (of ra­tio φ at 90°) for an en­tire ro­ta­tion, per­haps even more to­wards the cen­ter, and then quick­ly leave it, pre­cise­ly where some cham­bers fol­low, as if for set­tle­ment, ir­reg­u­lar with re­spect to all the others that will succeed; and that the final stage will probably tend towards contraction, like a folding in on itself for growth completed, around 30~33 compartments.
    In the two figures of Nautilus compared via console, it can be seen with the same parameters that the output profile of nautilus-windows is external to the more responsive one of the second one; but what matters, the whole body of both conforms sufficiently to our spiral 7.

    To be able to process it in the console with a minimum size of 600px, respecting the center of the spirals to which I should have moved it, and its horizontal extension I had to rotate it by 180°, then position it at x=131, y=391; after that I started the first attempts, which from 0.4 brought me to a step of 0.4634. This limitation will also be resolved in a new version.

    Since the 180° ro­ta­tion also ap­pli­es to the spi­ral, af­ter each first click @. would also be ro­tated, so for fur­ther am­pli­tude tests the spi­ral should be de­let­ed be­fore draw­ing a new one.
    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.
    3 Nautilus compared
    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°.
    spirale su Nautilus vivente con φ a 144°
    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.
    spirale su Nautilus con passo φ di 161.8°
    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.


    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.