This research and 3rd level domain are articulated to the study launched
at, dedicated to the total priority
of the π and the golden section, and follows a research that
from advanced solutions of operative geometry dedicated to
square and pentagon
, has fatally had to deal with the SPIRAL topic.
Preliminary critical considerations
5 real golden spirals, but not only…
perfect Solution of all golden spirals
play your golden spiral directly
try golden spirals on your images
(need to already know the above help)
the Golden Spiral par excellence
the Quintessence of the golden spiral
The secret of the 'spiraloid'
% The Vital Principle of the golden spiral

Triangles, golden section and false spirals

As from the starry pen­ta­gon, mentioning the in­trin­sic tri­an­gle that has come to be elected golden tri­an­gle par excellence, I can­not fail to bounce back on that incomparable gold­en tri­an­gle that has given me so much emotion since its first discovery, almost re­veal­ing itself as a link be­tween the 4th and the 3rd di­men­sion, say the mother sphere and pol­y­he­drons, not to tell cube, with a new question.
pentagon's 40 golden triangles In fact, the starry pentagon gives us an isosceles triangle (ABC in the figure) often referred to as 'sublime', for the golden ex­pres­sion of the base AB cor­re­spond­ing to the Φ of each side. To be more precise, it also generates BCD, e.g. CEx in re­duced version, as ExA is of ABC, also golden but in which the side is the golden section of the base, not con­sid­ered sub­lime; the reason is not well de­fined, but perhaps I will clarify it myself in the last stage of this research.
Compared to the true golden tri­an­gle, which I in­sist on de­fin­ing as the third treasure of ge­om­e­try, plac­ing it­self at the base of an in­no­va­tive re­search chan­nel, the penta-stel­lar tri­an­gles only half ful­fill the gold­en func­tion (while the com­bi­na­tion star and pen­ta­gon ad­mi­ra­bly re­flects it, re­peat­ing 40 times five types of semi­-gold­en tri­an­gles, clockwise: BAC, ExA / EzC, ECx / AEz, yxC, BCD e CBz), whereas ours expresses it doubly, in the most integral way conceivable: its base is in fact the Φ of the sum of the two sides that grow symmetrically above it, so it can be said that it personifies  the golden section in the most total, synthetic and essential way; it also follows that its height projected on half of the base gives rise to a right triangle whose base is the Φ of the hypotenuse.
This triangle that arises from the pure golden rectangle originating from the square, contains and defines such a series of harmonic and constructive correlations, as to make it difficult to remain closed-eyed in the face of such geometric majesty. In primis, from its circumcenter it is possible to cir­cum­scribe the cir­cum­fer­ence of which each quarter is equivalent in length to one of its sides, and therefore defines the π, much more likely than a pre­-fab­ri­cat­ed π, of which everyone is aware and gives for granted.
The Great Golden Triangle from the Giza Pyramid &
the 4 golden circles, also forming the Cosmic Egg

Not yet understanding that π is the square root of the golden section is prob­a­bly the most serious shortcoming in mathematics and geometry as well as in all re­lat­ed dis­ci­plines, from physics to astrophysics to nuclear phys­ics; not only to get closer to the mystery of life, but in order to avoid mistakes which, if on earth they may be irrelevant, over great distances can lead to considerable, if not irreversible, losses.

Perhaps this is precisely the keystone for which the same science that claims to demonstrate itself will have to annex an intuitive act, in the light of a clearly superior harmony, since, unlike any polygon, the circle is and will always be for it 'indomitable'.
My elementary essay «2x2=3,14» challenges anyone to prove the con­tra­ry, and frankly I don't think more can be done on this subject.
After all, the difference between the real and the simulated is only 0.9590…, I should have sufficiently motivated it, and in any case the true and the untrue will remain so forever.

Note that the only possible square inscribed in the egg has sides almost e­qual to those of the triangle(an ‘almost’ already suffered for an analysis very sim­i­lar to this one); and if for now it doesn't tell you anything, later on you will also be able to verify that the lower half of the egg's profile de­riving from this scheme, unlike a trivial half-circle, corresponds to half of any el­lipse ca­pa­ble of defining the classical golden spiral; moreover, the up­per half would con­tin­ue for 7-8 tenths alongside the same spiral.
This striking combination suggests how, passing from the starting square to the great triangle from which the circle, the creative expression translates the circle into the ellipse of equal diameter (location and presence), cor­re­spond­ing to the golden spiral (evolution), which the square com­men­su­rate with this (mat­ter) leads to dilate in the two upper quarters (in the fig.) by the measure of Φ2/2 in the circumscribed ovoid profile (waist), with the most varied be­long­ings.

In other words, the square with perimeter equal to the circumscribed cir­cum­fer­ence of the large triangle seems to be inscribed and nearly to con­fig­ure it, in a symmetrical ovoid figure, whose ratio between the minor and major diameter is roughly 0.79, not far from the gravitational thresh­old 0,78615 c.
To put it freewheeling, even the ellipsoidal shape appears, although en­er­get­i­cal­ly changeable and still unknown to terrestrials, of a spiritual atom, u­nique, eternal and indivisible; although that àtomos suggested by De­moc­ri­tus, who could look so much like, could be inspired by the physical per­spec­tive.
Yes, it's a bit like navigating without an itinerary, letting yourself be carried by the currents, but there are goals that cannot be reached otherwise.

If the 'golden' spirals, which we will also refer to the golden concatenation of con­cen­tric circles, are related to the development of living forms, from vegetables to molluscs up to expanding galaxies, my symbolic but strict­ly ge­o­met­ric reconstruction – only thanks to these golden components – of the universally known 'cosmic egg' (in turn misinterpreted in the most wide­spread figurations), as well as clearly outlining an ideal egg, designed with rigorous application of the golden ratios and superimposed for the occasion on a valuable marble sculpture for exibition, expertly realised, it also fol­lows with evident correspondence the profile of a section of the full human braincase, detected with scientific equipment (
Such and so many topics to dwell on… The reader should not be surprised if I ask myself about the hazard of such deductions; the thing is that I can't help but perceive and scrutinize how often the geometric language is inherent in the design of the creative Consciousness with traits that are as rapid as they are profound, and all the more unexplored; since it is Consciousness that we are dealing with.

A superlative transcendent choice the great triangle, for that magnificent por­tal between di­mensions, as is the great pyramid of Giza.

Great Golden Triangle and
golden circles into the pentagon

The puzzling side would be that it can­not be constructed or obtained alternately likewise it is from the square also from the pentagon, concave or convex, as if they were incompatible rather than shar­ing that privilege; nor can it be traced with ruler and compass from any other figure except by applying the golden proportion.
Of the four golden circles that are consonant with it, only the ex­ter­nal one and the 2nd internal one connect to the pentagram, since both circumscribe a figure at its vertices, both the concave and the convex inside it, which therefore stand in ratio 1: Φ; but they don't help to track them.
Nonetheless, the same figure that derives the pentagon from only two golden circles shows us that the golden circles constitute the 'native' medium for the transition from the triangle to the pentagon and vice versa, far more indicative of any extemporaneous geometric artifice.

To complement this, a survey following the last phase of the research, which will add special prestige and questions on the virtual structure of the sec­ond­ary golden triangle, leads me to reflect on the fact that it can be men­tal­ly traced back to a golden rectangle to which one of the two short sides has been subtracted.
A consideration that sends me to verify how this and the whole pentagon can be obtained with manual geometry from the same pattern that gen­er­ates the golden section and the great pyramidal triangle; since this would constitute, at least in one sense, the total communion of the three basic figures.
It is a concept that I had already introduced indirectly, illustrating it in this pre­vi­ous work on the golden ratio on this page
On the other hand, in the traditional scheme, which generates the golden sec­tion from the square, we note that the vertex of the secondary golden triangle, resulting from the same base of the square, touches with the ver­tex the same arc of radius AB which defines that of the major triangle.
Also interesting is the intersection of the arc AE with the center at L, in­tend­ed to cut the sides Av and Lv in inverse Φ ratios, which also meets the sem­i­diagonal of the basic square at the point E, equidistant from L and from v and lights up the inexhaustible range of triangles in golden pro­por­tion, like ALE.
Great Golden Triangle and the Pentagon
as from the Divine geometric Proportion

Touching the figure to the side, two arcs with cen­ters at A and B and ra­di­us AL (the base of the square) intersect in T, already sufficient to de­fine the sides of the con­vex pentagon with B and A, and of the con­cave one through the in­ter­sec­tions x, x of the two arcs with AB, ending the star sides reaching the arcs themselves in y, y.

As always when dealing with the golden properties of the pentagon, the clearly visible combinations are many and it is not necessary to enumerate them all.

Returning to the spiral that is attributed to it, one could already consider it a paradox to want to build a symmetrical circular function on a non­-e­qui­lat­e­ral and not a little unbalanced polygon.
It must be evident that it is not the spirals that grow on the trace of pol­y­gons, as in the special case of the rectangle born from the square to dem­on­strate the golden ratio algebraically, but it is these polygons that multiply on the rotational sequence of the golden section, like the squares that take turns in rotation, regenerating that rectangle.
I could argue that the spiral does not derive any origin from a triangular formation, which instead from a metaphysical point of view represents con­clu­sion and completion, whereas the square is a movement centered on an ex­is­ten­tial continuity that arises from opposition or cardinal harmony; where the pentagon is transformation and evolution, both compatible with the spiral trend.
At the end of this research phase it will be possible to better evaluate how the four phases, well marked by a symmetry indispensable for combining the quar­ters of each ellipse, are a strict index and enhance the virtually quadratic conformation of the better known spiral.
A premise not applicable to the triangle, which by itself does not provide us with the formation of the golden ratio (without which perhaps it could not even be constructed), drawing its quality from the pentagon, and being able to fully realize it only within the starry figure.

secondary golden triangle
and false spiral
Let it be said for now that, even if one is pleased to a­dorn it with a limping gold­en crown as rep­re­sent­ing the ab­so­lute, such a tri­an­gu­lar se­quen­ti­al­i­ty in my o­pin­ion should not even be pro­posed as ' spi­ral', not so much be­cause it does not con­form to the cir­cle, whose pe­ri­o­dic­i­ty is ev­i­dent­ly square, as are the cy­cles of each si­nus­oid, but be­cause the junc­tion points ap­pear more like su­tures, giv­en that the vir­tu­al ex­ten­sions of the arcs be­tray a cer­tain lack of cir­cu­lar ho­mo­ge­ne­i­ty, barely jus­ti­fi­a­ble be­tween quar­ters of a cir­cum­fer­ence, re­placed in­stead by a great­er con­ver­gence of each curve to­wards its cen­ter, con­se­quent to com­bined cros­sings of arcs great­er than 90°, which to the trained eye is al­most an­noy­ing.

Later we will also meet the spiral with a golden incidence of a third, which has much less to do with the circle; and above all of a fifth of a circle, which instead represents its golden soul; but the effective triangular one, which I will give the opportunity to construct directly, however superimposed, does not conform at all to the supposedly sublime one.
In the last part of this study we will also have the opportunity to admire the spiral of pentagonal cadence, and various correspondences with which it is ar­tic­u­lat­ed by intersection and tangentiality on those perimeters, a powerful and expressive network of concave and convex pentagons of all proportions, although evidently only indicative, given that the pentagon is the true re­cep­ta­cle of the golden ratio (and this triangle also derives from it; a bit like the hexagon, if I may, is for honey ;or); but nothing has ever been proposed for its spiral development based on that figure, too perfect in the golden framework to be able to apply an extraneous means of simulation.

A reiterated optical and mental illusion, which anyone with an attentive eye can also notice in many posted drawings, and which everyone, on the other hand, blindly copies and boasts of, professors, students and popular en­cy­clo­pae­di­as.
Finally, the two golden spirals, (rec­tan­gu­lar and tri­an­gu­lar) while ap­ply­ing i­den­ti­cal cri­te­ria, could nev­er co­in­cide; so what? An in­ter­est­ing tan­gen­tiality, in­ter­nal to the tri­an­gle on the three sides, of the real tri­an­gu­lar spi­ral (which does not co­in­cide with that claim) could con­sole, which I quick­ly an­tic­i­pate from a bit­map con­trol, but I won't go fur­ther since it seems to re­peat it­self with dif­fi­cul­ty on the only base of the 4th in­ter­nal tri­an­gle to scale; the same goes for con­tact with the three ver­ti­ces of the same mag­ni­fied bit­map curve, which is lim­it­ed to two of a base af­ter two rev­o­lutions. However, the two curves do not have the same center.
Those who wish can find here everything needed for greater verification.
The last word may never be said

I had started and partly already published this article with the mere aim of dismantling the geometric forcings that half the world attributes to this triangle as such, tracing those attributed to the rectangle - also fic­ti­tious, since they are not supported by any continuity in pro­jec­tion and it is only a stitching up quarters of circle, i.e. arcs with con­stant radius…; this is a mistake that I forgive myself for having na­ive­ly 'adapted' so many years ago, since I am working to remedy it...
After all, I had noth­ing else about the golden spiral un­til yes­ter­day, ex­cept the idea that it was wrong.
However, the knowledge and fundamental characteristic of the spiral would be enough, for which every point along its path can never be equidistant from the center like any other point; and this is true for both the rectangle and the triangle.
Nonetheless, just in composing some example figures of how the same con­cat­e­na­tion of arcs could be applied to any triangular figure, I came a­cross a sensational procedure, which undoubtedly evokes the spontaneous settling of vital proportions!
Having discovered it only now, in this phase of global review of the study, I promise to resume it after the work posted, oth­er­wise I will never get to publish it; because it's a topic that threat­ens to expand dramatically, and it's one that will require other days of attention.
Therefore, while maintaining all the critical structure of the case in this section, I announce that at the end, and precisely following this part of the analysis, a last unexpected reconstruction will import new emphasis for this spiral-shaped practice, revealing an exciting complementary side.
If this golden triangle cannot generate any spiral, I will demonstrate how a ‘spiraloid’ can extraordinarily redefine it, that is, recompose its pro­por­tions starting from any triangle.
As if to call into question all the premises, we will be a­mazed to dis­cov­er how every possible triangle, mul­ti­ply­ing and am­pli­fy­ing it­self side by side ac­cord­ing to a 'spi­raloid' ca­dence, is des­tined to trans­form it­self into the a­fore­men­tioned per­fect i­sos­ce­les golden.

Jointed and disjointed
An accredited abuse, in short, which borders even more on the so-called Fibonacci spiral, a real urban legend, since it is not understood what the curvature and circular continuity of hypothetical points can be based on, defined solely by integers with a clear solution of continuity between one and the other, and absolutely free of intermediate values.
I read on the web, on a specialist site:
the ratio of a number of the Fibonacci sequence to its previous one is, at the limit for n which tends to infinity equal to φ.
ie never! as well as human-made π can never reach actual π perfectly; π and Φ are one reality.
seen and considered that infinity is not of this earth, and even if someone's , fancy can imagine it, this does not make the ratio between any pair of numbers in the Fibonacci series equal to φ, if only because they are integer and rational numbers (They).
It's just an oxymoron.
In other words, nothing justifies drawing a curve between the number 1, 2, 3 and 5… which represent nothing more than deliberately pre-established units.
If the true golden section hadn't already been known, at least by some, its true and only value: (5± 1) / 2 could never have been identified via the Fibonacci series.
And yet, this series that took its fame from the golden section has now become a stepmother.
Moreover, this is a one-way numerical sequence, i.e. it is oriented towards infinity – provided that makes sense for the boot purpose – without ever re­al­iz­ing the authentic golden ratio, of which at the most it borders on a sur­ro­gate for defect or excess, whilst it cannot move towards the infinitesimal, or back­wards in a space-time sense, or in any case below the value 1.
It is evident that this was not in the creator's aims, but since man always seems to need some reference myth, his signature has nowadays become like a trademark, whereas the golden section is signed only by the Creator.
In the most respectful of cases, this sequence, linear and not spiral, let alone transcendent, should be represented solely by straight segments, which no π should or can support; this would undoubtedly allow a more realistic (and less distorting) vision of the trend of the phenomenon represented, and of its fluctuating zig-zag relationship (due to integers only).
It would be a bit like demanding a comparison between the discontinuity of matter and the potential manifestation of energy.

It's good to shed definitive light on the theme, abused in all ways
a goldsmith who doesn't distinguish two different curves! (too shameful to deserve a translation)
First, no or­di­nary com­pass can draw a spi­ral; while it can draw a cir­cle bet­ter than any com­pu­ter, which is con­di­tioned by the tex­ture of the pix­els or the lat­tice of the res­o­lu­tion.
The circular expansion - which is noth­ing else, if built with am­pli­fied quar­ter cir­cles as if they were ob­tained from Chi­nese box­es - in­volves rais­ing the ra­di­us Φ squared eve­ry 90°, as well as shift­ing its cen­ter from each quad­rant to the next, re­sult­ing in a vis­i­bly lop­sid­ed result if only the scaffolding is removed from view; and yet we pride ourselves on passing it as a spiral.

The photo reproduces the door of a shell, col­lect­ed in the Silhouette Island, Sey­chelles in 1986, reproduced in its nat­u­ral size in my treatise on Astrological As­pects of 1992.
Despite an occasional and unpretentious re­cov­ery, the image clearly reveals the development of a sharply ex­pand­ing spiral, but it seems to be only log­a­rith­mic, very well overlapping – but only up to a certain point – to that of the well-known Nau­ti­lus.
The golden spiral already expands with greater force in living organisms, but it rarely extends in that exceeding way which is containable to the eye, and perhaps also in nature for single structures, only at the first coils; the speech is different for stellar ensembles.

sham superpositions
Yet today the Web abounds in mys­ti­fi­ca­tions which repeats like these on the side (and I'll stop here), since it is not the task of search engines to discern the true from the false; nevertheless they speak for them­selves, and whoever publishes them can­not fail to realize it first, given that in most cases they cannot match… a myr­i­ad of wasted artifices to make what is not, culturally and even his­tor­i­cal­ly, seem true. Now you can refer to our gallery's solutions.
A pseudo-spiral based on rectangles, superimposed on the portrait of Monalisa, no matter if from the right or left side, or on any other portrait or painting, is widely used to make golden proportions appear which in truth mean nothing, do not have reference or specific correspondence with the parts, except natural anatomical proportions (such as e.g. those detected here, pls. repeat the click on image) which are not merit of pictorial com­po­si­tion, nor do they demonstrate that the author was aware of it, which I have clear­ly documented about Leonardo da Vinci.
About all if Leonardo had known it, he would undoubtedly have exalted it a­mong all his innumerable design works, what never took place.
Irony of the web, some illustrations of my site are reproduced by search engines as if I had in turn propagated the same nonsense…

Identifying a golden spiral in nature appears all too easy, if a geometric scheme is interpreted superficially, but at the same time too difficult if one really tries to apply the result to natural and biological formations, without considering factors of organic and functional growth and adaptation, since organisms face different needs at the various stages of development, plants or molluscs with environmental circumstances of adaptation, defense and so on.
Furthermore, there are various ways of tracing spirals, using parameters and algorithms referred to Φ, but approximate and unmotivated su­per­im­po­si­tions prove nothing but insipience.

March 2023
those interested will soon find a unique tool to experiment online
the type of spiral with real correspondence on its own image
uploaded to the console. (free, no subscription, no registering)
The point is not, in my opinion, how much the plot by squares can resemble the actual spiral, but it is in the most widespread conception of the Web, which distorts the deeper understanding of the golden section, regarding which there is still much to discover, and that due to its both natural and tran­scen­dent sacredness, the carrier of life and creation, it should be treat­ed as an object of meditation; instead, deformations such as the one stated above for the triangle are fed. So much so that the Fibonacci series, just for a marginal observation by Kepler, ended up making it look like his ad­dic­tion, when instead it is such a series that owes its glory precisely to the fact that it roughly resembles the golden section and makes it vaguely more un­der­stand­a­ble (even if it will never reach equate it)!
The Golden Section is the magic of creation; while the series assembled by the talented mathematician, however inflated it is only circus magic, agile in capturing the attention of [popular?] thought with greater ease, ready to guar­an­tee itself maximum evidence whenever the golden section was not yet eviscerated.
On the contrary, it seems that anyone who has not yet fully understood its profound meaning while remaining fascinated by it, prefers to use Fib­o­nac­ci's integers as crutches to make their way through.

the 5 true golden spirals

[ to the initial domain]