at https://pi-day.eye-of-revelation.org, 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 S
play your golden spiral directly
Triangles, golden section and false spirals
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 4 golden circles, also forming the Cosmic Egg
S
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.
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 (imaios.com).
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.
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.
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.
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.
 and false spiral |
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 ;
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
I read on the web, on a specialist site:
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.
(√5± 1) / 2 could never have been identified via the Fibonacci series.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
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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.
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 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
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to the initial domain]