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.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 pi, of which everyone is aware and gives for granted.
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
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.
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.
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.
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
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.
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.
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...
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.
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.
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 imaginative) 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
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 Seychelles Islands in 1986, reproduced in its natural size in my treatise on Astrological Aspects of '95.
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.
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]