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.

    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.