A geometrically fascinating sequence, but which can only concern the squares in their configuration of new rectangles in scale; applying circle arcs to it and then defining it as a spiral is an abuse, although probably born of an irresistible temptation. Indeed, it would seem that the very being attracted by this sequence of polygons in self-generating amplified rotation was the first step to inspire the idea of the golden spiral, with such enthusiasm as to then want to apply it to the triangle as well.

In reality, the fundamental condition and characteristic of the spiral is that 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.

In addition to ambiguous scholastic information, this incorrect practice has fueled bad information on the web also because in fact, except for a few mathematicians with their own and different formulations, the tools to commonly and easily trace this kind of magnificent curve have been lacking; sometimes even to understand its true meaning if not with the help of references to conformations of nature.

`π`

("«2x2=3,14»"), which originated from my first discovery in 2003 of the φ scaled circles configuring the majestic Golden Triangle from which everything is derived.
It was the inevitable stimulus that took my efforts one step further.

Following this research, a challenge that remained suspended but which I could no longer ignore after having mentioned it turned out to be precisely the authentic Golden Spiral, of which these circles have remained the background terrain.

By addressing 'Essential Geometry' solutions on the website that interests further and special developments, I finally delved into the correct, simplest and most definitive way to formulate and draw said spiral in each modality (since there is not only one) on a Cartesian system; I solved it at the root with a universal algorithm well illustrated and offered free in its web code, moreover revealing at least two 'secrets' concerning the geometric as well as the creative power of the endlessly open figure.

First of all, the precise and unprecedented relationship that binds or connects the spiral to progressive ellipses.

A summary of the results is clearly visible in the rich and advanced PDF offered here, explained in every detail at the appropriate domain.

After that I ensured that from today anyone can draw and segment the exact golden spiral, in various ways and perspectives useful for research, also with the display of the calculations, not only directly from a special console on the web page

https://golden-spiral.eye-of-revelation.org/Golden-Spirals_interactive-construction.html

but also in its own website (adapting my JavaScript code there), with a choice of more than 5 well defined and illustrated golden ratios.

But that is not all.

The research, with the initial intention of dismantling the unfounded claim to pivot a spiral on the golden triangle derived from the pentagon and on its golden sectionings, has led to the sensational discovery of a principle that we would never have suspected existed, since it demonstrates an implicit tendency of each triangular geometric combination, however irregular it may be, to achieve an ideal golden conformation, exactly that of said already perfect triangle, if only progressively amplified by spiraloid traits according to the dynamic layout that one improperly attempts to attribute to the so-called 'sublime' one.

How the Ellipses integrate the classic Golden Section Spiral ## https://golden-spiral.eye-of-revelation.org |

I have directly called it a 'spiraloid' procedure, not a spiral, carried out and illustrated step by step up to a sufficient precision limit on the site, which is always subject to updates.

This inverse application - where it is not the spiral that develops on the golden triangle subdivisions, but it is the triangle that perfects itself under the pace of the spiraloid - seems to positively reverse the effects of that geometric compromise whose diffusion remains under explicit criticism.