The way of the spiraloid

This unexpected discovery resulted from a first attempt to show that the fictitious spiral, generally constructed on the tenfold golden triangle present in the complete regular pentagon, can be applied with the same trick to any irregular triangle, without this demonstrating anything or generating a golden curve, and not even a true spiral; but I did not suspect its astonishing effect, when continued for anon increasing number of volutes.

On the other hand, having experimented with it extraordinarily demonstrated that starting from any triangular shape, applying the process with successive arcs as a criterion of growth and development, generates triangles tending more and more to the same golden shape as our derivative from the pentagon, almost revealing a sort of latent perfection! The figure to the side is only a first draft, hurried and only indicative of what occurred to me developing the diagram starting from an asymmetrical 1-2-3 triangle, and then transferring the blocks or triangles obtained to the base of the pentagon, in a scale contained by its base side or by its height, in order to be able to compare the progressive and ever greater correspondence to the regular perimeter of the star.
I'm reproducing it as it is, just to convey the same impression I had on first impact; but I will be more precise.

A sensational discovery indeed, as it undoubtedly evokes the spontaneous settling of natural and biological phenomena!
The figure gives an example that shows how each new triangle in the series, scaled and relocated to the base of the pentagon, gets closer and closer to the golden goal, according to a procedure that we would like to define spiraloid, not because it can or must simulate a spiral, let alone replace it given its constructive criterion carried out with ever-changing measurements, but precisely because it follows its pace.
Proceeding step by step
Spiraloid - basic step

We start with an irregular triangle (deliberately different from the one used in the first test), with angles 1, 2, 3 which in this case indicate the sides and their clockwise rotation. An arc is drawn for each [new] vertex, from the previous vertex until it meets (with radius the side between the two) the extension of the following side, in a continuous and homogeneous circular progression.
Starting from vertex 1 with radius 1-3, an arc ab is drawn, which reaches the extension of the subsequent side 2 with the shortest path. The segment ab is obtained, which establishes a new triangulation for vertex 2 with the new vertex 4 which, replacing vertex 1, now becomes for side 2 what 3 is for 1 . I don't want to make it complicated, but rather plan the logic that will move the next steps.
The key to the spiraloid consists in constructing a new triangle for each arc, so that each circle will end not in perspective of the first vertices 1-3, but from the new triangle 3 to 4, i.e. from the extension of the side ab already in place from the first arc, up to length ad.
In fact, it is with the center on vertex 2 that we will trace the following arc, until we meet the extension of the succeeding side 3 with the shortest path. And here is the segment bc outlining a 2nd potential triangle abc, which however does not complete the 1st round.
It is now up to vertex-side 3 to plot the next arc, until it meets in d the extension of the new ab, i.e. the last side following it, given the last triangle abc that the spiraloid has aired.

Spiraloid, steps 2, 3 and final triangle
Thus, at a first stage, the triangle bcd was obtained, the result of the spiral application on the three sides of the basic triangle.
The decisive maneuver consists in having fixed a pentagonal figure, on which to compare from time to time the triangles obtained from each spiraloid projection with 3 arcs.
The smaller base will match the base side of the pentagon, scaling the new triangular layout up to its maximum extension by base or height, first contained by the base, or in any case by the height of the fixed polygon as boundaries.
Spiraloid - steps 2, 3 -
and resulting triangle

Each control operation, for easily understandable reasons, must be carried out by dragging, orientation and manual scaling on the template, with the inevitable approximation of the entire drawing, using a type of software which allows direct manipulation; but not such as to jeopardize the certainty of progressive analysis. The same actions performed through calculations would involve unthinkable commitments and times, above all before knowing with certainty what the actual goal will be.
So all that remains is to proceed.
Since each full circle will expand approximately 4.3~4.4 times its breadth, to keep it containable and playable within the screen frame, it will also be necessary to further scale each new stage developed.
At each magnification an inevitable graphic adaptation and a new numerical approximation occur, above all as regards the combination of arcs of circles whose radius can vary with dragging according to the degree of destination, due to the ever greater jump on the display.
Even where the analysis of the finished figures should reveal discrepancies in this sense, the measurements were always performed on the entire precise circle.
Using a CAD program would require a specific approach and perhaps more time and repeated attention than writing a procedure for it; but in my opinion the illustrative functions remain precise enough to convey the idea in a certain and reliable manner.
Once you have taken note of the procedure that I describe in every detail, anyone who is passionate about this discovery, amplifying the case history, can always try their hand.

In order to guarantee an increasingly accurate and easily interpretable comparison, I will keep the dimension of the reference pentagon fixed in the graphic samples spiraloid-process-##.svg, on which to report the triangles obtained at every third arc, ignoring the intermediate ones.
spiraloid - stage 2
spiraloid - stage 2 non scaled yet

Spiraloid - steps 2, 3 -

Once the first phase is completed, we are faced with the new triangle bca, whose sides 1, 2, 3 will replace the previous ones, to help us repeat the same operation with 3 arcs.
We immediately see a notable gain in approaching the triangle (orange) to the profile of the vertical golden one of the pentagon, even if its maximum compatible height prevents its base from extending for the entire base side of the pentagon, with the left vertex of which it is aligned.

It's exciting enough to go and discover-verify what a third stage of this development has in store for us, even if afflicted by growing unavoidable freehand inaccuracies: a game that is becoming increasingly difficult to manage, to guarantee sufficient preciseness with an unequivocal outcome.
And the result is unequivocal! at the third turn of the screw, the approach to the golden perimeter becomes very strict with the green triangle, even if of course it is a smaller jump, proportionate to the ever smaller difference.
At the same height, the base remains reduced as in the previous case with respect to that of the pentagon.
Spiraloid, stage 3 partially scaled

The rotation of the spiraloid forces a smaller zoom of the contained set, but the starry template maintains the same size and a clarity limited only by the overlapping of several curves, which I preferred to leave in view.

Finally, here we are at the fourth stage, which can be considered decidedly satisfying, despite the even slightest difference with respect to the golden triangle of the pentagon. Such a small difference that it can be seen almost exclusively on PDF or filiform diagrams that I have set up alongside, since the overlap in the figure appears total.
The result is quite captivating, since the ever closer approach to the ex­pect­ed golden ratio leaves no room for anything but a further refinement of the trend.

SPIRALOID PROCESS step 04+outline
Get the PDF

Rather than go further, for a by now obvious result but equally demanding manually, at this point I took a step back, to check the first of the figures test­ed, stopped on the 3rd lap. Although not as accurate, by now I have lear­ned the technique and, without many preambles, here is the figure of the fourth profile in gold, although obtained from very 'free' precedents as can be seen (some in this test were intermediate triangles).
It should dispel any doubts, if there are any left. It is evident that the closer the triangle, which could always be the starting one, is to the golden ratio, the closer it will get to it with each successive increase.
First draft inaccurate, in its 4th stage

Although perhaps it is destined to reach absolute precision only to infinity (as one likes to declaim), the striking fact is that the destination reasserts itself with any starting figure, at least among those that I quickly tested at random.
It is a discovery that may add little to current applications, but also mean a great deal and in full depth.

In fact, as I have already underlined, the spiral movement is not a mere abstract symbol, but the dynamic vector expressed by life and by the existent in every manifestation, from the cellular to the cyclonic and galactic level, synonymous with the co­or­di­nat­ed development of the individual being and of the cosmic reality.
Given that the triangle expresses the foundation of any construct, the process represented here summarizes very well the intrinsic tendency to the growth, towards the perfection of the ideal and original balance.
In a certain way it reproduces the logistics of human thought which, starting from minimal sensations, made up of three people for the newborn, expands and grows with the accumulation of comparisons and relative experiences; to contain which with advancing age it will find itself resizing the perspective and conscience of an increasingly broader panorama of its knowledge and ac­qui­si­tions, with perceptions no longer comparable to the starting yard or pedestal, up to condensing the proportion and memory of them in an unlimited neuronic reservoir, where the cornerstones of experience are summarized in rapid sensations.
In short, it teaches us how every form can potentially grow and evolve to­wards a latent harmony and composure, which is then the same from which it comes; and it is precisely the pentagon that manifests and guards this secret, the first polygon that presents an internal winding as the first symbol of becoming and transformation, with the lymph of the golden section; an aspect that I had already intuited and developed in 1992, in my first treatise on Astrological Aspects.

from the spirals to triangle

All things considered, it would seem that it is the triangle as such that con­tains the seed of spiral projections, resulting from the effect of a syl­lo­gis­tic tension.
On the first page I premised that “ it is not the spirals that grow on the trace of pol­y­gons, … but it is these polygons that multiply on the ro­ta­tion­al se­quence of the golden section, …”
Also that “the spiral does not derive any origin from a triangular formation, which instead from a metaphysical point of view represents con­clu­sion and completion”.
I was referring to the play of golden rectangles that is used to rotate, in com­par­i­son with the golden triangles.
Where this appears contradictory, a clarification is useful: in fact, with the spi­raloid I applied the reverse procedure – to that pretended for the 'sub­lime' tri­an­gle – activating its potential immanence through the multiplication of ir­reg­u­lar triangles and according to a triangular logic.
It is important to understand that this combination as well confirms the as­sump­tion, as it is the spiral movement that leads to a perfect golden tri­an­gle, not the other way around, from which a true spiral cannot in any case arise, nor can it be applied to it.
Indeed, it is interesting to note from the figure of the 4th 'definitive' stage, that if the classic spiral (with four arches) is too broad for this scheme, the spi­raloids outlined at the various stages of development are quite close to that with three arches, i.e. in which φ proceeds at every 120° (here jux­ta­posed in gold colour), while not identifying sufficiently, at least in this scheme already developed.

In reality, if the concept of fulfillment is definitive in itself, the development that precedes it, and which it virtually contains, must be based on an initial model, capable of making explicit a potentiality.

Primum Mobile

Nothing that I know is capable of expressing it more and better than a triangle, and in absolute terms, than that Cosmic triangle, whose base mul­ti­plied by the sum of its two sides, i.e. its golden projection/expression/man­i­fes­ta­tion, equals to ONE, creative unity, source of all existence.

from the motionless triangle
to dynamical emanation

If it reduces its base by half, or opens and doubles its sides in length, such as from Ay to yC and from Ly to E and brings them to­geth­er at the vertex v, it generated the second triangle that we could conceive of as creation, passing from three to five parts that can be related to it, with the golden par­a­digms of becoming ready to rotate at the vertices of the pentagram.

Its immobility as a figure enclosed in the whole Self, MADE OF ITS CENTER, flows into the full mobility of the open form, whose perimeter fulfillment is achieved in multiplying into a pentagon.
If the great triangle is father-mother, the second is creature.
He that hath ears to hear, let him hear.
So here is how this study has brought the minor triangle back to its sublime guise, freeing it from erroneous vestiges and restoring to those who have intuited its supreme virtue – while remaining blocked by the convention – the most honorable perspective.

Let's retrace that concept of Essential Geometry, which enhances the mean­ing of the intrinsic synergy schemes that I have proposed or high­lighted.
Not a puzzle to be solved with tricks, but the effort to understand the me­an­ders and reproduce the heart of the profound, though unexplored, ge­o­met­ric language in every nuance that leads to the irreplaceable sense of its vital con­tents; since, whenever geometry speaks, it attests to the presence and mys­tery of a Creative Consciousness in all the realizations of substance.

The triangle is completion, but as such it is also the first expression of eve­ry existential structure.
The major one is the celestial imprint, which originates the universe sys­tem, governing it with the laws of absolute equilibrium of the π which it con­tains from the centre, and of the golden section, which it manifests in the pe­rim­e­ter (one side of which is ¼ of the circle which circumscribes it, thus destined for the quadripartite).
To it we ascend geometrically through the square, which is closest to being seen in tangible reality, while following it in numerical order, marking the matter in the 4 Elements, the 4 directions of space as well as the phases of time and of each wave frequency. It is e.g. the tetrahedron, base of DNA, and is multiplied by the sides of the triangle giving form and function to the zodiacal seasonal system.
It is followed by the pentagon, synonymous with movement and life, de­vel­op­ment and trans­for­ma­tion of every form and phenomenon, as shown by the path followed so far.

It is worth contemplating the great pyramid, of which the square is the base, the triangle the height and the pentagon the spatial communion of the four corners with the vertex.
tangentiality to the pentagon
by the golden triangle's circles

One last curiosity: unlike what we have seen in the figure on the first page about the golden circles in the pentagon, which sees the major circle cir­cum­scribed to the polygon, so only the 1st and 3rd touch apical points of the pentagram, examining their system com­pared to the major triangle obtained from the golden rectangle, and in the same magnificent scheme the pentagon derived from it, i.e. with side of the convex equal to the smaller one of the rectangle, and side of the concave equal to the larger one of the rectangle, it is possible to detect the tangentiality of the 2nd circle, in Φ ratio, with two opposite sides of the pentagon; as well as the tangentiality of the same 2nd circle whose center is displaced vertically on the circumference Φ3 thus resulting tangent to the circumference Φ2, to the two contiguous upper sides of the Pentagon.

to deepen the meaning of the spiraloid triangle

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