The way of the spiraloidThis 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.
Proceeding step by 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.
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.
spiraloid - stage 2
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.
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.
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.
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.
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 triangleAll 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.
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 O
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, M
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.
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.
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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