Draw a square of side AB
with invariable radius circles?
Only a circle of radius AB can accomplish it!
To the tracing for the equilateral triangle is added the bisector that joins the points C and Y of intersection of the two circles, then a circle with the center in C – passing through a and B – which intersects CY in X .
The section CX is obviously the same as AB , but usable only as a radius for a new circle with center in X , which will meet the two initial circles in D and E , valid to demonstrate tout court the completion of the square ABDE profiled by the [radius of] 1^{st} circle, delineating within it the exact quarter of the circle, a prerequisite for the final acquisition of the π geometry stated by this study.
An interesting aspect that derives from it is that this scheme offers the possibility, starting from an equilateral triangle ACB , to trace the square with equal sides.
I will not attempt an unnecessary algebraic proof, since my intent is to highlight how the circle makes use of an interesting horizontal and vertical symmetry together, the first based on the two centers, the other decisive on the circumferences, for a sort of parthenogenesis of the 3^{rd} from XC , as if to say a sign of the cross, synchronizing the use of 'itself' four times, ie with a single basic measurement, to give rise to what, according to a π that we will discover at the end of the treatise, it will correspond to a quarter of the circle, even if valid for the whole.
As in its immanent aspect the circular unit manifests its cyclical nature in four distinct phases, in an orbital circumnavigation as well as in a sine wave, so I believe that I have to conceptualize, or be induced to think of the four entities involved as if they condense their essence combined and reflected in each of the sides of the square, translating it into a finite unity, sufficient to identify and distil all the metaphysical key π, the art of combining the rhythms of time with those of space in becoming.
In this diagram, a very close connection between circle and square is undeniable: a circumference based on its own side multiplies by four in a cruciform movement, to produce a quarter of that square that geometrically envelops it and with which it will identify thanks to the π inferred at the conclusion of our research; almost as if its value were reintroduced by each of the four orientations.
As already stated on page 14 of the Treaty «2×2=3,14»
If the triangle is three [spirit], the square is four [matter] and the pentagon is five [transformation], we will have to think of the circle as One [Source of being]
with the center zero, and thus attribute the two to the extremities of the simple segment [duality, transit, polarity, comparison, connection].
Each point, like the extremes of the segment, is the center of a potential circle, which represents everything around it; but even if the number of circles is virtually infinite, only one is what defines the segment, and it is the circle passing through the other extreme, thus defining its own radius.
Therefore, the 'splitting' of the Unity into two circles can only take on an actuating value of tangible linear reality.
Even if it does not encompass its full width, as well as in any figure having a number of sides greater than 4, if I am allowed to consider the radius of the circle – the only measure that defines it, that draws it and calculates it – as its 'side', assimilating it to that of any other polygonal figure, that dynamic ratio from one to four manifests itself without compromise.
It is in this sense that the essential need for four circles of equal radius is evident to delimit the square and on the other hand, since the resulting figure is ¼ of each circle, it will take four to complete the entire circle.
Will it be just a play on words? maybe yes; but it solely applies to this solution, making it the most coveted.
It made me ponder for whole days, being the most efficient and compact of all the schemes for forming a square, and as far as I know it has never been reached before in such a way, unlike triangle and pentagon.
area: π r^{2}  circumference: 2 π r


It is foregrounded by the algebraic profile for which, referring to ¼ of π, the area is scanned by (r×r) × π /_{4} and the arc of circumference by (r+r) × π /_{4}.
Since the completion of my first issue of the subject («2×2=3,14», confidentially: "two for two equal to three and one four"), I am more and more convinced that to really understand it, the π should be related to the quarter circle.
I struggled in my mind to make such an insight evident, until geometry itself came to my aid:
It can be observed how the system of circles that incorporate the square is in turn naturally inscribed in a square that delimits them on all sides.
In fact, its dimensions depend on the double and equal formation of two circles, both horizontal and vertical, arranged so that
it corresponds to the projection of the virtual square built and centered on the cross CXAB , crossing its diagonals and vertices; a surprising solution for its uniqueness and which, respecting the contextual premises, appears unattainable without going through the square ABDE ; almost bound to remain a mystery.
Another noteworthy fact is that the side of the circumscribed square has a length equal to three times AB , as can be seen for the tangentiality to the two circles centered on its ends. The same occurs for the two by C and X .
With the four circles thus arranged, we had therefore virtually defined even the square that encloses them, which can be reinterpreted as the latent dominant of the whole.
Four equal and symmetrical circles (compass) – three centered at the point of an equilateral triangle with identical side;
two segments resulting in a cross (ruler) – the horizontal is for start, the vertical is resolutive.
It looks like a mandala to meditate on!
It is, however you interpret it, the most fascinating paradigm of how one circle could generate the square of a desired size, or that has a side in its radius, quite rather than passing through ruler and compass, for being due to its intrinsic virtual property.
To simply draw a square I could have used the method on the side, from AB to C and Y , and then from D and B to cross the vertex E ; or another occasional and plausible one, but what would they teach us? Following the vertices and sides of a figure is not always the same as exploring the creative Mind, to enhance the difference between matrix Intelligence and an improvised geometric path.
Whilst the former looks like a living integration of circle and square, the latter is just a drawing of a square with multiple uses of the compass.
An esoteric aspect not to be overlooked …
the VesicaPiscis hovers in the Vatican's p.za S. Pietro, albeit in a not exact way, since a satellite perspective distortion is less likely.

is the fact that this solution draws its foundations from the universally celebrated figure called Vesica Piscis (bladder of the fish, in Latin), also known as the "mystical almond", symbol of the Mother Goddess or the Eternal Feminine, handed down to the basis of different ethnic groups, from ancient Mesopotamia to Africa and from India to Asian civilizations, finally to various European cultures, for its plastic and mysteriosophic but also mathematical implications.
In fact, the splitting that from the circle One, creative, as it is governed by the triangle, generates the duplicity, regulated by the triangle at the centers and intersection points, as well as, at its horizontal center, by the same √3 – i.e. CY , on AB = 1 – and with it the polarity of positive and negative, Yin and Yang (or Ying Yang) is regarded as the sourcefather/mother, of all immanent forms,
and these brief passages already seem to confirm it; it is essential to note from this intertwining and for semantic coherence, how it is generated by the triangle, to then give life or substance to the square.
Not to mention that through appropriate grids obtained from its essential intersection points, and always according to the initial rules, it will allow you to geometrically measure the square roots of the numbers from 2 to 10!
In practice, the connection between the first two circles is reproduced in two others, giving rise to a second Vesica Piscis around CAB , pivoted at 90° on one of their meeting points, C in our figure below, which allows you to trace the inscribed equilateral triangle, without having to extend AB up to T .
It remains to point out what is probably the most significant evolution of the pivotal figure of two circles which, with the horizontal addition of a third circle centered on the circumference of the 2^{nd} and symmetrical at the 1^{st}, placed in a repeated rotation of 60°, develops in any direction the socalled
'Flower of Life'
From the metaphysical bladder to the fish
With the lateral extension of the two minor arcs until they meet the two lower vertices of our square – giving rise to those arcs of a circle that underline the π – the swim bladder is equipped with the tail that completes the profile of that 'fish', taken from the beginning as symbol of Christianity and called "Ichthys", that was "fish" in the ancient Greek, which became the acronym that concealed:
"Iesus Christos Theios Yios Soter" ie "Jesus Christ Son of God the Savior ".
An attribute which, moreover, in terms of the precession of the equinoxes, echoes the past 2000 years for the corresponding zodiacal era of Pisces.
Just examine the figuration, both in its ideographic form: and descriptive:, to realize the symbolic coherence, representative of what we are studying.
It should be noted that unlike the font reproduced here, in the classic iconography the two fish are connected by a double fishing line, or in any case by a sort of belt, a concept respected by the symbols albeit with a simple trait d'union, but almost completely ignored, by those illustrators who probably would not know hy to deal with it.
An apparent nonsense, that instead the explanation has it!
Here is a tiny survey captured from the web, rare though:
It is therefore that square in its position that gets actual the great potential of the symbol for the Christian era.
Also relevant from the same diagram is the all too unnoticed result of the equilateral triangle CYT , here originating from Vesica Piscis, whose side inscribed in the circle of radius 1 is the square root of 3.
As if to say that the side of such a triangle is equivalent to the side of a square with area 3 , a way that highlights how the science of numbers closely evokes the geometry and metaphysical implications of the square.
In general, the side of an equilateral triangle inscribed in a circle always has value √3×r and its height is ¾(2×r) or 1,5×r .
The framework revealed here guarantees its direct and immediate feasibility in an exclusive way, a much more laborious project to be carried out differently, even without following the standard proposed here; which seems to justify the proliferation of freehand drawings.
Indeed, it could be noted at this point that each of the arches measures 150 °, not far from the much argued number 153, which even in the Gospels [John ¶ 21:11] is mentioned as the number of fish in the net, for a miracle wanted by Jesus . Although it is not my intent to embroider on the remarkable mathematical virtues of 153, I cannot but stop to suggest to the researchers that a specular measure of 150 (whose numerological sum is 3, from 1 + 5 + 1 + 5), therefore interwoven and unified from a triple 3, said sum = 3, the square root of 3 and the carrier triangle, can be considered integrated in the special case by the number 153.
The triangular bearing that dominates the scheme led me to further investigate the construct, and here emerges an even more stringent solution made up of, and dedicated to only three circles, capable of creating and containing the same inevitable square in double order.
In fact, the base of the major equilateral triangle has suggested it, intersecting at the point X the third circle centered on the intersection of CY with AB (center of the whole figure); but in reality only as the result of a compositional graphic analysis which, although not practicable with a ruler and compass, being ideally educational under the symbolic profile, deserves to be focused.
As the diameter of the 3rd circle, passing through CY intersects in Y the 1st circle [centered in] A , so the diameter of this, passing through AX for the parallelism demonstrated by the symmetry of the two circles, meets in X the central circle where XY is AB/_{2} , to extend to E which joins D at the bottom of circle B .
It follows that the circle B develops the triangular symmetry that gives us the perpendicular to AB as the vertical diameter of the circle A , therefore the sides of the double square below and above, as well as the symmetrical ones on the opposite circle.
To be honest, I can't say which of the two pearls is the most precious, to demonstrate the square~circle symbiosis that I have set for myself.
I dare say that if the former illustrated a sort of gestation of the square, centered on the quaternary, the latter represents its double conception by reflection of the ternary; and that is precisely why I have intended to illustrate it.
Both cases disclose in a profound means the procedure suited for certifying the geometric and esoteric sacredness of the Vesica Piscis – emblem of the creative process – and of Ichthys, beyond the more or less improvised and fashionable gadgets.
At this point, however, a rigorous distinction is required: while the first solution – processed in the figure on the side – presents all the steps for the theoretical tracing, of the second it is not feasible to trace the 3^{rd} circle, lacking any reference to the point to which the compass should be extended, once placed on the new center.
Even if the rule of a fixed radius of the compass was stated from the beginning, using it ad hoc could contradict the canons of normal practice; however, I would not exclude that this contributes to enhancing the transcendental character of formation.
3^{2} + 4^{2} = 5^{2}
While not neglecting the fact that the square is the native casket of the Golden Section, to highlight which I had drawn the concave and convex pentagon by combining Φ and φ, the solution of the Pentagon could not be missing that integrated the assumptions just stated with equal elegance.
It has been solved, thanks to the 3^{rd} circle with center in Y , from whose intersections E and D crossing the x we reach the vertex~centers F and G and from these to the vertex of the pentagon naturally with five circles.
But that is not all. Even if four would be enough if you prolong the Y C until it intersects a fourth circle at the point w, I don't stop there, being able to obtain a star that almost builds itself, and that from concave it defines the convex direction, one more time with only three circles.
The intersection D of the 3^{rd} circle crossing the point x reaches the 1st circle in E , from which the segment EB – which is already one side of the star – intersects the 2^{nd} circle (center in B ) at the point e, which in turn allows you to extend Ae up to the apical point w and join two other sides to G , I let you choose how …
It is appropriate to specify that, in astrological symbolism, the two forms of the pentagon are synonyms of mediation, transformation, evolution: the convex one is creativeand creative, the concave is destructive or substitute.
At the conclusion of what we have seen, the rigor procedure proposed at the beginning, despite having always found its implicit solution for the triangle, and in more than one way known applications for the pentagon, had never encountered answers for the square, remained the object of easy as well as superficial constructs, regardless of its most emblematic and universal implications, cornerstones of reality at every level of study and knowledge.
Triangles, golden ratio and fake spirals
Nonetheless, mentioning the triangle and having reached the one traditionally considered the golden triangle, I cannot fail to return to that great golden triangle that has given me so much emotion since its first discovery, almost revealing itself as the link between the 4^{th} and 3^{rd} dimension, that is between sphere and polyhedra, not to say cube, with a new question.
The star pentagon gives us an isosceles triangle (in fig. ABW) often referred to as ‘sublime’, as a golden expression of the base AB which corresponds to Φ of each side. But even if we are pleased to adorn it with a false golden spiral as if it represented the absolute,
probably arisen from the inspiration of someone who thought it best to trace the one attributed to the rectangle – also a fictitious spiral, since it is not supported by any expanding continuity, but is only a collage of quarters of a circle; a mistake that I forgive myself for having 'fallen into it' many years ago, but only because I'm trying to fix it, in fact I never believed it… – such a 'spiral' based on a triangular sequentiality in my opinion should not even exist, since not only does it not conform to the circle, whose periodicity is obviously square, as are the cycles of each sinusoid, but the junction points appear more suture, given that the virtual extensions of the arches denote the total lack of tangential homogeneity, justifiable only between quarter circles, replaced by a convergence towards their center, consequent to combined intersections of arcs greater than 90 °, and it can also be seen with the naked eye!
A simple optical illusion, which those who have a keen eye can notice even in the many published drawings, and which all, however, are blindly copying and boasting, professors, students and popular encyclopedias.
It should be superfluous to add that the two golden spirals, whilst applying identical criteria, can never coincide; so what?
In any case, in order not to leave anything questionable, I reproduced it in PDF of very high resolution, easily resizable both on the screen and for the pro printer or plotter, where even the slight concave bending at the junction points cannot escape a careful zoom.
Slight yes, but sufficient to exclude that the curve can be defined as a spiral, even before contesting its path based on arcs with constant radius…
An abuse that goes even further into the socalled Fibonacci spiral, a real metropolitan legend, since it is not understood on what can be based the curvature and circular continuity of hypothetical points, defined only by integers with clear solution of continuity between one and the other, and absolutely without intermediate values. It is but an oxymoron.
In other words, nothing authorizes the drawing of a curve between number 1, 2, 3 and 5 … which represent nothing more than deliberately predetermined units.
In the most respectful of cases, this sequence, linear and not a spiral, least of all transcendent, should be represented only by straight segments, which no π must 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 relationship – however unattainable – with the authentic Golden Section, of which at most it is a makeshift.
It is time to make definitive clarity on this issue overused in all ways.
First and foremost, no ordinary compass can draw a spiral.
The circular expansion – which is nothing else, built with quarter circles as if they were made from Chinese boxes – involves the Φ radius squared every 90°! who would have ever decided so? and which would be the golden criterion? in addition to shifting its center from each quadrant to the next, with a visibly lopsided result if just is removed from view the scaffolding; and yet we boast of passing it as a spiral.
The figure gives an example of the golden increment applied to each full rotation (see also pages 10~11 and 26 of the treatise).
[soon continues with crucial data…].
Compared to the main golden triangle, which I insist on defining the third treasure of geometry, the stellar triangle does not fulfill the golden function that half (while the combination of whole star and pentagon reflects it admirably, repeating five times five types of semigolden triangles: ABW. ABE, AGC, ABx, Ax), where ours expresses it doubly, in the most integral way conceivable: in fact its base is the Φ of the sum of the two sides that grow symmetrically above it, so it can be said that it impersonates the golden section in the most complete, 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.
A magnificent portal between dimensions, as is the great pyramid of Giza.
However, the disconcerting side is that it cannot be alternatively constructed or derived from the pentagon, concave or convex, as if they were incompatible rather than sharing such a privilege; nor is it traceable with ruler and compass from any other figure except by applying the golden proportion.
Of the four golden circles that are appropriate to it, only the outer one and the 2nd inner one connect to the pentagram, since both circumscribe a figure at its vertices, both the concave and the convex inside, which therefore are in proportion 1:Φ² ; but they don't help plotting them.
from the Golden Section to the 3^{rd} treasure of geometry
If so far all these ruthless criticisms may have unintentionally compromised your good mood, and yet you have followed me this far, here is an unexpected exposure, which may offset some losses.
If there ever was an exhibition of 'geometric jewelery', so to speak, the next diagram would be worthy of a privileged display case, having never been seen before, but above all for its singular elegance and symbolic relevance.
Already from the correct reconstruction of the Cosmic Egg (see also «2×2=3,14» page 7), the virtual scaffolding of these circles has revealed itself as the key to impeccable proportions.
But now we are dealing with pure geometry in essence, and here is how we challenge their use, taking for granted the background triangle, which helps to identify them and from which it is brought into being.
Let's redefine Phi Φ on our page: Φ × (1 + Φ) = 1
It should be acknowledged that 1.618: φ is not the definition of the Golden Section: it is just Φ added to 1 , therefore in itself it is not a primary argument but is to be considered derivative, say as cause: Φ and effect: 1 / Φ .
To compensate for the apparent lack of communication between the two golden triangles, another preview, resulting from applications of the array of the four concentric golden circles with the diameters in Φ ratio, which indelibly accompany the structure of the Great Golden Triangle, an authentic mine of miracles.
The 1^{st} external circumference [diameter 1.000 ] comes into play in the most immediate definition of the pentagon, and therefore of the native golden triangles, if just drawn with the center on any point of the 2^{nd} [dmt. Φ], since it intersects the 2^{nd} in two points that mark opposite vertices of the pentagon virtually inscribed therein, symmetrical with respect to the center, naturally but not necessarily projected in the figure onto the external one.
It should be noted that the primary circle thus reported is also tangent to the 3rd [dmt. Φ²], and in turn to the sides of the large triangle (having a vertex opposite to the new center) which seems to want to reinforce the reference.
This note brings me back, with my own surprise, to the prospective investigation outlined in 2003, reported on the previous page about the pyramid of Giza from which all this study originated. A last minute enrichment, which could amplify the sense of that intuition for the resonance levels.
Naturally the coordinates of intersection of the two circles correspond exactly to the angular ones for 18°.
The technique traces in a certain sense the footsteps of the first search for the square, which started this page; but in this case we do not split equal circles, but we act on the dynamic golden ratio of only two, halving the necessary presence.
If we consider the pentagon as the vibrant kaleidoscopic expression of the golden section, the square is only its unmanifest casket.
Perhaps for this ‘rationality’ the procedure applied to the star does not indulge to the four sides, from which it appears contradicted in every direction for a difference of the radius of about 0.00192%.
In fact, the most attractive of the analyzes suggested that in a circle of diameter 1 , a circumference with a center in any point of its perimeter and a radius of a double Φ squared , four times the radius of the 3^{rd} golden circle, intersected it in two points such as to subtend a chord corresponding to the side of an inscribed square; and the first impact seemed encouraging.
Even a case of stimulating semantic relevance, which, however, would have led to the opposite outcome, and which at this point of the walk I do not give up considering worthy of capturing the curiosity of scholars, as it was for mine.
The SVG figure shows it, but given the little manageability of this new format, which I use here only at an approximate illustrative level; also in this case I provided a PDF of very high resolution and precision, which demonstrates the absolute correspondence of the pentagon, against the almost unexpected irreducibility of the square to any attempt of its descent from the gold implant.
In fact it is the prime bearer of the Divine Proportion, and does not need it to be traced.
It seems to hover at a balanced distance from the intersections of the golden circles wide enough to define it from the outside, or commensurate within, and even with symmetrical differences.
The application of the Φ^{2} circle also protrudes in the same direction and always with negligible distances; tangential brought to Φ^{6} , potentially on 4 sides; and that of the circle Φ^{3} tangent to itself or to the base of the large triangle.
I tried and doublechecked the various modes – after stumbling upon two flaws in the PostScript language, which altered the dimensions of the square subsequently rotated by 45 °, and whose native command approximated the outline of the circle (imperceptibly), keeping me in check for whole days, given this precision control, until I realized it and applied algebraic functions, more laborious but exact like a CAD software.
In fact, the detected deviations are almost invisible to normal sight, and become decisive only with a micrometric accuracy verification; I suggest the maximum zoom starting from the dashed circled area, the intersection that defines the pentagon, to see what errors this investigation technique may reserve; a zoom greater than 600% is needed to begin to distinguish the two curves, of which the yellow one is the primitive one and not sufficiently correct.
On the opposite side, the exact intersection data of the two circles, one of radius 1000.00 and center in 0.0 , the other 618.033988749894848 .
The PDF is designed with the utmost attention to exhibit and demonstrate the smallest details, so it is built with fine, sometimes very thin lines (0.01pt, Acrobat ver.5 may not suffice).
In the PDF the major red curves (green in the svg figure) demonstrate the attempts to intercept the vertices of the inscribed square and the resulting approximation, the first at the top flanked in green by what should be the correct trace , ie from the radius 763.9320225 to the same + 1.465.
Minor dashed reds, always derived from the golden system, could at sight appear tangent to the sides of the square… but none satisfies it, a constant proportional anticonnexion prevents it, which seems to challenge the mirage.
This procedure may not be conventional, since I do not dwell on the usual demonstration that others can carry out, but I make use of tools with which the search runs fast, and deserves to be reported, even within its limits.
On the other hand, PDF has become a precious and almost irreplaceable means of communication due to its potential (which I have personally developed since the early 90s) and portability and consequent 360° accessibility.
In the figure, two rectangles naturally inherent to the golden scheme appear to complement the whole: the vertical tangent to the circle Φ², ie with a base equal to its diameter and height equal to that of the great triangle; and horizontal, tangent to the circle Φ, i.e. of base equal to its diameter, and height measured by the golden section of the height and/or side of the triangle, therefore distant Φ³ from the top.
As I see it, a scheme that increasingly evokes a musical score, bearing melodies that alternate and intertwine whirlwind, but perhaps are not satisfied with a Euclidean space.
It is not risky to argue that the golden system makes the figures outlined in these processes intercommunicating, given their properties, not only inscribed in the first circle, but also related to its concentric proportions; fascinating modulations so little random that you can't ignore them.
It should be clear that in this case I am not attending to a manual geometry task, but to highlight and deepen a pronounced golden synergy between the fundamental polygons (whose drawing with ruler and compass is however practicable).
That of a living geometry which speaks through elementary archetypes and reveals its mysterious connections to the roots of becoming.
Not to mention that it is the only source from which to draw the true π.
However, returning to the square, we can deduce its total nondependence (or descent) from the golden section, which several attempts have revealed to reject, repulsing it as if it were a magnet of equal polarity, even if each time for minimal approximations, almost invisible in a common aided drawing.
It is that with respect to the circle, the square is a figure based on the perpendicular intersection of two diameters, or of sides in parallel symmetry or at right angles, all aspects that have no need for the golden ratio, which indeed are able to represent with only two lines, that express their total intrinsic mastery:
half side (= 1) and a semidiagonal (√5 ).
Four is an even number par excellence, and everything in the square falls within a static equality and autonomy, distinguished from the polyhedral facets of the pentagonal star, and perhaps precisely this very appearance denotes a transcendent reality.
All of the above, not without a naive initial disappointment, seems to want to accredit the principle according to which the square cannot derive from the golden section, which in turn cannot be constructed without resorting to the square; even if it is used to reduce it to the triangle CDE, which in reality is only a fourth of it, but which in any case does not participate in this kind of classification, since it is nothing but an arbitrary artificial figure.
On the other hand, obviously, no figuration that exhibits the golden section, or is built on that base, can be considered a matrix or a background from which to deduce it.
Thanks for participating.
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