Draw a square of side AB
with invariable radius circles?
Only a circle of radius AB can accomplish it!

SquareFromCircle-©The Watch Publisher 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] 1st 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 3rd 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 r2circumference: 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 CX-AB, 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!

Sample square From Circle 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 Vesica-Piscis 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 2nd and symmetrical at the 1st, placed in a repeated ro­ta­tion of 60°, de­vel­ops in any direction the so-called '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 3rd 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.

32 + 42 = 52
Pentagon From Circle 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 3rd 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. Pentagon star From 3 Circles
But that is not all. Even if four would be enough if you prolong the YC 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 3rd 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 2nd 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.

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