Laws that take a lifetime to understand,
geometry solves them at the very moment when
depicts them. Where does that Intelligence reside?

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

Square From Circle
it is by repeating itself four
times that the circle gives
shape to a square
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 treatise «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 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, starting from the to circles A and B.
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.

new insert 2023 – one year later
Mathematics is not made of numbers but of concepts,
that connect the phenomenal reality to numerical factors.
Nonetheless, if the above solutions represent a good example of practical ge­om­e­try, from my advanced research – introduced in the following par­a­graph – the greater value of what I will define ESSENTIAL GEOMETRY, art in nature, will emerge, being the scenario in which the Golden Section ex­press­es its con­tin­u­ous and dominant function, of which the most eloquent synthesis is condensed in this figure:
complete golden system of the 5 fundamental polygons
click to highlight the pentagon's best golden solution by
the multiple golden correspondences with the rectangle

In addition to the passage from the basic square to the golden rectangle, the scheme, shown here in absolute preview, contains the construct of the pas­sages ABC, from the golden rectangle to the great golden triangle with vertex C, but first with ABv at the vertex v of the golden triangle ALv which is in­ter­nal to it (and virtually synthesizes its property), and then from the in­ter­sec­tions x of the arcs ALE with center at A and its symmetric from L, define the line x–E whose extensions intersecting the arcs themselves point out the vertices of the pentagon implicit in the applied proportion.
Not to forget the simultaneous construction of the lateral triangle, for ex­am­ple: AEv also golden and which I will call tertiary, with inverse func­tion­al­i­ty: not the base golden section of each side but each side golden section of the base, which will be decomposed into two other equivalents with the bases on the sides of the convex pentagon.
Even more elegant for both, it would have sufficed to intersect the arcs from L and from A with radius LA with those from L and from A with radius AB.

Like a pair of spread wings…
I'm keeping the description narrow, so as not to weigh down this magical fig­ure, a real monument that speaks for itself. Note also that the + sign inside the figure, visible with one click, indicates the exact center of the golden rectangle.
Applying an inverted golden scheme on the left side, with the side square cor­re­spond­ing to the height of the triangle ADC, right represented on the site indexed in the following paragraph, we can define the cir­cum­fer­ence that cir­cum­scribes the great triangle, with all the range of concentric golden cir­cles, won­der­ful for their mutual properties.
the golden rectangle “irregular polygon”?
The study that I strive to present is that of an integrative ge­o­met­ric pro­spec­tus, aimed at fo­cus­ing in an ever more com­mit­ted and ad­vanced way the lit­tle-ex­plored val­ue – de­spite the mul­ti­ple fic­ti­tious re­pro­duc­tions that pop­u­late the web – in math­e­mat­ics as in ge­om­e­try, and bet­ter to say in the nat­u­ral whole, of the gold­en sec­tion, not by chance de­fined Di­vine Pro­por­tion.
Since 2002, with the cre­a­tion of the first Sri Chak­ra yantra ge­o­met­ri­cal­ly ex­act over the cen­tu­ries to­geth­er with the par­al­lel dis­cov­ery of the ex­traor­di­nary com­ponents of the Great Gold­en Tri­an­gle in­side the pyr­a­mid of Giza, at the do­main 3rd level domains have been added: sriyantra com­ple­men­ta­ry to the geometric esoteric study of the mandala, gold­en-ra­ with a first review of the topic, together with a study on the tricks implemented by Leonardo da Vinci to make ends meet for the famous Man of Vitruvius, with my actual golden solution a­chieved; the pre­sent pi-day.eye-of- which develops the main theme of π, whose further development has led to a subsequent golden-spiral.eye-of­-rev­e­la­ which brings the integral theoretical and prac­ti­ca­ble on­line so­lu­tion, as well as unexpected discoveries regarding the development prop­er­ties of the golden spiral and the 'spiraloid'.
The exposition involves a 'primitive' geometry, hinged on three figures: 3, 4, 5, foundations of every more or less elementary construct, starting from the the­o­rem of Pythagoras; it does not refer to the concepts of infinity, which prob­a­bly dwells only in the mind of those who imagine it, since in the slowed­-down di­men­sion (that is, of the matter in which we are immersed and sci­en­tif­ically be­liev­ers) nothing infinite could exist.


For the same reason, the straight line doesn't exist either – if not a figment of the imagination – but only the segment, i.e. the shortest section that joins two extremes, however it is intended to be calculated.
As I see it, arguing about the parallelism of two straight lines is a waste of en­er­gy that goes beyond complacency; and if so, many have been spent, with­out changing the initial assumption…
Reality must be understood for what it is, while inventing it can only deviate from it, as well as from the rigor that science itself imposes.

We know that by definition the regular polygons are only the con­vex e­qui­lat­e­ral and e­qui­an­gu­lar ones; but e­val­u­at­ing the priv­i­leged, en­tire­ly u­nique re­la­tion­ship of the gold­en rec­tan­gle, as a me­di­a­tor be­tween square and con­cave pen­ta­gon, the need now a­rises spon­ta­ne­ous­ly to fo­cus on a new, ad­e­quate vi­sion and dis­tinc­tion of the three fig­ures, en­dowed with par­tic­u­lar "reg­u­larities" at the base of all manifestations of reality.
In fact, although the rectangle is equiangular but not equilateral, instead of this impeccable qual­i­ty it boasts the no less sig­nif­i­cant qual­i­ty of a gold­en ma­trix which i­de­al­izes its func­tion, since in ad­di­tion to con­tain­ing the square, it gives shape and spe­cif­ic cor­re­spond­ence to oth­ers in an even more sig­nif­i­cant key than of al­most in­ert flat figures.

This diagram presents a somewhat innovative map, but seen from a more likely perspective of reality.
classification of basic polygons for
classical and/or essential geometry
and the 40 composing triangles

In fact, if at the foun­da­tion of Eu­clid­e­an ge­om­e­try (whose sym­bol­ism is what mat­ters, even if space is curved, since their sym­bol­ic pro­jec­tion is in­al­ien­a­ble) we clas­si­fy three reg­u­lar pol­y­gons de­rived in max­i­mum syn­the­sis from the in­ter­ac­tions of [a] cir­cle[s] with con­stant ra­di­us, as we have dem­on­strat­ed: tri­an­gle, square and dual pen­ta­gon, at the same time we can­not o­ver­look the im­por­tance of the fig­ures ex­alting the gold­en ra­tio de­duced from the gold­en rec­tan­gle: on one side the su­preme gold­en tri­an­gle, on the oth­er the con­cave pen­ta­gon, which show it be­fore and up­stream of each oth­er re­al­ization and reg­u­la­tion.
While the first class con­cerns reg­u­lar and stat­ic fig­ures, the se­cond in­tro­duces liv­ing and dy­nam­ic pro­cess­es, as they are more in­ter­est­ed in the de­vel­op­ment of nat­u­ral phe­nom­e­na; represents the three distinct ways in which the golden section takes shape, in the geometric and symbolic figures fundamental to existence.

Of the square, which does not express it directly but contains it, there must take place that sort of 'birth' which carries out its metamorphosis into the gold­en rec­tan­gle, the quadrilateral such as to act as a bridge to the con­struc­tion of the other two.

In my meta-vi­sion eg. it is the pen­tag­o­nal star that ar­tic­u­lates pro­cess­es of growth and trans­for­ma­tion through con­struc­tion and de­struc­tion, where­as the con­vex pen­ta­gon ex­press­es ver­tex af­ter ver­tex only the de­riv­a­tive (but does not for­mu­late the gold­en sec­tion by it­self), and makes use of the in­ter­nal plot of its pe­rim­e­ter to cre­ate max­i­mum har­mo­ny, just as the square refers to its di­ag­o­nals to de­fine the cross of the four cardinal directions and the pro­cess­es of di­a­lec­ti­cal op­po­si­tion; the na­tive pen­ta­gram pe­rim­e­ter de­fines 10 sec­ond­ary + 5 ter­ti­ary gold­en tri­an­gles, to which if we con­nect its ver­ti­ces with the pe­rim­e­ter of the con­vex pol­y­gon, 5 are add­ed for each side, of which 3 of the ter­ti­ary type of two different sizes.
The clockwise list in the above figure is:
5 × BAC, 5 × ExA | × EzC, 5 × ECx | × AEz, 5 × yxC, 5 × BCD e 5 × CBz
A collection of unrivaled synergistic richness.
Do you perhaps know of a way to draw a pentagon, which makes more sense than this crystallization of the golden ratio?
In this most re­cent ac­qui­si­tion of the im­pli­ca­tions of its sec­ond­ary gold­en tri­an­gle there is an unparalleled confirmation.
As for the great triangle, I have been and will still be too explicit in declaring it the container and revealer of the authentic π and measure of the circle that incorporates it. Given a circumscribing circle with a diameter equal to Unity, one of its sides is equivalent to ¼ of the circumference and is Φ, revealing the π in the ineffable and total harmony of the rules.
Mathematics: science that uses logical and deductive methods for the study
of the properties of numbers, figures and geometric configurations,
formal structures and evolutionary processes, and which with other scientific
and technological disciplines serves to define models of the aforementioned entities
to solve real problems of broad scope

Compared in a more in-depth synthesis, the first potentials are better com­pre­hended.

  1. The equilateral triangle by all sides equal does not present any basis, no orientation, no dynamics. It is a mere geometric symbol, of complete harmony, which getsmajor sense in the tetrahedron.
  2. On the other hand, the first great triangle is the maximum, direct and es­sen­tial ex­pres­sion of the Divine Proportion: as it presents a base on which the two sides form an agreement, related to the base in full and exclusive ratio φ. As such it is capable of representing the splitting of the primordial Unity, in the maximum conceivable har­mon­ic e­qui­lib­ri­um.
    Not for nothing it is a recognized universal symbol of the divine.
    It is the Creative Principle that generates the two opposite com­ple­men­ta­ry poles, Father and Mother, purposeful and receptive, expansion and contraction whose product is equal to 1.
    It makes vertical sense; although of little importance, if it is halved ver­ti­cal­ly (from the vertex to the middle of the base), the golden rectangle can be immediately obtained, from which all the annexes.
    It is precisely in trying to graphically demonstrate the primary gold­en re­la­tion­ship that I discover a new method for integrating the large tri­an­gle with its circumscribed circle in the most direct and immediate way; giving rise to the multiplication in an explosion of four other triangles of the same type, two of decidedly larger dimensions and two a lit­tle small­er than the basic one, which in turn is duplicated by its sym­met­ri­cal vertical projection. The procedure is elementary:
    draw the great golden triangle
    in the circumscribed circle
    with ruler and compass

    doubled side AC to AB – let's say with a circle of diameter =1 – an arc with center A and radius AD intersects AB at the point Φ golden section of the sum of the two sides, and more: if extended up to E, where it meets the a­fore­men­tioned circle with cen­ter at C, it will give rise to the new base DE of the golden triangle DEB perfectly inscribed in the major circle – a proof that gives us the shortest path to construct it with ruler and compass – whose 4th inner golden circle is tangent not only to the base of the triangle but also to the arc ADE precisely at the point Φ. while the 2nd as well as obviously to the sides DBE is tangent to the side of the golden rectangle of origin.

    It doesn't end here: repeating this last operation vertically, i.e. on the side DC of the triangle [CLICK], the segment xy between the two in­ter­sec­tions of the major triangles [blue-green] together with xF and xE the sides of two other new golden triangles xyE and xyF with vertex at x, slight­ly smaller in size than the initial triangle; as if to feed the magical deducible harmony.
    What the magic of this figure consists of is soon explained: since the side DB is normal to the base DA, the ratio DA/DB , representative of the rectangle that appears holding the mouse on the clicked figure, is 1.272022 or rather theφ.
  3. The square, conversely, has a horizontal sense, it is the space [of the] cre­a­ted, which distributes and set against the four Elements on the ma­te­ri­al plane.
  4. The complete pentagon is a complex figure, vertical and horizontal as a syn­the­sis of the two: 5² = 3² + 4² !
    Each of its vertexes is reached by the other four, graphically simulating an almost pyramidal per­spec­tive, paradoxical in the elevation, but inviting to a log­ic not to be overlooked, which re- proposes itself in rotation.
    It directs the vortex that produces movement and mutation, through a double direction of rotation, as well as a sort of spiral deriving from au­ton­o­mous­ly re­pro­duc­ing itself, something that neither the triangle nor the square do.
    If in the pentagon the transition from one vertex to the next, e.g. of the base, occurs with a side AB in an anticlockwise direction, in the star the two sides AC, CB concur in a clockwise direction, which de­fine the secondary golden triangle together with the side AB which is its base.

    To move from one vertex to the next of the pentagram, the pentagon instead needs two sides in a clockwise direction (the shorter), thus generating the tertiary golden triangle AEC, whose base in this case is the side of the star.
    Therefore, the fifth element is identified with it, the ether, permeating every aspect of life.
    We can distinguish the two sides that co­ex­ist in the polygon by the vertical, cen­trip­e­tal, positive attributes, and vice versa.
    This is how Intent is configured, with the revision principle which directs everything, action or event to completion according to those laws of cause and effect which lead to advancement through achievements; the law of return, course and recourse is im­ple­mented, which is counterpart and com­pen­sa­tion, called karma in the East, e­nun­ci­at­ed by Newton's third law in the West:
    “for every action there is an equal and op­po­site re­ac­tion”, which translated to the vi­bra­tion­al and energetic sphere of the earth­ly human means 'you reap what you sow'.

    Electricity and Magnetism
    Unlike the great first triangle, in the 'secondary' and 'tertiary' triangles the golden ratio is dou­bled, the golden ratio exists only between the base and each of the two sides separately.
    Both in pentapolar symmetry outline the com­plete pol­y­gon, but while the tertiary is more oriented towards the perimeter, i.e. peripheral fringe, as if identifying only with the convex fig­ure and without concerning the area of the internal convex pentagon which is extraneous to its area, the secondary is concentrated on the internal field of the figure, it delimits and contains it with its development.
    So while the tertiary represents expansion and centrifugal motion, the secondary represents contraction and the gravitational principle.
    By tracing for each of them the two arcs with centers at the extremes of the respective bases and having the side of the pentagon as their radius, their rays at the intersections with the sides outline the presence of 2 +1 triangles of the opposite type and 2 of their own type; the latter in the 1st case are added to the overlap (Yon), in the 2nd they are subtracted (Yin).

Whether or not it is agreed in the time to come, these figurations show an in­tel­li­gent and essential concatenation, the foundation of an esoteric, almost id­e­o­graph­ic geometric cosmogony.
The power of 5

Anyone wishing to delve into a scientific discussion in support of what has been retraced in this report, fo­cused on the geometric aspect of the pentagon, will find on this page matter of reflection and appropriate ref­er­ences to the discovery of the MAGNETIC ATOME, dis­closed since 1940 and published in 1954 by the fa­mous researcher Pier Lu­i­gi Ighina (formerly Guglielmo Marconi's as­sis­tant).
You will find further detailed descriptions regarding the dynamics of triangle and square within astrological analysis.
That page bears a recent datetime due to forced updates of some web ad­dress­es from http:// to https://, but its content dates back to the first AstroTime production prior to 2010 (when the golden spiral was not yet really explored by me). Retracing it for a useful translation, I realize that quite a few concepts expressed and coordinated in those years are strongly integrative of the cur­rent ef­fort, precisely from the ex­cel­lent dialectical horizon of astrology, which offers the most significant food for thought, due to the close correlations that link the dimension of matter to the unfolding of events.

Although Ighina's statements have not found the probably most adequate historical ratification, which from the example of other similar cases is not surprising at all, the central revelation of the magnetic atom fits perfectly into my repeated analyses, then and now, in particular with regard to quintile and pentagon.
I must therefore specify that I, in my turn, came to know about Ighina precisely because I matured said correspondence. The specific descriptions of the astrological aspects developed for users of the AstroTime service, published in the treatise «PER UNA CONGRUENZA DEGLI ASPETTI PLANETARI» – © The Watch Publisher, 1992 – ISBN 978-88-904390-3-2, were and remain independent, without prejudice to the synoptic, both astrological and geometric, as well as esoteric validity of this prospectus, which finds support in every side of the natural kingdom. If we then wanted to, we would have no difficulty in recognizing the abstract meaning of the single elements in the three figures, it will suffice to mention the most salient and indicative for a quick comparison.

  1. The triangle is Fire, and like any flame it has a base, height and point up­wards, in a single direction. Burning produces energy; in one way or an­oth­er it permeates any organism, determines its compactness and can sublimate it; it has no dimension of its own nor any measure: a match or a spark is enough to ignite the largest fire.
    It is therefore and above all Light.
    Its three sides can represent the principles of at­trac­tion–re­pul­sion–pro­pul­sion that characterize the dynamics of existence, but only in their potential, the effect being integrated with the other two figures.
  2. The square, on the other hand, is Earth, with a rigid and measurable ma­te­ri­al dimension, just like the time that accompanies space, in four fun­da­men­tal physical or seasonal directions.
    Flat or four-dimensional (time included), it is the scenario in which things and events find their place and relative stability. Its first characteristic is the contraposition, inherent in the parallelism; its two opposite vertices reach another two opposite each other.
    It is a source of complementarity, of overcoming and continuation through comparison, but also of friction and stagnation, since its nature is fixed, static, not self-propelled but only dispositive, precisely of at­trac­tion­-re­pul­sion.
    In astrological language, quadrature (or square, 90°) is synonymous with friction, impact and discomfort, as well as multiples (135°) and sub­mul­ti­ples (45°, very insidious). These references may not be of interest to mathematicians, but they have enabled me to predict for the orbital an­gu­lar relations between planets (* see below) dozens of documented seis­mic events, available online.
    After all, it is enough to observe the specifics of the seasons, opposite and complementary.
  3. The pentagon then presents a double expression: passive and active.
    From what has been described above, it is easy to deduce the man­i­fes­ta­tion of a component of Water (or liquid state), always mixed and con­vert­i­ble into one of Air (or gaseous state) and vice versa under the action of Fire; components capable of transforming and transmuting the state of the Earth, which in itself is inert and immobile without this incessant push.
    So the propulsive reality of the magnetic atom manifests itself not by chance in the directions of five absorbing atoms, which the photo shows us, even if probably Ighina was not personally aware of the principle in review.
It is interesting to observe how the presence of a tet­ra­he­dron [triangle in solid] inscribed in a cube [square in solid] subdivides it into a total of 5 tetrahedrons, each having as base one of the faces of the internal tet­ra­he­dron and as vertex one of the vertices of the cube.
The 3 faces of each exterior tetrahedron × 4 times are 12, as many are the faces of the dodecahedron [the pen­ta­gon in solid].

In short, it would not be risky to consider the three figures under ex­am­i­na­tion emblematic of the attributes of Cardinals, Fixed and Mutable, under whose groupings the distribution of the 4 elements in the zodiacal circuit is repeated.
Unlike the square, which breaks it down into four phases, the pentagon is closer to the circle and establishes that factor of continuity which is its rotation in double direction.
It should be noted that while in the square each vertex reaches just two contiguous vertexes, but not the opposite vertex (potential complementarity, but on the basis of the contrast), in the odd figures each vertex reaches all the others, excluding a possible conflict. In the triangle this passage is single and immediate, in the pentagon it is multiple and mediated; in any case it ex­press­es an ever-present dynamic impulse.
We saw at the entrance to this page how a circumference is multiplied 4 times to generate the square; but how can the golden triangle, with its spe­cial prop­er­ties, derive directly from it? it's simple nearly predictable: while 4 equal circles reiterate the sym­met­ri­cal and total flatness of the square (which is equal in side to the radius), we meet here three concentric circles (internal to 1st, thence 4 in scale) as if to exalt the verticality of the tri­an­gle, since they introduce the vortex of a spiral to thought, or even their de­vel­op­ment to­wards the apex of the great pyramid, already mentioned in the entry page; in fact aligned in triple gold­en re­duc­tion since the divine ratio is the one to be re­pro­duced in this case.
It is thanks to the detection of these circles that I was able to discover and understand the supreme triangle, highlighting and publishing its specifics in 2002 in various formal directions. By means of that triangle, and only that one, the circumference advises us with its half with the relative Φ.
It could be deduced from this rich experimentation that if the four it can be seen, the three and the five are impalpable, like the soul of reality.
By imagining the absolute that everything contains, this is a certain truth: the circle means containment of everything, it is the π ! and by this way it in­forms us about, since as I said, half of the circumference will be e­quiv­a­lent to the upper half of the great golden triangle (I consider as such the two sides united in their golden ratio with the base), as if to underline the way to show its unity to the tangible world.
A precious gift for those who know how to read.
Thus every quarter is the true π, center and periphery, root in essence of the golden section and all that follows from it.
Its digit governs the grav­i­ta­tion­al e­qui­lib­ri­um between subatomic particles.
How can one not grasp so much perfection and magnificence, once noticed? Replacing it with a π artificial and unquestionably imperfect, to the point of needing so much clamor (of geometric impossibility) to assert itself?

It is worth mentioning that on the simple and direct application of these three basic figures of the geometric language to the processes of ce­les­ti­al me­chan­ics, more simply to the angular ratios in the orbital movements of the planets in the solar system, I have developed for about four years (2011-2014) daily seismic forecasts anticipated on the server of, too often occurring the following day within minutes or only seconds to ignore them, then archive with all other events in advanced PDF tables always available; still systematically downloaded for the obvious purpose of study by scholars from various countries, despite the correct reading of the technical pages updated on the web, on non-changeable dates, has been compromised over the years by the functional mutations of the browsers, from which I asked for the maximum, given the huge volume of graphical data in sight.
On 05/31/2012, I jotted down a first list of among the most precise forecasts that is, they hit the spot in less than a minute, but then I didn't bother with it anymore, being the work in real time rather tight.
When I thought that was enough, I ceased this commitment without respite, but I didn't give up posting forecasts via Twitter, often addressed to INGV, most­ly sub­ject to undoubtedly shocking results in the most serious cases.
Unfortunately, however, predicting earthquakes is hot ground, reasonably al­most for­bid­den, which will not make Astrology (the serious one) and science go hand in hand as easily as I intended. But the data remains, resulting from equations and algorithms based on triangles, squares and pentagons.
Looking back at that home page, with the celestial table that photographs the system at the mo­ment of the strongest of earthquakes, I can't help but notice that a terrible spiral is really enthroned.

The final finish line
I had come to mature these last reflections after discovering the power of the spiraloid, capable of bringing irregular figures back to the perfect pentagonal triangle.
Therefore I preferred to continue them on this page, which introduced the re­la­tions between the basic polygons; this can cause discomfort in reading, but the matter is amplified in real time on its own drafting, and I could not re­con­struct the entire discussion for each new addendum.
Now, after having meditated and studied this last theme in depth, an inspired deduction forces me to return to the last launching ramp, the Quintessence of the golden spiral, to develop what will be the most coveted of conclusions, a true last tile capable of completing the mosaic of this unknown world, fulfilling all expectations in a pragmatic way, to finally be applicable to the natural world.
The work is in progress, those who have the patience to follow me will not be disappointed.
end insert 2023

Triangles, golden ratio and fake spirals

If this paragraph had confined itself to the false spirals, to which the star above reminds me alas, it could still appear on this page; but the creative law of which the golden spiral informs us at every level of reality, at this point made me dutiful to deepen and document, which has taken on the dimensions of real research, with innovative analyzes and answers to topics so fas­ci­nat­ing as to give prominence – in purely geometric terms – to the dedicated subdomain
The fact is that the golden spiral generated by the Divine Proportion is in­ter­preted and presented by various mathematicians as a simple kind of log­a­rith­mic spiral; perhaps to simplify something that gets out of hand.
In reality the opposite is true, since while the logarithm is a mathematical de­vice formalized in the seventeenth century, which gave rise to the for­mu­la­tion of geometric spirals of various kinds, the golden spiral has always rested on a completely autonomous principle, centered on the immutable and ir­re­place­a­ble ratio of Φ, the factor that has no equal in any other ex­pan­sion­/con­trac­tion formula, and which, as we shall see, is realized in more ways than one.
It is an expression and is expressed by a single principle that permeates eve­ry­thing with balance and harmony. revealing a force at the root of phys­i­cal ex­ist­ence and its biological evolution; miracle of unceasing perfection in be­com­ing, self-defined in the past and in the ever accelerating future.
This rank of his is therefore neither naturally amenable nor historically re­ducible to a case of common spiral, even though it can be calculated with the same formulas used for the latter; if the other spirals, by comparison, are little more than ideated geometric structures, the golden spiral is an ideal geometric structure. So let's see the reasons.
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." – da Bertrand Russell, “A History of Western Philosophy”
It may be thanks to the topics I'm dealing with,
but personally I find it of a warm and vibrant beauty!

from the Golden Section to the 3rd treasure of geometry

If then some ruthless criticism may have unintentionally muffled your good mood, and yet you have followed me this far, here is an un­ex­pec­ted ex­po­sure, which may offset some compromises.
If there ever was an exhibition of 'geometric jewelery', so to speak, the next diagram would as well be worthy of a privileged display case, having never been seen before, but above all for its singular elegance and sym­bol­ic relevance.

If from a historical point of view the rule of 'certainty', deriving from prac­ti­cal geometry with straightedge and compass, has been decisive, today that cer­tain boundaries, including speed of execution, are superseded by pro­gram­ming languages, it becomes essential to delve into the most intimate, intrinsic construct of the fundamental figures, their rhythmic, creative and dy­nam­ic re­la­tion­ships, since natural laws and synergies emerge from these po­ten­ti­alities, which, not being made of geometric expedients, can make us aware what's behind the scenes.

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.
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.
Pentagon defined through just
two golden circles: r1=1, r2
The 1st external cir­cum­fer­ence [diameter 1,000] co­mes into play in the most im­me­di­ate definition of the pen­ta­gon, and therefore of the native golden triangles, if only repeated tangentially to the sides of the triangle, so from the two points of in­ter­sec­tion with the original cir­cum­scrib­ing, the di­am­e­ters pass­ing through the center will define the vertices of the pentagon at the opposite end. It should be traced with the center on the lower point of the 2nd [dmt. Φ] (which is also the cen­ter of the 3rd translated tangent to the low­er point of the 1st as well as to the base of the great triangle), and intersects the 2nd in two points which equally mark the opposite vertices of the pentagon virtually inscribed therein, symmetrical with respect to the center, of course but not nec­es­sar­i­ly, projected into the figure on the outer circumference.
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 3rd 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 the square is the independent 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 Φ2 circle also protrudes in the same direction and al­ways with negligible distances; tangential brought to Φ6, potentially on 4 sides; and that of the circle Φ3 tangent to itself or to the base of the large triangle.
I tried and double-checked 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 anti-connexion prevents it, which seems to challenge the mirage.

Intrigued, almost provoked by such an incompatibility I want­ed to probe another case, inspired by the strategy of the 4 circles that define the square, precisely because of their vaguely tri­an­gu­lar arrangement, and at the same time framed in a cir­cum­scribed square: by tracing the circle inscribed in it, it im­me­di­ate­ly arises to verify whether its intersections with the diameters of the two initial circles delimit the base of the great golden triangle inscribed, which at first sight does not seem to fail, in conjunction with the side top of the square at the base of an equilateral triangle.
And instead nope! again for minimal difference.
Since the gap I see is too small and unreliable, I reconstruct the graph by passing from the more fragile illustrative sphere to PostScript programming; but the distance is confirmed accurately (click the figure to watch it).

These tests and related considerations, apparently useless, at least dem­on­strate to a philosophical thought that the square of matter, while bearing the golden imprint, does not appear to take a direct part by integrating itself into schemes deriving from the light (triangle) or from the electromagnetic field (pentagon) of the divine proportion, but can only take effect as a starting gate.

The procedure set out here may not be conventional, since I do not dwell on the usual dem­on­stra­tions 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 non-dependence (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 semi-diagonal (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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