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!

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 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 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 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!

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 2^{nd} and symmetrical at the 1^{st}, 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 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 one 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 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 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 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 …

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.

first 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:

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 eye-of-revelation.org 3^{rd} level domains have been added: sriyantra complementary to the geometric esoteric study of the mandala, golden-ratio.eye-of-revelation.org 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- revelation.org which develops the main theme of π, whose further development has led to a subsequent golden-spiral.eye-of-revelation.org 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 subsist.
The idea of infinity adopted by the academic world is almost parallel to the image of God as the priests tell it.

B.C. – PARALLEL LINES AND NON-EUCLIDEAN GEOMETRY

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, although in turn endowed with 'an infinite number of points', however it is intended to be measured.
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.

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
– https://sapere.virgilio.it/parole/sinonimi-e-contrari/matematica

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

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 gets major sense in the tetrahedron.
Like any other triangle, which can always be inscribed in a circumference, it is Uniqueness, as each vertex is connected only to the opposite ends of the adjacent sides in the single perimeter path, articulated on three points.
Related to a physical sphere it becomes a tetrahedron, with four triangular faces, without any deviation from the principle already exposed, since each vertex can only reach one of the contiguous ones.

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.

November 8, 2023

THIS MEANS IN ALGEBRAIC TERMS THAT ITS BASE [1] IS TO THE SUM OF THE TWO SIDES [2] AS THIS IS TO THE ENTIRE PERIMETER OF THE TRIANGLE [3].
Extreme thinkable synthesis of the numerical ratio: Φ : 1 = 1 : φ .
As such it is suited to admirably representing the splitting or subdivision of the primordial Unity, in the greatest conceivable harmonious balance.
It's not just this that surprises us, its area couldn't be any less.
Four months later, a specific study on the relation π and Φ led me to also delve into the parameters of its area, from which it appears that:
THE AREA OF THE TRIANGLE IS IN DIVINE PROPORTION Φ^{2} WITH THE AREA OF THE CIRCLE THAT CONTAINS IT WITH A PERIMETER OF 4 TIMES ITS SIDE; AND IS EQUIVALENT TO THAT OF THE CIRCLE WITH DIAMETER Φ TANGENT TO ITS TWO SIDES.
Can you think of a figure of greater synthesis, essence and essentiality than such a concentrate of the Golden Ratio and sacred π?
Perhaps it could be the way to recalculate it with real precision, through the partition of the circle.
So if the second golden triangle – which we will see better by analyzing the pentagon and then discover unknown latent properties in the in-depth study of golden spirals – is now loudly defined as ‘sublime’, this one which I have revealed since 2002 in various publications as the Great Golden Triangle can without a shadow of a doubt
be admired in its majesty, complemented by the emblem of creation “THEORY” [The Holy Eye Of Revelation Yard], a perfect version of the Śrī chakra yantra with its immense vibrational scent, itself recognized as global symbol of the divine], and therefore worthy to be considered the "third treasure of geometry" that claims to be called, or rather acknowledged as

THE DIVINE GOLDEN TRIANGLE

enclosing {and enclosed as well in} the Eye of Revelation.

Inscribed in a circle with diameter = 1, its height is Φ.
The four basic circles which have special tangentiality arguments on its sides stand in ratios of Φ^0, Φ^1, Φ^2, Φ^3; a cosmic spiral departs from the center compensated by the anti-spiral of containment, related in the my study on the vital principle of the golden spiral while the same spiral expands in absolute or universal form (experimental range = 24°).

One may even not contemplate the figurations added to the image to enhance its scientific or yet broader potential, but the absolute primacy of this triangle in every kingdom is now striking.

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, and to the ONE.

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:

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 the√φ.

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.
Unlike the triangle, it involves Comparison, since each vertex is connected not only to the opposite ends of the adjacent sides, but also to the opposite vertices, thus articulating its essence on four points of the perimeter path, plus two diagonals forming a cross; six is the same number of faces it will have, projected into a sphere as a cube, with four faces, let's say vertical, plus the two above and below.
By joining a vertex of the cube [square in solid] with each of the opposite vertices on the three contiguous faces, and the same from the vertices reached, a perfect tetrahedron [triangle in solid] is inscribed in it; and it is interesting to observe how this arrangement in a cube divides it into a total of 5 tetrahedra, four of which have one of the faces of the central tetrahedron as their base and one of the vertices of the cube as their vertex.
The 3 faces of each outer tetrahedron × 4 times are therefore 12, as many as the faces of the dodecahedron [the solid pentagon].

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.

In fact, it recalls the composition of √ 5 from the factors 1 ( apex) and 2^{2} = 4 (square base), integrated into the most recent anamnesis of the golden section.

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 alternate 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'.

In this regard a new paragraph has been added in my analysis, unprecedented combination of the symbolic and mathematical value of the figure 5, properly illustrated in the context of the golden section, in the domain dedicated to the spirals, and which I highly recommend examining.

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).

According to a more recent geometric discovery of mine, this triangle represents as well the destination of any triangle, however 'irregular', which is transformed according to the spiraloid progression described in detail in the chapter “The secret of the spiraloid”.
It is appropriate to specify that, in astrological symbolism, the two pentagonal forms are synonyms of mediation, transformation, evolution: the convex one is creativeand creative, the concave is destructive or regenerative through replacement.
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.

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. IT ISCONSCIOUSNESS 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.

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 ISMODALITY 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.

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. IT ISIMPLEMENTATION 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 here.

It is worth referring to a passage in the paragraph “° ASPETTI DI QUINTA ARMONICA” from my Astrology Lessons (Italian only in 2009)

The forms of the Quintile seem to have been entrusted with the task of dissolving the apparent fixity of matter, well represented in the cross of the elements. The progress of the figure in fact resolves the rigidity of the square by transforming its static tension into a continuous flowing flow: there are no pauses, nor conflicts in its Aspects, whether positive or negative, nor on the other hand is there impeccable completeness and self-sufficient of the great Trine, in which each vertex communicates with the others.
Unlike the previous [families], here each Aspect only sees the consequent part of the whole, towards which to move or the antecedent, from which to draw movement, but never as an opponent.

A final extraordinary discovery of marine biology concerning starfish leads us back onto this path and reaffirms it.
Research published in Nature reveals that these marine organisms do not have a trunk and that the head is actually all over the body, articulated and distributed in the central area of each of five equal arms, eyes at the ends, which connect in a pentaradial symmetry unique in nature with a single mouth in the center.
It seems documented that this symmetry, which makes it "a head that crawls on the bottom of the sea" is the result of a specific evolution from bilateral.
In essence, 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 diametrically opposite vertex (potential complementarity, but on the basis of the contrast), in the odd figures each vertex reaches 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 1^{st}, 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?

Agreement [pentagonal] and disagreement [quadrilateral]
in orbital modularities for the Hamas-Israel conflict

An important notion that still escapes the majority of astrologers, despite having been published and made known by The Watch Publisher since its first Personalized Astrological Agenda (1992), then disclosed by AstroTime (2012) accompanied a particular historical prediction based on orbital relations fixed by modules of the pentagon.
When it came true a few days after being posted on August 13th together with other subsequent updates, given its special evidence I reported the details via email to some of my correspondents.
As with the other articles, the originals of which I duplicated on the server in a /pre_vers/ folder with certified timestamp to document the actual predictions,
his one is also still readable and verifiable here.
I did not imagine such an astonishing impact with the historical developments of November 2023, during which we witness an almost identical configuration of the orbital aspects, but regressive from the quintile range to that of squares with all their devastating potential.

A truly historic teaching from the dynamic-geometric condition, that I purposely present in this non-astrological research site, so once again en-passant, but which could well be a prelude to a targeted study on the problematic developments between these two populations and their related astral components over time.

Astro-seismology

(*) 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 seismic.astrotime.org, 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."
from 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 3^{rd} 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.
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 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 2^{nd} [dmt. Φ] (which is also the center of the 3^{rd} translated tangent to the lower point of the 1^{st} as well as to the base of the great triangle), and intersects the 2^{nd} 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 3^{rd} [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 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 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 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.