– you can download the PDF of this page, but it may not be up to date –
abstract
This page documents the integration of the Divine Proportion in the Great Pyramid of Giza, i.e. the absolute and complete synergy of π and φ until now only partially resolved by the mathematical and Egyptological tradition.
Finally exact precisely because it hinges on the authentic π . (download).

After having declaimed its absolute importance, as geometric and in­de­pend­ent of the monument that revealed it to me, precisely in order to high­light its extraordinary properties above any debate on architectural meas­ure­ments, and having illustrated step by step increasingly more exciting characteristics for the mathematical world – which is why the 4 concentric and tangent golden circles represent only the starting stage – at this point it was worth taking a step back and returning to the actual pyramid, since in turn there offers a three, perhaps four dimensions, no less important.

The most typical condition of doubt or dissent concerns just the squaring of the circle, and in parallel the definition of π, which seems to many almost pre­sent in the proportions of the monument, while leaving one's mouth dry like a mirage in who is thirsty to know and understand.
If on the one hand the squaring of the circle has been declared impossible, on the other the pyramid seems to offer us at least one proof, or an un­der­tone, so stimulating and compelling as to provoke the minds and am­bi­tions of schol­ars, irresistibly attracted by such a miracle, as if its evidence nec­es­sar­i­ly had to make sense; therefore be possible, not to say obvious.
It thus appears easier and more sensible to accept the challenge than to give up and be left with a bad taste in the mouth; even if the solution appears un­at­tain­a­ble.

This page unfolds in five arguments, three of which are primary, the fourth, unexpected and disconcerting, final and clear proof of the lack of precision of the official π finally, conclusive and synoptic for an overall acquisition of the philosophical and representative potential, and a historical­-con­cep­tu­al place­ment related in part on the next page.
They are:

transcendence of the circle
the deity of the triangle
the immanence of the square
redraft or defeat of π?
icing on the cake?
the pyramid, power of the five

April 2024 -from Divine Triangle to Great Pyramid

Thousands of pages describe and hypothesize on this topic, mostly re­trac­ing what others have deduced or written; many improvise their articles to spread, if not discover, what mysteries the Great Pyramid hides; and since its entire construct inevitably takes shape on the Web with ap­prox­i­mate and ex­per­i­men­tal arguments, everything becomes plausible...
However, since the numbers never add up (nor could they), several fas­ci­nat­ing hypotheses run aground on the 'very near' issue; and this is because we don't focus on the first and greatest of these secrets, which connects them all: it is the Divine Golden Triangle, which the more than monumental construction has bestowed upon us in the silence of millennia, brought back to view by me – since it has certainly not always been unknown – and proposed at 360° since 2002 as the 'third treasure of geometry'.
A geometric and mathematical expression of the Vital Power, which in the fol­low­ing years has strengthened in the eyes of my research to the point of assuming absolute validity, as it contains the major compositional forces of phe­nom­e­nal reality: the Golden Section and the π.
Of this figure, which I have already demonstrated to be unique and ab­so­lute­ly special due to its immediate characteristics, two other pre­rog­a­tives of fun­da­men­tal importance emerge precisely from traditions relating to the pyr­a­mid in defining the Squaring of the Circle, for which I have started this latest research of mine.
It is therefore through it that everything becomes harmonious, responsive and unequivocal. Let's retrace its most salient components.
transcendence of the circle

At the beginning the question one might ask is right behind the scenes: what does the circle have to do with the pyramid?
First of all, we need to focus our attention on what the circle is, something that we all know but on which we never dwell too much.
The circle is ONE, the principle of every expression, energetic, grav­i­ta­tion­al, undulatory. As a figure it reflects the one, being made up of a single line, which moreover has no beginning or end of its own, as its tracing can start from any point and, once complete, there remains nothing to distinguish in its perimeter, except by referring to external space topics.
The circle requires a center to be produced, a center which in turn dis­ap­pears since it has no geometric consistency in any dimension.

I introduced its graphic dynamics in my essay «2x2=3,14» pg. 5-6

Immediately following the circle, with 2 lines or sides we observe any angle, which in a certain sense modulates and relates to the circle every ge­o­met­ric figure with a greater number of sides. 2 is not a symbol of a whole, but indicates subdivision, as symbolized by the two sides that depart from the vertex of the angle.
The most representative ideal angle could be that of 38°10 which, touching the inside of circle 1, due to the symmetrical property of the 2 sides that meet it, generates the perfect 3 that we will see.
I have already mentioned the properties of the subsequent fundamental figures with 3, 4 and 5 sides on the page Square-From-Circle.html; each of them has a beginning and end according to a rational definition, which al­lows calculation, modification and comparison with others, where the circle is pure transcendence and containment.
It is also interesting to note how the circumference can act as an in­stru­men­tal unit of measurement and tracing alongside the single straight line, re­porting and establishing distances using a ruler and compass.
Here we are interested in the pure transcendence of the circle, as – ac­cord­ing to an archaic wisdom that has been lost, but whose echo has for­tu­nate­ly not died out – it can descend to the immanent plane.
Evidently such a very high level process had to and could be expressed through a geometric structure, and therefore implemented for its own pre­rog­a­tives, condensing its presuppositions in the correlation of flat and solid fig­ures with perfect symbolic correspondence to the ritual contents.
This long speech crystallizes in the expression ‘Squaring the Circle’, which from what I will explain could actually be the title to be attributed to the Great Pyramid.

the deity of the triangle
A marvelous figure that I could have discovered only by insistently re­-ex­am­in­ing the Great Pyramid, trusting in the prospect that most likely re­pro­duced its profile; even if the conditions in which 4 thousand years of history and geological transformations leave it offer us asymmetrical sides and base in terms of measurements and inclinations.
Even though they are fundamental for the study of the construction and the mysteries it hides within it, from the first graphic findings that guided me to the heart of its ideal canon – which shines through and reaffirms itself more and more as its development advances – I preferred to follow a logic that started from the inside rather than the outside of its real measurement.
A measure which for the writer is synonymous with essence, entity, vi­bra­tion and symbol, even if the etymology adapts to both sides of the research.

According to the surveys obtained from the most accurate CAD outlines of the Upuaut Project by Rudolf Gantenbrink, who managed to instrumentally ex­plore the pyramid inside and out, thanks to advanced robotic devices, I ex­trap­o­lat­ed the triangular profile from some models available at the time, trying to optimize it for my first attempts to integrate it into the creation of a Śrī Chakra yantra (which only years later I would have programmed, for the first his­tor­i­cal time complete and free from errors, dedicating more than one web domain to it for pure study).

I report some screenshots of CAD taken from the web in 2002, which today seem unreachable from the project sites themselves.
The first image of still limited size can be enlarged off the page in PDF vector format, to verify in detail the correspondence of the red hatching of the mag­i­cal profile, superimposed with precision.
The second view also in vector format, at that time practically cut by the screen, being almost quadruple in size given a starting graphic format: width="4238", height="3238" points, to allow a numerical verification I had to complete it with the red path, which highlights even more near the ver­ti­ces the pre­ci­sion of the triangular section.

triangle in red —: 1481,162 mm |: 942,025 mm, ratio ÷ 0,786158
dashed triangle —: 370,853 mm |: 235,866 mm, ratio ÷ 0,786151

Compatibly with the approximations forced by the graphic tools used, with pas­sages from one to the other starting from lengths to three decimals, they give impeccable testimony of the relationships tending to Φ with a cor­re­spond­ence up to the 5th decimal of the golden ratio ½base÷height of 0.78615, more than sufficient to indicate the underlying presence of the Great Golden Triangle, which we will see below in a unitary way, and com­plete­ly surpass improvised measurements, or those documented by wide­spread drawings for schematic purposes.

I soon became aware, let's say with insight, of the first truly unusual, in­deed absolutely unique, potentialities of the large triangle, which lit up be­fore my eyes, then and there astonished, like four spotlights with a light of increasing intensity.
An extraordinary choreography that could not fail to surprise me, maybe witness to a hidden memory.
They were the first messengers of the total Divine Proportion of that su­preme triangle, always ready to give much more than appeared at first sight.
I was immediately certain that unlimited wisdom took shape in this figure; but at the moment my commitment was oriented towards the '5 Rites' and the 4 Elements that they evoked, so much so as to make the Great Pyramid symbol of it, yet I overlooked the Squaring of the Circle of which only the myth echoed, without realizing that I was outlining it in the entire work in progress.
 the Third Treasure of Geometry

Let's interpret its most salient qualities once again:

Inscribed in a circle which is the ONE mentioned at the beginning, with diameter =1 and circumference =π, therefore labeled Φ0, we can immediately note that the 2nd concentric circle with diameter scaled to Φ, is tangent to the two symmetrical sides of the Triangle. The 3rd circle of diameter Φ2 follows, in apparent suspension, but which, brought down into the area below in tangency to the external circle, is in turn tangential to the base of the Triangle, stating its distance from the circle, as well as to the 4th circle of diameter Φ3, since the latter is also tangent to the base of the internal side Triangle, stating its distance from the center.
There are many things to read in this diagram, of such apparent simplicity.
The height of the triangle is Φ, which is equivalent to the radius 0.5 + Φ2÷2; you can see it by moving the 2nd circle upwards, tangential to the external circle, with an effect on the base that mirrors that of the 3rd circle below. Incomparable vibrational synergy.

This golden concatenation in itself already led to further developments, mentioned from my first approach, in the development of an essay that was not intended for this, but whose overall contents would take me far in the following 22 years.
I had to fix and make known these stimulating features, obviously rejected by Wikipedia, and I did it at Scribd.com, and then Academia.edu; but I realized that although I found them hypnotic, they had very little effect on those who learned about them, not knowing what to do with them or what they could get from them.
Personally, I immediately applied them to a new concept of the golden spiral (which years later I would developed extensively not without unpublished discoveries), and above all to the expansion plan of the Śrī Chakra yantra , which I was finally able to propose in a complete, exact and non-improvised guise on the external bands of the 9 central triangles, revealing themselves to be the same intersections of these marvelously related to the same waves of increasing golden amplitude towards the final perimeter of the 4 doors.

Let's first point out what else this figure has in store for us, which I finally dared – or I was taken to – define DIVINE.
This first Great Triangle is the maximum, direct and essential expression of the total Divine Proportion: as it presents a base on which the two sides constitute an agreement, related to the base in full and exclusive ratio φ.

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.

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 Primal Oneness, 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 new study on the π and Φ ratio 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 Φ2 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.

It must be understood how this Principle integrates into the Great Pyramid, manifesting its generative & regenerative, physical and ultraphysical Power.
Therefore, the many approximations or intuitions that have run aground due to presumed, even small, lacks of precision, while in reality they could have been completely successful by applying the right parameters, matter little.

In reality the general error is upstream, and I really hope to have high­light­ed it in my introductory treatise, followed by far too many indirect evidences.
Nonetheless, this document will provide a direct and definitive one.

It is the calculation of π considered definitive at present, which is lacking, con­sti­tut­ing an insurmountable barricade to any step forward
A minimal flaw, but enough to make one reject it, or give up any aim for accuracy so as not to contradict its academic definition; even if in this case it is precisely the one that has gone overboard, with its claim to assert itself precisely where it feels least safe.
Many doors and adjustments will open, in addition to those presented here, once the π 3.14460 will be officially recognized.

When it comes to ascertaining dimensions such as those of the monument, worn down over millennia and eroded by natural agents and man, and the measurements are obviously inconsistent, depending on the methods and instruments used, it is very difficult to raise certainties that do not boil down to just opinions.
Even more so since the base of the pyramid, in addition to presenting different measurements of each side compared to the others over time, due to possible disproportions caused by the soil and/or the climate, is not a regular square, but each of its sides flexes the entire wall in two halves as door leaves inwards for approximately 27', which makes the base an octagonal star figure, in­tro­duc­ing a further doubt on the way of verifying certain occult proportions inherent in an architecture in which nothing has the ap­pear­ance of be occasional.
Furthermore, the current appearance of the pyramid is stripped of its precious covering, which should absolutely be taken into account by anyone who intends to investigate the occult or initiatory potential of the unlikely tomb.

the immanence of the square
To continue relating to the Φ ratio for the purposes of a direct and immediate demonstration, I will now not attribute a unit value to the circle in which the triangle is inscribed, but to the one traced with the center at its vertex, and radius equal to its height, which will therefore be = 0.5.
From the 1 sublimated in the Circle, which is for the created = Φ × φ, we have seen the primigenial Unity to split admirably into the Su­preme Triangle, through which we witness its manifestation in the dense world, configuring itself in the square of its roots, or the four sides of 7.8615, which make a perimeter of 3.1446, equal to the cir­cum­fer­ence of diameter 1.

Since Φ it is the golden number, I proposed right during an exchange of ideas with the astrophysicist Richard Ravenhall at academia.edu, to call 7.8615 the 'platinum number'. As if on purpose, the atomic number of platinum is 78! and so here I propose it again publicly, and here I will attribute the letter – taken following the initial P which in more than one way already belongs to π – suitable for remembering the relationship between half the base and the height of a triangle.
By the way, the monogram of the letters P and L is a symbol in astrology and astronomy of the planet Pluto, whose orbit surrounds the entire solar system.

I then drew a triangle with the circle tangent to its base, also reproducing the square base, both with the center at the vertex of the triangle.
In this way we will observe the ratio between the circumference and the square of the base, kept constant, varying the height of the triangle as nec­es­sary for a comparison between the two versions of π: 3.14159 or 3.14460.

To keep the new measurements re­fer­ring to a cir­cle with di­am­e­ter =1, it can be not­ed that in the pre­vi­ous fig­ure the cir­cle with the cen­ter at V (ver­tex) would pass through the cen­ter C, as a tan­gent to the red dot­ted di­am­e­ter of the in­i­tial circle.
The points where this intersects the sides of the golden triangle will de­lim­it the base of the new triangle, scaled from a height Φ to 0.5, from which derives a base of length , which corresponds to that of the oblique side of the larger triangle.
Now, from the three fundamental arguments: base, side and height of this triangle we can deduce two other quadrangular figures of specific unique­ness and correspondence if associated:

  • a rectangle having as its base ½ the base of the triangle and its oblique side for height (easily portable following the arc of the protractor);
  • then a square with a side equal to the height of the triangle.
It is quickly observed that while the area of the square, referred to the circle of diameter =1 is 0.52 = 0.25, the area of the rectangle, whose base is also the golden section of the height, is given by ½ × (½ ÷ Φ), in numbers:
0,39307568887871164303477928666004
× 0,63600982475703448212621124099997 = 0.25, as well as
from the base square of the golden rectangle to its total area: ½2 × φ:
0.15450849718747371205114670366897
× 1.61803398874989484820458681467588 = 0.25.
That said, the best is yet to appear, since of the two equal areas, that of the rectangle can be easily divided by a diagonal that divides it into two right tri­an­gles, each of which corresponds to half of each face of the Great Pyr­a­mid.
In fact, placed side by side on the base along the shorter side (triangle without profile, purple background), their repeated rotation of 90° on the axis and the inclination of 51°49'38 brings them together from each side of the base square to the vertex of the triangle which becomes pyramidal!
So here we have reached the solid state; all that remains is to verify what has mysteriously been handed down over the millennia from known and unknown sources.
The scenario is soon defined: the perimeter of the square with side is 0.78615 ×4 = 3.14460. If the circle with diameter =1 has the same length, i.e. trivially 2 ×0.5 ×π = 3.14460, then π is 3.14460!

In short, if inscribed in the circle with radius Φ, the side of the great golden triangle is = ¼ of the circumference, compared to a circle with radius its height, is its base which corresponds to ¼ of the circumference!

Given the potential of the great triangle, the force of this axiom is too sig­nif­i­cant to be ignored. The busillis is only that If..., before which we could also stop, for the sake of many academics; but if we have reached this far, it being understood that the π it is elusive from whichever side you approach it, it is worth carrying out further verification of the facts.
To this end I have implemented the opposite path: to make it appear in the same figure of indisputable precision, also the variants that derive from ap­ply­ing to the π the current value 3.14159.

Please keep in mind that I have purposely used very thin lines in the PDF di­a­gram which reproduces both (you can open or download it by clicking on the figure), in order to allow the vision of the most accurate details and the right proportions through adequate zoom on a good reader, for dimensions that the browser may not be able to render sufficiently.

Always tangent to the unchanged base of the triangle, a new circle must be added to the first one, already with perimeter 3.14460 and radius 0.5, whose radius, in order to obtain from π 3.14159 the perimeter 3.14460 will become 3.14460 ÷ 3.14159 ÷ 2 = 0.5004790.
573 / 5.000 This radius, i.e. the new height of the triangle, will give rise to a square with an area 0.250479q greater than 0.25q, whereas the rectangle flanked by the new length of the oblique side will no longer be golden, nor will its area ever be equivalent to that of the square: even without pro­duc­ing cal­cu­la­tions, it will be a good exercise to imagine the dis­pro­portions of a rec­tan­gle with a fixed base, with a square of multiplied height, for an area always larger, or vice versa of minimum height, until it is reduced to a small fraction of the hypothetical rectangle.
Details of the enlarged figure, at the vertex we can observe the slight­est dif­fer­ence in the tracing of the triangle with the height of the se­cond hy­poth­e­sis; at the top the almost symbolic difference between the two circles, which if they were not tangent to the base, would even be halved from view.
All this can prove very useful in helping us evaluate how the diagrams pre­sented and proposed since past centuries are responsive on one side, but mis­lead­ing on the other. A few thousandths are enough to disable a theory, while now untraceable data are needed to support it, except... the dominant ab­strac­tion of our ‘Great Triangle’.
We are faced with a scheme that should inspire on its own, and in truth cannot do more: a single height of the pyramid can make the circumference with e­qual radius equivalent to its base perimeter, with a square of this side to its tri­an­gu­lar faces, and a single π can allow it!
Therefore you just have to choose which side to take, whether to adopt a perfect math­e­mat­i­cal and geometric manifestation, adapting to what proves to be ex­traor­di­nary and immeasurable, or to submit and condition the fu­ture of re­search to an ambiguous constant that is never defined except by ap­prox­i­mate pro­gres­sions, and which precludes any possible advancement.
One is a living letter, and will remain so forever; the other is only an ar­ti­fice, useful up to a certain point, but constituted by a misleading tran­scend­ence.
As I have already argued, the circle itself is the transcendent being.
redraft or defeat of π?
Precisely in terms of the reverse path, a detail comes in handy that could help to better focus on certain implications in the choice of π.
It deals with a numerical event that left me literally speechless for at least a day and a night... for the absurdity it represented.
In the new figure we immediately notice that the two intersections of the cir­cum­fer­ence with the base of the square take place with remarkable pre­ci­sion on the two sides of the triangle, as if to establish the absoluteness of the nu­mer­i­cal and geometric relationship.

It must be said that the most widespread drawings for reproducing this re­la­tions do not highlight this crucial meeting point for the three figures, since they do not realize that the circle belongs, due to its symbolic, celestial value, to the top of the pyramid, the vertex, not the earthly center of the base; a detail that gives the figure what we could consider a notable added value.

Except at an appropriate magnification, the intersection appears with im­pres­sive accuracy exactly where it shouldn't, since it is the external circle, the red one resulting from the artifice of π 3.14159, which intersects the crossing of the base of the original square (pale green, horizontal) with the profile of the artificial triangle of greater height (in grey).

The base of the square centered on the new height is far from these two screens, as their mag­ni­fi­ca­tion is 6400% of a 2.52 me­ter base di­a­gram! but it is clear­ly vis­i­ble dot­ted in red in the PDF [click], which can ren­der quite thin lines for a thor­ough ex­am­i­na­tion.
All of this is nonsense; nonetheless, after such a display of perfect geometric and mathematical combinations, uniqueness uniqueness uniqueness uniqueness uniqueness high­light­ed in this long and careful work, it is really difficult to accept a concept of mere randomness in such an example.
Anyone who has ever attempted to draw a Śrī yantra knows well what I'm talking about.
Perhaps the π classic claimed its validity? unthinkable with promiscuous fig­ures, no train can run on the tracks of two different railways.
Having rechecked my error-free code, all I had to do was rely on such pre­ci­sion for the statement of an error in the basic software..
A pressing need to get out of this trap reminded me of a relevant gap, pre­cise­ly in the definition of a circle during my initial study on this site (link above), of the relationships between circle, square and Φ.
The problem was connected to the use of geometric path PS operators such as arc and arcn responsible for tracing a circle with a code of the type: x y ang1 ang2 arc, placed in the case with: x y 0 360 arc stroke.
I at once replaced it in the drawing with direct personal instructions, reworking it according to the example:
/circle {/radius exch def gsave translate	
radius 0 moveto
0 .05 360 {dup cos radius mul exch sin radius mul lineto} for 
stroke grestore} def
0 pyramidHeight dup circle
It shows the precariousness of π 3.14159
for purely geometric view
and here the bor­der is re­de­fined, in­cred­i­bly per­fect like the first, but now it is the 1st gold­en cir­cle that in­ter­sects the meet­ing of the base of the square with the pro­file of the o­rig­i­nal tri­an­gle, even if this does not have an im­me­di­ate val­ue in the dem­on­stra­tions that in­ter­est us.
Since any an­gu­lar op­er­a­tion is set to π, how I sup­pose the de­fault 'arc' op­er­a­tor is han­dled at a low lev­el, like­ly Bézier curves, a ra­tio be­tween two legs and a hy­pot­e­nuse wins out (al­though greedy for el­e­men­ta­ry in­struc­tions).
By alternating the two figures by touching the image, you can see how this shift depends on the correct tracing of the two circumferences, which in the first phase are more com­pressed and clos­er to each oth­er; this is the rea­son why the view ap­pears shift­ed, even though it was ob­tained by keep­ing the zoom win­dow fixed.
With the same mag­ni­fi­ca­tion in the PDF I made vis­i­ble on the right side at 3 o'clock, a­bove and be­low the ver­tex of the pur­ple square, the 0.5° step be­tween the start and end of the cir­cles drawn by my code above.
Almost nothing that can be seen with the naked eye under normal con­di­tions, but only with this enormous magnification; and I wouldn't rule out that the cause lies precisely in the use of π 3.14159. It may help to understand that the PDF file in which a single circle is obtained with the arc operator is 28Kb, com­pared to 145Kb in the accurate file; but the problem π remains at the forefront.
For this reason I entrusted the above complete SVG image to the lighter stan­dard method (126Kb versus 265Kb), given that its illustrative purpose requires clearly visible thicker lines, for which a comparison, although pos­si­ble, would still be less than satisfactory.

Even if the definition of each angle in the above algorithm can suffer mi­cro­met­ric deviations by applying 3.14159 or 3.14460, according to Pythagoras Sine and Cosine the x, y coordinates of a Cartesian axis system must deal with a constant Radius, at which the instruction 'arc' does not adjust, squeez­ing thus every quarter of the circumference.
Considering the accuracy and structural efficiency of the magnificent Post­Script language, it now comes clear to me that what I had interpreted as 'a basic software error' is instead the glaring verification of the error pro­duced by the use of π 3.14159, even in the probable use of Bézier curves (of which in fact the approximation is taken for granted, although perhaps theoretically it shouldn't). While aware that the PostScript language is not a CAD but was created to satisfy the typographic world – the reason I document it is the scope of this particular condition; which is why I could only re-produce it by isolating it in a code fragment suitable for a simple, direct and exclusive comparison, free from any possible interference.

In the same work area of 7.143 square points, I entrusted two separate rou­tines with the task of tracing two circles, one in red with the standard arc meth­od (for the whole circle), the other in green using the already men­tioned algorithm, but in this case concentric, to facilitate the comparison, since in the diagram referring to the triangle the two circles rest on the same base and this can be confusing.
I saved the code to the PS_circle-Test.ps, distilling which you obtain the PS_circle-Test.pdf, which I still inserted into the PDF of the general model.

Scrolling through the figure with a 6400% zoom or by means of progressive enlargements on the interested areas, you will easily notice two compound effects:

  1. a normal correspondence by superposition of the two curves at the car­di­nal angles: top, bottom, left, right.
  2. a progressive distancing of the same which, according to the tangential thrust exemplified in the specialized site above, tends to flatten the cur­va­ture of an arch outwards near the ends, and then compress it on the contrary in the central area, as if to compensate the subtraction of cur­va­ture imposed on the circle by the polygonal methods of calculating π, invisibly redistributing its effect in each of the 4 quadrants.
I could reconfirm the process by converting the PDF to SVG (too big for a browser), whose trans­par­ent format allows you to read the executive code.
Apparently an algorithm not based on individual degrees, as I did, but on the global length of the circumference, to distribute it more quickly along arcs defined by fewer instructions, being based on a π which reduces it by 6.2 thousandths, will never be able to compress it in the axial angles, in which the ray has greater incidence for x or y, but will com­pro­mise the ra­di­us in the intermediate areas.
If it is easy to trust in the absoluteness of the calculations, it is another thing to see them projected into real and effective space; but who will dare to blame the π?
Yet, a more explicit demonstration could not have been had! that's enough to derail an aerospace project.

I even verified that the deviation is different from one dial to another (prob­a­bly due to the play of ±), but I wouldn't want to make it more cum­ber­some than necessary.

All this stresses that whatever the drawing method used – (Bézier curves or NURBS: Non-Uniform Rational B-Splines, or other) – we can only deal with this ge­o­met­ric complex through theory.

In any case, the extremes are clear for anyone who wishes to try their hand at this puzzle, for which an explanation remains missing that manages to com­bine the second triangle and its squared circle with an unchanged base, but centered on the vertex of the first.
In fact, I wonder if we can exclude a reason why this dual effect manifests itself with such precision exactly at the meeting point of triangle, square and circle, underlined precisely by the angular participation of the internal [green] square, as well as where the gap between the two curves is de­ter­mined by the ratio 3.14159÷3.14460.

  icing on the cake?
The game is not over yet, since at­tracted by this stim­u­lat­ing com­bi­na­tion I want­ed to in­ves­ti­gate oth­er re­la­tion­ships that this scheme could re­serve; so I shift­ed my at­ten­tion to the 45° an­gu­lar side view of the py­ram­i­dal struc­ture.
It was e­nough to su­per­im­pose a pro­file with a base the di­ag­o­nal of the square and a height that of the pyr­a­mid, a com­mon ver­tex to both a hor­i­zon­tal and ver­ti­cal views.
Each diagonal of the square centered in this way intersects the circle with two opposite edges of the pyramid, say seen from above.
In the new lateral view, however, the two opposite edges of the transversal profile cross the square at two points, joining which the base of a new tri­an­gle is delimited, proportional to the profile of which it is part, but scaled to the width of the square.
Well this base crosses the intersection of the diagonal of the square (edge of the pyramid) with the circle equivalent to its perimeter, with renewed ac­cu­ra­cy. Another victory of perfection, it seems to me like the icing on the cake (or should I say ‘pie’?), however little it may matter.
With or without that icing the cake remains cake, a cherry can add little, but put in the right place it can make the cake unique.

Or do we want to make it an Art Gallery?
It is the turn of the circle circumscribed to the Great Triangle, which I traced in order to examine what the inverted triangle itself, i.e. vertically mirrored, reserves within it; and even in this case it did not disappoint me.
The symbol of the two opposing equilateral triangles is certainly not new, but it has never been applied to this formation, at least four times special:
  1. first of all because the sum of the four sides of the two triangles, excluding bases, is equivalent to the perimeter of the cir­cum­scribed circle, therefore they are vir­tu­al­ly the four sides of the square which is e­quiv­a­lent to that cir­cle, even if dif­fer­ent­ly op­posed.
  2. Then because the segment that joins their me­di­an in­ter­sec­tion points is e­quiv­a­lent to each of the four sides and is π÷4, I mean 0.78615 or on the di­am­e­ter =1 of the cir­cum­scribed circle.
    This is quickly demonstrated by the internal ar­chi­tec­ture of the Great Tri­an­gle in­scribed in that ref­er­ence cir­cle. As can be seen by re­trac­ing the em­blem­at­ic fig­ure, its ½ base AB is the gold­en sec­tion of the side AV, a ra­tio com­men­su­rate with the height BV that is pre­cise­ly Φ, which breaks down as we have seen into 0.5 from the center of the circle to the vertex, and 0.118 from the center to the base of the triangle; therefore 0.618.
    Now, if the ½ base AB is slid upwards, parallel to itself, keeping the end point B along the height BV, its length will be cut at the intersection with the oblique side, according to the ratio established by the height achieved. Therefore, if the height-distance from the vertex from 0.618 has been reduced to 0.5 having reached point C (center of the circle), its proportion with the oblique side will now no longer be Φ, but 50%.
    At this point, since it is the center of the circle, doubled horizontally it will constitute the axis that unites the two points of intersection of the four sides of the two triangles, a precious reference module for everything that can derive to geometry from the management of the golden section; something I had already introduced in this curious article: “Genesis of the π, as seen by Astro­Time”.

    That means even more simply that the Gold­en Tri­an­gle con­tains the ‘Plat­i­num’ con­stant = 0.78615 also in the seg­ment par­al­lel to the base and pass­ing through its cen­ter which is the same as the cir­cum­scribed cir­cle, ex­tend­ed to meet the two sides.
    The di­rect squar­ing of this cir­cle de­rives from this. LEGENDA
  3. More, because while the sides of the tri­an­gle (this ap­pli­es to each of the two) are tan­gent (pur­ple hatch­ing) to the cir­cle with the cen­ter at the ver­tex of the op­po­site tri­an­gle, in­tend­ed for quad­ra­ture, this oc­curs ex­act­ly at the point of in­ter­sec­tion of this cir­cle with the square of the base cen­tered on it, as well as with the circle hosting the triangle, at the top of the base of the inverted triangle partially visible in the figure; an in­cred­i­ble cor­re­spond­ence of 4 lines, two of which are circles!! (only in the PDF page 3 can you clearly watch the discrepancy of the cir­cum­fer­ence in red, due here too to the incorrect π…)
    From another perspective, the two circles, the container of the triangle and the one to be squared on its base, meet at two points that exactly intersect the square of the base, wherever it is positioned.
    The distance between these two points is equivalent to the base length of the triangle, and being equidistant from its center it also turns out to be the base of the same inverted triangle.
    The large circle with its center at the vertex and tangent to the base of a tri­an­gle is tangent to the two sides of the opposite triangle.
  4. What remains is the base of the inverted triangle, which in addition to pointing like a laser at the points of tangency of the sides, in turn is tan­gent to a reduced circle in Φ scale. from the opposite one of quadrature.
Nothing that is not connected. Mirabilia.
Will they all be cases? or indices of a grandiose harmony that distinguishes at each passage what is pure harmony from improvisation?
Who could conceive a plan so proportionate in its most peripheral details?
Mathematicians should rejoice, for this is pure poetry, in their domain!
I've inserted the graphic in the already prepared PDF.


the pyramid, power of the five
The Great Pyramid of Giza is not a simple tomb, but a device designed and positioned to condense and convey vital and regenerating energetic vi­bra­tions on a planetary scale.
The construction of the only one of seven wonders left in the world, there­fore incorporates the surprise that allows access to its metaphysical ar­ca­na: it is the Divine Triangle that emerges resplendent from its building as the key to its and many others mysteries that are not only mathematical, by­pass­es any attempt at reconstruction or theoretical corruption, since it encloses the exact answer to any type of approach, unifying the principles of π and Φ which are essentially the same thing, even if this is still ignored, since the equality × 4 = π and 2 = Φ is ignored. Any effort to understand the Great Pyramid without taking its incidence into account can never be successful.
We have seen so far how the most striking secrets of the pyramid arise from this perfect figure, which represents its ideal profile, like a physical and ultr­a­phys­i­cal blueprint.

Other mysteries are woven around these basic qualities, both within it and from it towards sidereal spaces, even if they are not the subject of this study.

In the present work I do not propose archaeological tasks, nor as an E­gyp­tol­o­gist, nor do I care to delve into the numerological apparatus deriving from the ap­pli­ca­tion of specific units of measurement, without taking anything away from their merit or their sacredness though.
I am only concerned to assert the closest and most direct link between certain ge­o­met­ric factors and the current sacred numbers such as π and Φ, against a confined know­ledge that underestimates their essential relations.
Therefore I will only use measurements in meters, degrees and decimals with the necessary min­i­mum approximation; and above all through them, and thanks to the Divine Tri­an­gle which is the Driving Source, bring to the right so­lu­tion some of the most dis­put­ed and unresolved statements.
On the basis of this solution, it will perhaps be easier and more truthful to cor­re­spond to certain geographical and astronomical, planetary and lunar re­la­tion­ships which do not interest this office, but which endlessly intrigue hordes of researchers.
Probably, just for the record, the same could help for certain contents of this first scheme, which I developed and made public in 2002, from which I may have inspired more than one theoretical-editorial application in the following years, which arose without naturally citing any source.
To avoid erroneous interpretations in hypothesizing the relationship between the Zed, the spinal column of the pyramid, and the human figure, I only have to point out that it is not the pyramid built to the measure of man, if anything, it's the opposite: the laws that transpire from the monument are prior to the cre­a­tion of the human being, as well as the egg [cosmic, like it or not],was born before the chicken.

The reason for such a unique and irreplaceable architecture, as well as its geographical and geophysical location, did not aim to demonstrate advanced mathematical knowledge to future generations (which they probably would not have understood; after all, that concept of the past and future in which the past is a symbol of beginnings - have you ever asked yourself the ques­tion if anyone could ever consider us to be in the first steps of de­vel­op­ment? ), but rather to build a cosmic instrument that would integrate the main lines of force of the universe for an energetic, or historical and ev­o­lu­tion­ary, if not also tech­no­logical purpose of the material world; and we have not yet seen the con­clu­sion.
What denotes the astrophysical location of the great pyramid being aimed at expanding its functional value beyond the usual earthly boundaries, or tomb boundaries like many others, is precisely that numerical synergy be­tween all its dimensions that does not represent only numbers, but val­ues and prop­er­ties that recall other values, according to paths that con­sti­tute a real challenge for our unprepared reasoning
Essentially responding to the functionality of the π in its occult structure, which I attempted to mention in the essay under the section: § - “The π: 4th di­men­sion & gravitational interaction”, they are like an energetic alphabet com­mon to the entire universe, a metaphysical connection of essential ef­fi­cien­cy in the in­tel­li­gence of creation.
Only a targeted device could convey the necessary wisdom; and this is cer­tain­ly the primary reason why it captivates every kind of scholar, even the most sceptical; because even scientists have a soul.
If for today's researchers hidden chambers and secrets kept inside the pyr­a­mid are the most intense challenge, the mathematical aspect is no less; indeed its importance can bring more benefits than the discovery of rules and truths that could remain incomprehensible, except to those who are destined to receive them as events mature.
That of a science that has been lost for more than four millennia, or why not? more recently with the de­struc­tion of the Library of Alexandria in Egypt, together with the casing of the pyramid – or perhaps I rediscovered it firsthand, through memories that bypass various reincarnations of mine.
In any case, my re­e­val­u­a­tion of the π it is a theory that reserves only ex­act­i­tudes and no con­tra­in­di­ca­tions, and yet ready to assert itself on the basis of a fundamental state­ment:
If the height H of a pyramid is such that the area of each face is = H2,
the perimeter of the base will be = 2×H×π.

The 1st assumption makes the ratio between ½base and the height of the faces to be Φ.
The 2nd assumption makes the ratio between ½base and the height of the pyr­a­mid to be Φ.
Of all this, the π would be the only unknown, which has kept generations of scholars in check, but which in the current time, I repeat, is perhaps the most important discovery after that of fire.
Nonetheless, even without taking into account the π, with the changing his­tor­i­cal and geological conditions naturally none of the architectural con­fig­u­ra­tions proposed by various authors and researchers over the decades cor­re­sponds with indisputable precision to the triangular profile already described.
Also because being known just for its local traits, more than for its global mean­ing, no one tries to emulate it by adapting its measurements to that pat­tern. Indeed, we have differentiated findings of the sides, angles and in­cli­na­tions from each other and we cannot glimpse any for­mu­la­tion that can unify them, other than the mathematical average, and not only of the basic pe­rim­e­ter, but even between the measures proposed by the various authors; average and perfection are two entirely different things.
So let's see, flower to flower, how they fared.



An­to­nio A­lessi © The Watch Pu­blisher, 2003-24

[ back || or to the main page ]