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

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).

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.

Let's interpret its most salient qualities once again:

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.

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 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.

the immanence of the square

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.

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.

½ × (½ ÷ Φ)

½² × φ

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!

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.

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

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.

Studying this figure we can realize how certain orbital calculations, even in Astrology, can undergo differentiated deviations according to the quadrant and arc involved and the functions used.

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?

Or do we want to make it an Art Gallery?

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
   The figure demonstrates the cen­ter of cir­cle, tri­an­gle and square, with an il­lus­tra­tive meth­od that re­peats the tri­an­gle in two po­si­tions, green and pink, ro­tated by ±90°, with their two left and right sides perpendicular to the left and right sides of the Golden Triangle intersected at the midpoint. Also in this case you can note the great har­mo­ny where­by the green and pink sides and an ex­ten­sion of the same makes them converge at 4 of the intersection points of the circle with the square. Furthermore, having brought the lower vertices of the two on the base line of the Golden Triangle, the arcs with center at the two extremes of the base and radius its half, which is Φ of the oblique sides, reach ex­act­ly the ver­ti­ces of the two service triangles in less than 90°.
   It is worth summarizing the composition of this and any other scheme that con­cerns the pyr­a­mid, ac­cord­ing to the syn­the­sis path that starts right from the cen­ter.
   The quickest and most direct way to ob­tain the Triangle is clearly visible in the figure on the side.
   Once built vertically the golden rectangle that we all know, an arc having a larger side as its radius will reach the other side at a point [vertex] which, joined to the center of the arc, will form the side of the isosceles tri­an­gle sought.
   All you have to do is double the base of the rectangle in symmetry with the vertex and join its new end to this to outline the triangle.
   Having identified its circumcenter, we can draw the circle whose di­am­e­ter cut out from the triangle will be the [non-diagonal] symmetry axis of the square.
   Here is the notorious Squaring of the Circle, without tricks or cum­ber­some ex­pe­di­ents. It will be enough to ignore the false π, to discover which it has been so sought and denied, then look forward.

   It's just a matter of time now.
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.

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.

So let's see, flower to flower, how they fared