February 2024 -A proof that cannot be refused

I propose it a­gain, as an echo of my es­say, in a sym­bol­ic in­tro­duc­to­ry scheme, which seems use­ful to me to re­pres­ent the re­la­tion­ship be­tween square, nat­u­ral π and cir­cle, de­lib­er­ate­ly con­sid­er­ing the π as re­lat­ing to only one quar­ter at a time, i.e. 3,14…× ¼.
Based on a radius R we can construct both a 90° arc and a square with side R, which circumscribes it.

Every medal has its own reverse

If you can accept for a moment the value of π natural [3.1446055…] proposed by my research, you will find that not only a square

with perimeter:	3,14460055110296931442782342932803
whose side is:	0,7861513777574232860695585
has area Φ:	0,6180339887498948482045868

Of course, a square with the same area as the circle will have the side = ⁴√Φ

Focusing on the term "square root", one can easily deduce its double mean­ing, referring it to the figure of the square, the only emblematic one for this type of operation: 'root' indicates the side, 'square' indicates the motion im­plic­it in the relation between the side of the square and its surface, which multiplies the side by itself; but as in nature, it also in­di­cates its e­mer­gence.
From what we are observing in the two figures, two faces of a shared truth on the crest of the golden ratio, we could argue a first extraordinary statement for which:

A circle with the same area Φ as the square has radius = 0.4433258966 and circumference = 2.788170071589; but in particular, the area of the cir­cle with perimeter π i.e 0.7861513777574232860695585 divided by the ra­di­us of the circle with area Φ: 0.44332588966 is = √π.

april 2025 - Like in a mirror

Four years after the beginning of this effort, I am inspired by a renewed curiosity, which could better focus a definitive statement.

Before proceeding I must appeal to the patience of the reader, since this, and other pages of mine, are now a mixture of ‘before and after’ articles, not always as ordered as I would like. The fact is that new verifications flow continuously, faster than writing them down in texts and figures, and while at each step I should restructure a good part of the work, in fact I can only give precedence over everything else to their importance, contenting myself with inserting them where it is most pertinent.
Otherwise nothing of this extensive research, from the Squaring of the Circle to the Golden Spirals with reciprocal rebounds, returning to the Great Pyramid with unexpected annexes and connections, could have taken shape.
The green text areas here delimit the latest contributions.

What I have done above was started for the perimeter measurement of π, which can be considered a symbol of the circle. I thought it was a finished job, if only for its evidence, while reviewing it at a distance from my own surprise, I realize that I have neglected the other side of the coin, the one that sym­bol­izes the square. No sooner said than done: if 3.14 represents the circle, 4 re­presents the square.
And here, applying the value 4 to the circumference, yet another proof of unrepeatable correspondence; perhaps obvious at this point, but which forces us to reflect at the utmost.
In the face of said square,

In short, the square with side = 1 and perimeter = 4 has area = 1;
a circumference = 4 delimits an area = √ φ,
while the square circumscribed to it has area φ.
The ratio, as in the first case examined, here is √ φ / φ, that is, always 0,78615137775742328606955858584295 [√ Φ]; with the difference that if in the parametric case lower than unity the ratio examined was of square/circle of equal perimeter, (which is obviously also attested with 1 / √ φ – the monad of the π) now, under the impulse of a new push, we have added the circumscribed circle/square ratio, giving rise to an even more significant statement:

At perimeter = 4, the circle is root of the circumscribed square

A sort of eutectic point that reinforces the intrinsic relationship be­tween the essence of the two figures; almost magical, like the golden section point, below or above which nothing of such importance takes place.
Different cases distance the values of the areas in more or less ex­po­nen­tial progression, while maintaining the aforementioned ratio con­stant .
Key to the union between the circle, undifferentiated unit of perpetual motion without beginning or end, a concentrate to which all geometry refers, and the slowing down to the fixity of the expanding form, in the 4 fundamental directions.
Elementary, like every essential truth. From a linear point of view:
diameter × π ÷ diameter × 4  |  for the areas: radius2 × π ÷ diameter2,
lead to the universal constant 0.78615, for which I was pleased to pro­pose during an exchange of opinions at academia.edu the de­nom­i­na­tion Platinum and the symbol , which is √ Φ and ¼ of the π.
It is superfluous to complete the table by specifying that if the circle has area = 1, the area of the circumscribed square will be 1/ , i.e. √ φ.

	dividing the perimeter by our π, we will have
circle diameter:	4 / 3,1446055110296931442782343433718
that is yet the √ φ	= 1,2720196495140689642524224617375
and /2 gives radius:	0,63600982475703448212621123086875
Given my vision of the close relation between the quadrants of a circle and a square, I just have to multiply × 4, or × π, the area of the square having as its side the radius of the circle with the same perimeter 4. This modality would not be necessary, having the diameter as the virtual base of the square obtained with the circular sum of the 4 quarter-quadrants; which is nothing other than the square circumscribed to the circle. In any case, it follows the classic formula for the area of a circle:π × r2
for the square:	4 × 0,40450849718747371205114670859141
we have the total	= φ, to compare with the area of the circle
which is precisely:	0,4045084971874737120… × π = √ φ

In addition, I will now try to delve deeper into the unusual statement in ev­i­dence, perhaps the major achievement to which this procedure was in­tend­ed to lead.
I think I can maintain that this is precisely the most significant path to follow, to discover the profound square-circle communion.
A communion based without a doubt on the factor 4, which supports them both, providing with two axes (Cartesian, just to be clear) a quadruple com­mon hinge. This means that the transposition from one figure to an­oth­er must not be sought through contractions of axes, side or radius, as in the case of equal area, but for a conceptual projection that enhances its met­a­phys­i­cal character, or interrelation between concentration and dilation, which respects the basic cross by exploding the force of the circle up to the area delimited by the vertices of the square.
The potential of the circle as a dynamic subject is a promise of expansion, not contraction; the angular vertices of the four directions of the Elements, opposite and complementary, will provide containment.
Of the 4 phases that make up the circle, each quadrant arc shoots its arrow endowed with a precise and distinct valence, mathematical and physical.
If the fiction is clear and well-accepted, it is permissible to propose the circle as if it could virtually never resolve or dissolve into a square of equal extension, but only in a sort of balanced transposition, from North to South, from East to West.
This is what can be said to be the emblem of Squaring the Circle with a sense worthy of the Divine Proportion and belonging to it; any other perspective represents nothing but a geometric puzzle, until yesterday destined for dust.

work in progress …

Nonostante questo risvolto esoterico che ha radici in profondità (gioco di parole non casuale), nell'impulso collettivo ha sempre prevalso l'im­me­dia­to bisogno di aver ragione della sfida, di aree o perimetri uguali dall'una al­l'al­tra figura, complicando una mera speculazione, la meno adatta a con­dur­re da qualche parte.
La prima figura ripete quan­­to di più ov­vio e dif­fu­so nel­le im­ma­gi­ni di pre­sun­ta qua­dra­tu­ra del cer­chio, an­che se nes­su­na di esse, con il 3.1415, potrà cor­ri­spon­dere al vero.
Costituisce però uno stimolo vivo, credibile – poiché il cer­chio è vivo – e l'idea di poterlo tradurre in quadrato è affrontabile senza ulteriori valutazioni, se non l'ambizione della riuscita.
Nel caso in esame, al contrario, è come se l'arco BD dividendosi per scoc­cas­se da C la freccia del suo potenziale fino al 4° vertice E, che delinea l'area con la quale il set­to­re tracciato CBD sta in un rapporto regolato dalla costante, come 3.14460 ÷ 4.
Una favola difficile da formulare e rac­con­ta­re, an­cor più da ren­de­re ac­cet­ta­bi­le, ma che di­ven­ta mol­to cre­di­bi­le a con­ti fat­ti, an­che se il gli­fo che ho in­sce­na­to la ren­de co­me può. Ha il pre­gio però, di far me­di­ta­re un mag­gior si­gni­fi­ca­to del­la ma­e­sto­sa Rosa dei ven­ti, su cui non ci si era mai fer­ma­ti; ma dove la si­ntesi di cer­chio ­/ pun­ti car­di­na­li è di una e­vi­den­za innegabile (per noi an­go­li e lati del qua­dra­to, fuori dal cerchio).
L'immagine a lato dal museo galileiano pare precorrere il nostro assunto. Oggi, ove il raf­fron­to tra e­spres­sio­ni del­la na­tu­ra con ge­o­me­tria e nu­me­ri è più che mai in auge, me­ri­ta­no at­ten­zio­ne so­prat­tut­to le for­ze mag­giori.

Si rivela così il sublime momento matematico, di quel rapporto Assoluto in cui l'UNO si divide con la Φ per potersi produrre in una realtà sterminata di forme in equilibrio, che si mol­ti­pli­ca­no secondo φ, e si riflettono animate dal π che è all'origine di ogni mo­vi­men­to. Per approfondire questo condensato di im­ma­gi­ni dall'aria esplosiva del Big Bang, posso solo proporre l'articolo e­sclu­si­vo dedicato ad una con­ce­zio­ne ap­pro­fon­di­ta della Sezione Aurea. Tutto questo rende la Quadratura del Cerchio non solo la cosa più naturale del mondo – e così deve essere – ma un autentico ca­po­la­vo­ro del­l'In­tel­li­gen­za Creatrice, poiché definisce più che mai il rapporto intrinseco e de­ter­mi­nan­te tra il cer­chio ed il quadrato.
Infatti a parità di ampiezza (equivalendo il diametro al lato del quadrato) il paradigma soddisfa appieno il risvolto semantico già accennato in quanto l'area del cerchio elevata al quadrato diventa quella del quadrato cir­co­scrit­to, e questo sul valore assoluto della sezione aurea.
Va saputo riconoscere in quanto tale, ma è una cosa che potrà essere ap­prezzata sol­tan­to una volta compreso ed acquisito il legame tra π e Φ.
Tanto da poter sostituire con que­sta nuova prospettiva il concetto a­bi­tua­le di quadratura del cerchio, alla ricerca di una corrispondenza paritetica che non a­vreb­be condotto alla maggior comprensione; e lo sottolinea quella fu­or­vian­te esibizione ac­ca­de­mi­ca ormai storica di Lindemann, che pur facendo comodo a molti, non poteva colmare l'implicita lacuna di 0.00301.

• If the circle of circumference (and π) 3.14460..., which is perfectly re­flected in the square, is worth more, what the current advance of the π should have at least suggested.
• or you opt for a circle with circumference 3.1415… which does not cor­re­spond to the square, ignoring tout court the wonder exposed here?
  give up such a melody in the name of a makeshift construct?
  What will happen e.g. in calculating spatial trajectories with approximate or incorrect figures?

may 2025 - the Gap

Again as I see it, the π current after much pilgrimage in the most con­vo­lut­ed and illusory meanders, he has reached the walls of the longed-for goal, but he lacks the key to access, since the key is precisely the π.
Yet it is not a demerit, if we recognize that π represents a mathematically indefinable goal, if not an insurmountable bulwark for rational intelligence; and not for reasons of transcendence, since even the Φ it is, and yet there is nothing easier to demonstrate and reproduce.
Returning then to the topic of ruler and compass, if we associate algebra with the ruler, the compass is undoubtedly the π, and no ruler can replace it.

At the current state of progress, it deals with a gap of only 3.01 thou­sandths (0.1965 for the π by ¼),, as yet insurmountable in light of the traditional po­si­tions.
One may wonder whether there are too many to be acceptable, or whether there are too few to worry about.

Try to visualize a beautiful and simple soap bubble, splendid, de­fense­less and extremely rare expression of congenital equilibrium, which hov­ers in the air reflecting the environment and the colors of light on its brilliant liquid veil, kept in balance by the gravitational or elastic laws manifested by π native. Now suppose for just a moment that you can replace it with the cur­rent π precalculated; can't you imagine what would happen?
The bubble would explode into a myriad of micro-droplets, such as to sug­gest more than the 131k side-segments of the last simulation polygon inscribed in the circle.
Is it not likely that the molecular or even inter-atomic dynamic con­ti­nu­i­ty of the surface can never be supported by the many flat faces of the hypothetical polyhedron, whose only vertices would come into contact with it?

The academic world can only make use of indirect verifications, such as the pro­di­gious rendering just represented, which is of in­dis­pen­sa­ble and ir­re­place­a­ble evidence; and even before that, that of the Divine Golden Triangle, first revealed in 2002 with my treatise «The Secret of the 5 Tibetan Rites»" in its most evident properties, removed by a Wikipedia expert because it is inadequate, but only 12 years later understood and illustrated by me in its deepest sense, elevated and indisputable.\
However, it was not such a damage to affect 20 centuries of knowledge for just 11 years of delay...
In spite of the actor, in fact, such an exhibition of 'learned ignorance' could only exalt the concept of self-evidence, ie of those special first fruits as seals of the creative Truth, which Nature bestows on those who know how to read it, and who led me to the most decisive goal.
A bit like Isaac Newton's apple, where nothing could be seen.
Inscribed in the same circle of radius 0.5, it in fact bears at the apex of the shape, both Φ and φ, and the right π which it enunciates in its sides.

It may be objected that all this does not involve any demonstration, but how many other randomities do we know that can complement or dilute these, reducing their un­a­void­a­ble im­pact?

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.

Could we deny, if anything, such a complete correspondence between this triangle, the third treasure of geometry, and the circle with area defined from the π par excellence, the only one which can determine their golden pro­por­tion?

Alessi Antonio Nothing could have been condensed more concisely, completely and ex­clu­sive­ly than in this cosmic symbol.

work in progress
Past or future, the potency of 4 is in the π × every season

We know well that the definition of π it arose from the need to calculate the circle seen in its entirety; but if this does not prevent us from discovering that a fourth of it is equivalent not only to one of the aforementioned com­ponents of the unit, but also to a constant that regulates

Table for our three models, with π = 0,78615…

It can thus be seen how the true π carries out its balancing function not only in physics with the motion of particles and so on, but in the same way in the world of numbers, as a matrix of the golden section in the dual man­i­fes­ta­tion of increase and decrease, in a spatial and perhaps temporal sense.

In any case – recognized the Π fundamental – the Unit of Measure to refer to turns out to be a single Source, the key that encloses and originates both our 4π and Φ and φ: it is the figure
0.78615137775742328606955858584296 for which 5 decimals seem to be enough, if only because the following two di­min­ish and justify the ap­prox­i­ma­tion, in a research in Physics at its first steps.
As I proposed at the beginning, approaching the circle according to the rhythm of 4 quadrants denotes that balance is formed from contraposition.
By the way 3,14460 / 4 is exactly = 0.78615, which when squared re­spects the first 5 decimals of 0.61803, two more than those commonly applied.

Pi Day fans might be happy to realize that it's about tasting slice after slice of the pie, and not all the cake in one go ;o)

By the way, I cannot help but notice how curious it is that on the date I am writ­ing, the planetary and astrological con­fig­u­ra­tion is so similar to that of the day on which the Pi Day was born, as for the rare unilateral arrangement of the planets on only one half of the cir­cle. It lasts a long time and to make everyone happy I'm reproducing the one for next March 14th.

1. It uses the first three to designate the month and day of the anniversary, institutionalized 21 years later at an international level.
2. Similarly, I have adopted the following two figures to indicate the Hours and Minutes of the event, necessary for the purposes of the symbolic cal­cu­la­tion of the Card.
   If nothing else, it is the most coherent of the various numerical ma­nip­u­lations, proposed as folkloristic hypotheses to offer this pi day to the com­mu­nity.

Exploratorium museum of science – San Francisco CA

In any case, it is the extraordinary nature of the 3°14' Ascendant that in truth can only leave you speechless: 1/21.600 possibilities, for an as­cend­ant at 3°14' of Aquarius, the zodiac sign that characterizes the ev­o­lu­tion­ary era long awaited for the precession of the equinoxes; whereas the as­cend­ant in Sagittarius does not have any particular meaning in the theme.
Not even its creator could have imagined so much.

If one can give prominence to the fact that Albert Einstein was born on March 14 – a fact that few posts fail to mention – it is worth considering that it was 1÷365 possibilities, while the one we are seeing by sticking to the first three digits of the ascendant, is 1÷21.600, in addition to the correspondence of the precessional Sign; if we then evaluate how much easier it is to obtain 46", we are at 1÷1.296.000!
Should I perhaps ask myself why just last night I watched a cheerful film from 1937 (“TOPPER”), whose Italian title is translated “The Way of the Impossible”, culminating in a hotel room number 314; or whether the English name pier, as the residence of the new Museum has an affinity with pi and pie?
As I said, it doesn't prove anything and will never have an explanation.
It is even inconsistent, since the aforementioned Museum headquarters was relocated in 2013, the Pi day conceived in 1998 and declared official in 2009; but here a cocktail is served between heterogeneous ingredients, symbols focused on the numerical and historical value of the figure, combined as in an unpretentious abstract puzzle, stimulated by curiosity following the pi day, but with an amazing and unsurpassable result.
There is undoubtedly something for all tastes!
It happened and anyone can reproduce it with the proper parameters and app.

Perhaps mathematics is not just the calculation and regulation of numbers made up of all undifferentiated digits, but conceals a metaphysical paradigm of values and events whose intimate connection escapes flat rational logic.

Wishing you a happy Pi Day, with a fair pie forever!