“Smaragdina Hermetis Tabula” detail - by Johannes Petreius ed. “De alchemia” 1541
As with the Golden Section,
to 'draw out' the π, it was
necessary to compare it with the ONE.
preface
After having written and published the treatise «Pi-drawer» at ZENODO(CERN infrastructure), and having communicated it on March 8, 2026 to the media@ of the Exploratorium Museum as well as to the Official website of the Pi Day organised by the Ministry of Education and Merit, at the University of Turin (Italy), while waiting for the mathematical community to surrender to the evidence, my refined maturation on the topic invites me, rather than translating locally the treatise at first impact intended for researchers, to make intelligible from the middle school level the simple path necessary and sufficient to be able to define and become aware of the actual π.
Perhaps too simple for scholars, but it has held all the greatest minds in check for millennia, forcing them to fall back on approximate simulations, and since 2019 to celebrate them.
So at least the youngest, as they grow up, will not be confused by the insidious webs of knowledge.
prepare your mind
To be able to delve into a true understanding of the problem, it is first necessary to free one's mind from the conditioning due to the main connotations of circle and π, sedimented over the centuries and by computers, which only distance one from a correct interpretation of both, if not even distort it.
Are you ready for such a truth? or do you believe that the Creator had to use a polygon to give core to the transcendence of the circle? Because if this mirage were true, the π of supercomputers would be correct, but it isn't.
From the pretense to conceive it as "a regular polygon with an infinite number of sides," which will never be the case;
to the irrationality of the π to attribute to it, as if it were an exclusive that strengthens its credibility;
to describing its circumference as π × diameter, when the diameter is not a determining factor, as we will see;
to the irrationality of π to attribute to it as an exclusive that useless trail of decimals to strengthen its credibility, which exist for many square roots of integers, and no mathematician cares about them, since the use of a symbol even for an irrational number is sufficient, not only as √Φ, but as the same
√2.
I have every respect for those who undertake the task of memorizing hundreds of decimal places, both for the effort and the methodology certainly put into it. However, Creative Intelligence deserves far greater consideration, and it is only by trusting in Its Impulse that I have been able to achieve what was missing to solve, for the first time in this civilization, the perennial problem of pi..
The above and other apparent truths throughout history have only led to makeshift solutions.
The circle is irreducible; like a self-protective shield; anyone who tries to conquer it by external means is destined to remain cut off; it was necessary to move from within.
The first and most important truth to realize is that up until now we have been talking about a nonexistent π, an invented approximate simulation forever, not present anywhere or in any geometry of creation,
an artificial substitute even less precise than the 22/7 division of integers reported by the Egyptians, perhaps for practicality, being in turn inadequate to the architectural perfection of the Great Pyramid of Giza;
The mere fact that it cannot be traced with a ruler and compass attests more than to its transcendence, to the fact that it is a makeshift solution, the fruit of an artifice, which, for lack of anything better, has been arbitrarily conferred the attribute of universal constant, as if it were authentic and absolute!
The main fact is that a natural &pi must exist, like any other perfect geometric construct; and, contrary to what everyone now trumpets, it does exist as the root of a polynomial equation with integer coefficients based on a radical value; we just had to go and get it.
Its very extraction, albeit belated, will demonstrate that it can be constructed using compass and straightedge in a finite number of steps, and achieve that absolute precision that numbers alone could not represent.
To be able to tackle it more correctly, in this work initially we will take into account the only certain data that can be leveraged: the fateful figure 3.14, and starting from it we will solve up to the point where, for those who have followed me carefully, the π will no longer have anything to hide,
After some necessary introduction, it will develop in three linear steps, of which I anticipate the extreme introductory summary:
"Platinum" in three steps
Set a unit of measurement at ⅛ of the unit circle, a 45° arc»angle.
Virtually rectify its arc into a segment of length L[Platinum] defining it on its tangent; we will call it curvature coefficient of tan(45°), and this will be tan(a).
Focus on the value l that satisfies the equation where l = ¼π for any radius.
In that unique case, the extraordinary correspondence of sin(a)
= tan(a)2 will occur, giving rise to the Squaring of the Circle without any approximation; I'm talking about none other than the golden section and its square root, and perhaps not even those who have already studied the Pi-drawer treatise expected it. And here's why and how to verify it
.
The exposition on this web page follows and integrates the Pi-drawer treatise already published (by Zenodo), aiming to better illustrate its dynamics and proof, without resorting to trigonometry, used here for greater descriptive autonomy, but also to stimulate a certain curiosity, before the diagram based on plane Euclidean geometry.
This does not exclude that the same solution can be obtained directly from the above formula; but here I will stick to the intuitive and descriptive path I have followed up to now.
the nature of the circle
The circle represents 1, the Unity without boundaries other than itself.
2 is dualism, a segment with its extremes: distance and/or separation.
Each subsequent digit virtually introduces regular closed polygons, inscribed and deriving from this subdivision of the circle (but no matter how high, the polygons will remain polygons! and it still amazes me that no one has ever realized this).
Unlike all polygons, the circle can only be drawn with a compass, while a ruler is needed for any regular figure, even if it cannot be drawn without a compass.
It expresses a direction that operates behind the scenes, unapproachable in its sacred, metaphysical, or transcendent essence. Comprehensive of all rhythms and frequencies, the Absolute is imprinted within it.
It reproduces the sphere in which the universe condensed at the moment of the Big Bang,
First of all, its formation in matter can be seen as the expression of a gravitational field, and this study will calculate its beating heart.
The circle has no beginning or end; it exists on the basis of only two parameters: the center with any coordinate, and the radius
that rotates around one of its ends, making it the center; and neither is visible.
Rotating first to one end and then to the other, the radius triggers, with two circles, the figure from which can be derived the square with only four equal circles, which represented the first challenge, introducing my concept of ‘essential geometry’ at the beginning of this study. The only way to define the circle is with a Cartesian coordinate system whose point 0,0 is its origin.
Therefore the circle is the result of its radius, of which the diameter is nothing but the derivation; referring to the diameter means failing to understand the circle.
Only the radius in fact can be considered the basic unit of the circle, just as the side is for the square, and in the following demonstration it turns out, as the dominant one, to be the true direct connecting bridge. read:
Any reference to diameter for calculation, which in fact does not apply to area, is to be considered inappropriate.
Although it obviously indicates the width as double the radius, lacking a center, it cannot define or trace any circle.
This is a slight correction to what I stated on page 5 of my first treatise, in the first steps towards Squaring the Circle, to which I address you for a more precise didactic correspondence of the relation between radius and the method of measuring the circle.
Nevertheless, we will see the indirect importance of its measurement, and its virtual relation with area as well.
Why focus on the radius
While the diameter is a static and passive measurement, the radius is the dynamic side of the circle, as if in an unstoppable motion that renders it invisible, since, if stationary, it would mark the extremities of the circumference, both the beginning and the end, and a specific orientation, arguments incompatible with the nature of the circle. Therefore, the most suitable way to represent both is to indicate the cardinal directions with 4 or 8 radii.
The radius is, in a certain sense, the circle's only link with normal geometric figures, or rather, the bridge between its absolute curvature and any regular polygonal construction.
On the radius, it is natural to construct a square with equal sides, which therefore circumscribes each quadrant of the circle itself.
Starting from the radius [which in this context will always have length = 1], we can construct two closely related figures: the circle C and the square CBED with side =1.
The square, which will thus be circumscribed to a quarter of a circle, reproduced for each quadrant will give shape and content to the square circumscribed to the entire circle, which therefore will have side =2, like the aforementioned diameter.
Our Intent
Our aim is to plan a inquadrature of the circle, using a geometric device to focus the actual measurement of the π with an algebraic equation representing a comparison of the parts.
The fundamental difference between the Pi-driver and the classical conception of squaring the circle, consists in extracting from the comparison of both – due to the extreme and natural compatibility of the square, superior to that of any other polygon – the reduction (or curvature) coefficient from the first to the second, since this is the π; or vice versa, that is the process determined by 1 / π.
meaning of π 3.14
Let's start by reviewing the numerical ratio between perimeter and area, between the circle and the square that circumscribes it.
The π is independent of the circle; based on the radius, it calculates its perimeter and area. We will try to represent and combine the two procedures.
Just as the square with side =2 has area =4, so, for the normal formulas [π×1²], the circle reduces its area from 4 to 3.14 r².
Just as the square with side =2 has perimeter =8, so, for the normal formulas [π×2×1], the circle has perimeter =6.28 r.
read: Both calculations derive from the radius, the first with r², the second with r×2, which rather than the diameter, could refer to the two external sides of each of the 4 squares, e.g.: BED, since this, let's say from a circular point of view, would better express the reduction, from 8 radii to 6.28.
In practice, as the area of the circumscribed square is reduced from 4 to 3.14, the same applies to ½ the perimeter [4] of the circumscribed square.
The π therefore affects ½ the circumference, and can be considered as the ‘curvature coefficient’ of ½ the perimeter of the square containing the circle; but this does not satisfy any dialectical congruence.
Recognizing L [Platinum] as the unit of measurement or elementary module of the circle, we can state with semantic certainty that for the quadrant of the unit circumference "L m curvesr m" and for the area "Lm² curves r² m²".
the unit of measurement
That being said, from ½π equivalent to ¼ of a circle DB we move to ¼π for ½ arc AB of a quadrant, which is the perfect candidate to adopt as a minimal unit of measurement, since it is mirrored on the diagonals of the square-quadrant, thus dividing the circle 8 times.
Other modular subdivisions would not be equally suitable, although not excluded, deviating from the perfect symmetry of the square with the orthogonal axes of the circle and its significant 4 phases.
The reason for this simplification is essential: to section the circle so that its unit module [⅛ circumference] can be related to the external side of the square circumscribed to the quadrant in question [⅛ perimeter].
In this case, the unit curvature coefficient will be ¼π, which for years I have defined as the constant l and called "Platinum." This constant is a fundamental key to various models of existence.
So the next step is to project, with sufficiently low accuracy - a little more than the length of the chord of the unit of measurement AB - onto the side of the square to which it is closest.
Or by defining on the side BE of the square the point e, distant from B by a measure l, close to our ¼π to the hundredth.
We can assign l its presumed value, to avoid suspensions of analytical thought, or we can also consider it an unknown variable; the accuracy of the Pi-drawer diagram will not be affected.
In short and in minimal terms:AB = l × EB where
the Pi-drawer will serve to resize it to the exact unlimited match.
Assembling the Pi-Drawer
phase 1 – from circle to square
As I said, ignoring everything that followed antiquity without achieving improvements, I will start from the formula transmitted by the Egyptians to define π = 22/7, whose result, 3.1429, will be closer to the true one than ours (Archimedes was also closer, but only up to 3.1419).
The basic concept is relatively simple:
Conceptually adopting the length 0.785 [22/7/4 or even just 3.14/4] for l, we cut it on the side BE at the point e, so that Be = l.
The external side BE of the square circumscribed to the quadrant would be reduced by the true coefficient l to AB which, from the formula ¼π × r or l × EB, since EB=1 will be = l.
Even if this is a rough attribution hypothesis, since it concerns thousandths it is more accurate than is necessary to be able to evaluate it in a graph.
Having thus outlined a right-angled triangle CeB, by the Pythagorean theorem we set the first reference datum in the length of Ce = √ Be² + 1².
After several weeks of tormented research and turbulent dissatisfaction, explanatory version after version, here I am at a final turning point which, for its simplicity and essential evidence, disconcertes even me who conceived it; and I wonder more and more how the academic world will take it..
phase 2 – from square to circle
Given a diagram as described in 'phase 1',
the question is:
what must l do to confirm the curvature coefficient of EB at ⅛ of circumference AB, that is, ¼π?
The Pi-drawer treatise published on Zenodo on March 1, 2026 describes the debut of my theorem, retracing its kaleidoscopic combinations from various points of view, with the aim of an increasingly stringent and effective intellectual exposition, since its linear principle tends to become elusive.
However, in the light of the most careful maturation, to mathematicians all over the planet I can finally present the theorem in its essential form.
As an exact rectification of AB [¼π] in eB, L must be such that Ce = 1 /l.
In this way, Ce as a radius will trace an arc cb as an effective enlargement of AB [l] to cb of length =1, precisely as ⅛ of a circle of length =8, as the perimeter of the square circumscribed around the unit circle.
It is a mere mathematical statement, containing everything it needs to be true and complete.
If l is a rectification of AB, its projection onto 1 in the manner described below will give shape to a square whose side = 1/l will be curved by the new coefficient with length =1, identifying itself with the aforementioned cb, an arc which likewise intersects BE in e.
Conversely, Ce × l will give the radius =1, as CA, which traces the arc AB in its natural form.
We can doubt l, but not 1, so in its ratio the Pi-drawer will construct a mirror counterpart, which allows us to verify that l reproduces the function of ¼π at every scale, and graphically demonstrate the actual value of 1/l.
It will allow to calculate and confirm the value of L that meets all these requirements, from the actual rectification of AB in l =¼π, to that of cb at the segment bf=1, simultaneously performing in e the squaring of the circle via EB and cb.
It's an indirect geometric proof, reconstructing cbfrom the outside via 1, rather than calculating from the inside via l, balancing the three factors:
1 ×l » 0.78
reduction from 1
1
1 /l » 1.272 enlargement from 1
To this end, 1/l will be the base of a triangle resulting from the enlargement of CbE into Cbf, such that its height l (in the figure) becomes bf = 1.
The transformation will be set by multiplying the three sides × 1/l, from which 1 ÷ l = Cb[ ÷ CB= 1]
Thus Cb will also become the base of the square whose height bF = 1/l will have to be reduced to a curve by the scalar coefficient bf in an arc cb=1.
The equality of the said arc with that defined by the radius Ce assumed 1/l, will demonstrate the consistency of the initially established ratio of l with BE curved in AB, as well as the validity of the value of l.
Only the right value of l will have produced a square in which its projection at 1 in bf will curve the side 1/l to its own measure of 1.
For any contrary event, it will be sufficient to consider the following:
while the scaling of l to bf, that is × 1/l will bring bf to the length =1 whatever the value of l, always remaining eB ÷ CB = fb ÷ Cb, with a value of l different from the premise [of making Ce = 1/l], the bf obtained will no longer be able to represent the curvature coefficient of Fb in an arc =1, since Fb would become, more or less visibly, larger as l decreases, or vice versa, curving in a proportional arc other than ⅛ of a circle.
In any case Ce would no longer be equal to Cb.
Basically, to draw an arc cb [= EB] as an enlargement of AB [l] at length =1 we need the radius Ce = 1/l, in proportion to the radius 1 of AB, as per the basic self-sufficient premise.
S. of C.i.e. [AB× 1/l] – Note that the intersection of cb and EB in e will give rise not only to a subdivision of BE such that BE / Be = 1/l, but also to CeE as ½ quadrature angle [cf. EB] of the circumference [cf. cb], that is ⅛ of lengths =8, circumference and perimeter of the square circumscribed to the base circle, of which BED is ¼.
If l establishes the only and definitive ratio between the circle and the square of equal perimeter, the perimeter ratio between the circle and the circumscribed square is1/l.
For those who still haven't had enough, what I report in the following paragraphs is only a part of the convolutions that preceded the above summary, even after the treatise's first publication.
If nothing else, it will give an idea of the incessant neural effort that accompanied this research, due to the need to make it comprehensible through dialogue, even before formulas.
As an integrative crowning of the setting: we have developed the expansion of AB not by calculating the radius Ce, but by amplifying l geometrically to give rise to an arc cb=1, i.e. ⅛ of a circle, which introduces its squaring.
Thus, two interactive phases coexist, in a certain sense mirroring the circle and the square:
an arc drawn from the outside, which must be identified with the projection of AB enlarged from the inside with radius 1/l, as above.
The equivalence of the two arcs is necessary and sufficient
to confirm the constant function of the π for any radius and arc cut.
In essence, while the 2nd phase geometrically defines 1/l and the arc cb as curvature of bF from 1/l to 1, and is also the enlargement of the arc AB from l to 1, this confirms the absolute value of l as the mediating modulus of π. It is a stringent and condensed labyrinth of perfect correspondences, which induces repetition from various points of view and analysis, without however managing to say it in all possible ways.
It's all more complex to explain than to understand – because, as I've always maintained, intuitions come quickly and first, explanations later and slowly – especially for the simplest things that inevitably turn out more complicated.
Anyone who has read my exposition published first at Zenodo, in a vain race with March 14, will have noticed that it was much less simple and more painful than the present one after a month of decantation.
I started from the solution, and it wasn't easy to break it down:
a process that made me verbose; but I spared no expense in order to emphasize the grandiose congruence between the parts, in a game of interlocking parts that was anything but linear, an extract of a pure curve, not produced by a function, and this is what makes the π so difficult to enucleate.
Nonetheless, it's there, like a flower ready to be plucked with just two fingers.
I'm certain that anyone who wants to delve into this whirlwind of thought will encounter more than one subjective tangle to unravel on their own, for this is the powerful driving force of an extraordinary truth.
The Pi-drawer device guides and makes manifest beyond any possible doubt the way and the only case in which the value of l [Platinum] corresponds to its assigned function, that of defining ¼ of the much-desired π, literally the π, rectified.
Any formula that leads to a different result is nothing but imagination.
"kaleidoscopic combinations?"
"Kaleidoscope" from the Greek kalós 'beautiful', eidos 'image', and skopéo 'observe';
literally meaning "to contemplate beauty."
It is a tool used to decompose and recompose images with the same fragments but from different angles, to explore the complexity of converging effects.
That a new arc bc with radius Cb is equivalent to the scaled enlargement 1/l of the measuring arc AB, both passing through e, will occur only if
the segment fb set to length =1 is equivalent to the curvature coefficient of the side bF in the new square, which in proportion must measure 1/l to be converted into an arc as long as fb.
It is essential to keep in mind the fixity of the key factor bf = 1, which can and must only reach the extension of DE, whereas any other argument can vary according to l, invalidating the mathematics of the scheme, even in the most distant of decimal places; since this is the strength of the PI-drawer device..
In short, a lowering of l, in addition to no longer being able to curve EB to its own length, in shortening Ce would lengthen Cf, expanding the larger square, but not fb, thus circumventing the essential proportion.
In fact, a discrepancy would arise due, for example, in case 0.7857, to an anomalous lengthening of CB in Cb, distancing an arc bc (smaller if l exceeds, or larger otherwise) from the ideal enlargement of the arc AB, given a radius different from the presumed Ce.
The opposite hypothesis would result in a negative process, called "Reductio ab Absurdum," the extremes of which I reiterate,
even though we won't need it given a solving algebraic equation: read+
bfF is the mirror of BeE in proportion 1/l, where bf is always =1 while bF can vary.
1/l will be for any value of l the ratio to project l to bf of height =1; but the greater distance will apply it to a square with an out-of-focus arc.
If, therefore, we do not have Ce = 1/l, but for a smaller measure of l an excessive length of Cb, the fundamental convergence will collapse from every point of view.
This base and its new square will give rise to an arc cb greater than 1, canceling the expected effect of the factor bf on bF.
On the other hand, we would obviously have AB>l, since the arc remains fixed at ⅛ of a circle as the first module, independent of l.
With l< ¼π (we are not interested in examining the opposite case),
Ce shortens while Cb lengthens, generating two different arcs, neither of which is the inverse scalar curvature of BE for the coefficient l or the corresponding curvature of bF for a coefficient =1.
Therefore, the ratio is Ce = Cb.
To clearly highlight the anomaly that is taking shape, in the first illustrative example I use a value of l that is much lower than the actual value, not to exaggerate the effect but to make the scheme easier to read.
Giving l [Be] the current value of ¼π = 0.78539 instead of 0.78615, it wouldn't even let you see the difference; but the difference is there, and it is substantial; I will give precise proof of this in the paragraph "appearances can be deceiving", which this graph will help you read, as it is free of notations. given the large resolution.
chiudi:
A length of L less would make such equivalence of a specular reflection from bf impossible.
since it lengthens Cb and shortens Ce, altering the proportions of the arcs between them due to measurements deriving from presumed but invalid modules of the π, giving rise to a sort of astigmatism.
In the figure it can be easily seen that the semi-arc of the new square, at approximately bE, is considerably larger than bf, which excludes that bf can correspond to the curvature coefficient in 1/l scale.
wrong l simulation
Starting from a length certainly shorter than Be, like that of the string AB, in figure Bd¹ deliberately much lower than the presumed Be i.e. lto ensure the readability of the graph, but which could also be 3.1416 /4 with the effect of making the figure perfect at this illustrative level, it is easy to visualize a process parallel but inverse to the one applied for millennia, progressively raising the point e to bring the two arcs, Bd² and Bd¹ closer together, until they coincide at be.
We have already established that this is possible.
the Squaring the Circle
In fact, with this derived arc, a circumference of length 8 is set, equal to the perimeter of the global square circumscribed to the circle of radius 1. In practice, if everything is clear (and I do not hesitate to repeat myself to make it so with certainty), the most direct conceivable formation of squaring the circle has been configured, which when all is said and done we will discover we can easily construct with ruler and compass.
the extraction of ¼π, the Platinum constant
Bringing everything back to the pre-established lowest terms, Ce as the radius of the enlargement of AB to 1 must equal Cb as the side of the square depending on the amplified height of l. therefore the radius of cb is derived from bF.
If 1/l constructs from both sides the enlargement to 1 of AB, this means that eB is the natural rectification of AB.
I have tried to rely on dialectics rather than algebraic formulations, since everything, especially in this concentration of simplicity and complexity in one, requires immediate and deductive mental participation, whereas mathematical expressions, to be understood, may need to be translated in the mind into a fluid language.
But it's time to unload the greater weight accumulated, on the initial notation which is then the synthesis of a more laborious approach, exposed in my treatise «Pi-drawer»
In an elementary algebraic form, this magic which is the extraction of the π, almost a symbol like the first slice of a wedding cake, read: once summarized with Ce = 1/l, is solved with the formula: √L² + 1² = 1 /L.
When faced with a cake, you wouldn't try to eat it all at once, or even half of it in one bite... you cut yourself a slice ;o) and yet that's been the approach up until now. Those with a sweet tooth may have had an indigestion and call it transcendence.
All that was needed to find the correct answer was to fit the two factors together in an equation, to obtain the only value that reconciles the two perspectives, giving l [Platinum] its radical function of ¼π.
And finally, here is the true ¼π to the tune of this score:
√L² + 1 × L = 1
A mathematical distillation between the physical and the metaphysical,
new Light on clear Knowledge
The projection interface revealed ¼π and this is:
l = 0.786151377757423286069558
It wasn't that difficult; it was just a matter of finding the right path through a more appropriate concept.
I have already described the historical approach as inadequate, alien to the circle, and almost in conflict with its nature; perhaps this is the reason why the salvageable part stops at 3.14, already betrayed from the third decimal place onwards.
In its place, the correct ‘modulus’ ⅛ of a circle, to be multiplied by 4 for the current use of π = 3.14460, seems to suggest that the figure for defining π should be identified precisely with this Platinum constant, L as the minimum direct converter of the radius itself, which is the basic unit for every [¼ of] a circle and the squares that mark its four natural phases.
the Unit of measurement
The use of ⅛ of circle deserves further mention.
Far from being a minor perspective, it is not by chance that we end up observing the circle in a way based on the number 8; it was not an extemporaneous arbitrariness, since it is precisely its modular value that opens the gate to exploration of π; which in my analysis is nothing other than the constant L.
For me, this is the monad.
read:
The 8| , cut in two parts by the vertical axis, generates two symmetrical S, like opposing wave-like figures, a symbol of completion in balance, and from the horizontal axis, two entire circular paths, communicating in perpetual alternation, with opposite directions of rotation that merge into Unity. This gives rise to the image of boundless infinity, just as the 8 is horizontal in mathematics, in a form even more dynamic than the hieratic circle, whereas the square is merely its static extension in the physical dimension. See also the TAO, below,
music of the spheres
I entrusted the development of the equation to a valid online calculator, it was not my task; nevertheless, in the development of this complex formula, it almost seems to me that it has stumbled upon 4 solutions, somehow 'auxiliary', characterized only by alternating ± signs that it seems unable to interpret, but which immediately make me think of the coordinates of the 4 quadrants.of the circle.
I saved in PDF the calculation of mathcracker.com/it/calcolatore-equazioni. It includes comments in English interspersed with clarifications in Italian, but it's the calculation that counts. I reproduce its result in figures and a graph, which clearly shows how the coordinates of the two arcs diverge drastically if the length L, Be is incorrect.
The input was: "\sqrt{\left(x^2\right)+1}\times x=1"
inally, the formula is perfect and absolute: an equation
with one irrational but not transcendental coefficient!
After centuries of painstaking research,
we find ourselves faced with the unique real index of π.
Everything else is water under the bridge.
In the end, it wouldn't have taken so much calculation effort, just check, by substituting the unknown, if and which of the few variables already known satisfied the calculation, especially ¼3.14159 or ¼3.14460.
But since we have gone back to the 22/7 of ancient Egypt, so as not to tell the Babylonians who also knew the square root,
it is worth adding that even without tackling the algebraic solution, it would be enough starting from 0,700000… to increment each decimal place of the lower 0.785398… by the maximum unit from 0 to 9 that keeps it < 10, to obtain an exact result
ever closer to 1, in practice bringing the length of the segment Be up to and not beyond the actual length of the arc AB, not an approximative one for the type of calculation applied, but the exact one rounded only to the decimal point
for the desired precision deemed necessary for scientific or technical use, and allowed by the software.
I condense an example taken from this simple calculation algorithm (code EUPHORIA), which I limit to the first 12 digits for obvious reasons of space, whereas the Postscript offers 8, while some astrophysicists declare that 6 are sufficient, and in a certain sense they are right.
In 1881, astronomer Simon Newcomb declared that
Ten decimals are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimals would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope.
“Elements of Geometry” in 1881.
Who knows if the fraction of an inch didn't contain the gap of 3.01 thousandths.
The fact remains that 2 reliable are only good for Pi Day!
--- ======
global object digit, L
digit = 0.1 L = 0.7
--- ======
function digitsteps ( object L )
atom Platinum integer count
Platinum = L count = 0
while Platinum < 1 do
count += 1
L += digit
Platinum = sqrt( power(L, 2) +1 ) *L
end while
return ( L - digit )
end function
--- ====== love the digits?
-- L= 0.78615137775742328606955858584295892952312205783772323766490197
-- π = 3.14460551102969314427823434337183571809248823135089295065960788
for i =1 to 12 do
digit *= 0.1
L = digitsteps (L)
[ ? L , L * 4]
end for
--- ======
In the case in question, after all, a single and defined constant must be sought; not as I had to resolve (again for the first time in history) the complete architecture (and without errors at any level of magnification) of the extraordinary Śrī Chakra yantra.
. read: Śrī Chakra yantra - Complete Architecture
in 7 golden circles, error-free.
There, it was a matter of seeing triple-linked intersections coincide in the same diagram, moreover not based on a fixed parameter, but rather on more than one, responding to a potentially unlimited range of schemes, predefined by subjective and variable settings, to which I had dedicated an application of mine available to the public since 2014. In this regard, I recall a test up to 640 decimal places, even too many for the precision allowed by the highest graphic resolution, but for any future eventuality, expandable at will by my program.
Furthermore, it is time to ascertain how the process absolutely re-proposes the figure of the Golden Section, a solution that I have highlighted since my first steps, as the close and natural relationship between π and Φ, in which the solution is identified in its final notation, being able to reverse into the more widespread one:
√(√ 5- 1) / 2, cioè sempre √Φ.
“The Origami of Power”
Naturally, the resources of L also extend to φ (which is then = Cf, being the intermediary with its square root
[ φ×L= √ φ ] and above all in relation to Unity, generating in the case in question a balance that cannot be replaced by any improvisation:
CB ÷ Be = Ce [1 / 0,78615… = 1,27202…] pivoted on 1 and
CB ÷ Ce = Be [1 / 1.27202… = 0.78615…] at the maximum fusion level:
Ce × Be [1.27202… * 0.78615…] = 1 or, knowing the golden ratio:
√φ×√Φ = 1, the keystones of the whole, in the wake of Φ × φ = 1.
The mystery of that message transmitted to me from another space-time appears ever clearer to me, ever since my first considerations on the great pyramid of Giza in the drafting of the treatise on ‘the 5 Tibetan Rites’; it was repeated by heart by a partner (who however did not grasp the meaning)
“The Origami of Power”, already mentioned on this site, but not yet so pregnant of meaning.
The traditional art in various cultures around the globe of folding paper as if it were an authentic geometric reality, to give shape and expressiveness to all sorts of models and symbols, sometimes spectacular, representing a myriad of living or abstract things in a 360-degree way.
Knowing the correct result at this point, we can enjoy the privileged and wonderful property of the golden ratio, which allows us to rotate numbers and symbols with unparalleled ease, to jump to a result that can almost be read directly:
√ Φ + 1/l =
√ φ + 1,
√ φ/
√ Φ =
√ φ²
Cf = 1.618
Ultimately, e is the point in the square that manifests and guarantees the continuous balance between radius and angle in a circle inscribed in any dimension and, to begin with, the key to squaring the circumference.the key to squaring the circumference.
THEDIVINE PROPORTION
COULD ONLY FIND ITS ROOTS IN THE HEART OF THE CIRCLE!
It must be understood, to the credit of pi, as with the golden ratio, that Universal Intelligence, which is also mathematics, knows well how to follow its course, without needing to be exalted by billions of decimals, which only limit it, proclaimed only to obscure a concept of π flawed from the outset, and which therefore lead to nothing certain and useful.
Those who have studied my articles, which in four years of research have paved the way up to this point, will not be surprised to discover that the unique solution to this equation, more troubled than it seemed going through a 4th degree polynomial - refers to the digit 5 in the form:
1 /√ 0.5 × √5 + 0.5
and in terms familiar to us:
1 /√1.6180339887… or to simplify √ Φ, hich is precisely 0,7861513777… Q.E.D.
It is evident that this result makes all research and speculation connected with the number of decimals and what they could possibly reveal superfluous, since as for the true power of π we would gain no advantage by knowing an interminable tail of it, instead of the harmonic laws it expresses and of which π is the cornerstone, together with the golden section, and which I had re-emphasized in the latest addendum of December 2025
If no one worries about the myriad decimals belonging to the Φ, it's precisely because they're focusing instead on what each of its expressions represents in creation, attributable to the divine, letting the the unfathomable fraction carry out its work of perfection undisturbed.
The same goes for the π, which from now on we will be able to contextualize properly, as part of the golden ratio.
There will certainly be no need to memorize its tail, since we have an integral π at our disposal, and its genesis will be sufficient to select and use it with absolute safety at any application level, both physical and astrophysical.
As should emerge from the actual discovery, π represents much more than a geometric ratio between diameter and circumference.
Have you ever seriously wondered what purpose all those millions of digits serve in life? They won't help you turn a cylinder rounder, yet the authentic ones are indispensable to guarantee the cosmic Balance at levels so deep and high that earthly science will never be able to fathom.
Super-powerful computers in comparison gave nothing but the [pathetic] illusion of getting ever closer, assigning compelling realism to the disposable factitious constant.
Here's instead what the Pi-drawer gave us, like an explosion of synergy:
The complete ¼π extractor: √l² + 1×l =1
It is evident that no other hypothesis of π will be able to replace the current outcome.
This vital parameter should be updated as soon as possible in all calculation systems – unless they are content to display figures with more than 30 characters knowing that from the 3rd onwards they are all notoriously erroneous – whether research institutes, teaching institutes, scientific organizations or others, in the interest of the entire Earth Community and of the Knowledge of all time.
Hopefully, before AI takes over! The true Intelligence is that of the Creative Consciousness Power, ours is a limited and yet conceited effect of it, a consequence because we are the fruit of IT; not a merit, but a supreme gift to be experienced.
Is the alternative perhaps to ignore it and stop at 3.14?
We can be satisfied with four decimal places (our fifth is zero, almost an invitation as confirmation), but for minimal scientific precision, not two.
Furthermore, while rounding a 0.0015 may be acceptable, rounding a 0.0046 is much less so; not forgetting the 6.28, which should become 6.29.
In any case, all operators should be urged to no longer distribute misleading info, and numbers for which there is no longer any justification.
Yet the news, or should I say the demonstration, arouses no interest at the Exploratorium Museum's media@ service, which is avowedly waiting for 'a story worth telling', nor at the "Sito ufficiale del PiGreco Day organizzato dal Ministero dell'Istruzione e del Merito", at the University of Turin.
However, since the network exists, it cannot be circumscribed for long.
Fortunately, mathematics is a law that stands above politics and any religious belief, even though Indiana Bill No. 246 of 1897 already attempted to do so, defining the value of π as a rational number.
Politics and religion are merely side effects; mathematics governs the Universe.
From one paradox to another, it turns out that the intuition of its promoter, Edward Goodwin, was correct; even if the disputed legal value of 3.2 was unfounded. Ultimately, the current state of π, in light of this direct experience, appears not to be entirely distant from a dogma, of which the very recurrence of adjectives like "consolidated" is a testament to mathematical uncertainty, to opinion; no one would dream of claiming that 2 + 2 = 2 × 2 is now consolidated.
I believe that a true mathematician would just have to look at the cover of the «Pi-drawer» to discover the arcane.
Appearances can be deceiving
I mentioned that π / 4 would leave the figure perfect at this level of illustration, but if that was enough to draw up a concept or a project, numerical reality is another matter. I couldn't resist a graphical verification, which was, after all, much less demanding than when I had to present the discovery that the 90° golden spiral was made up of quarter ellipses; a compelling, almost creative theme, while this one is far too basic and, as already explained on the page detailed below, simply seeks to highlight the inevitable error.
However, since a visual comparison between the two versions is warranted at this point, it's worth examining to see what and how much the mind can be led to miss.
I have highlighted the resolution on a board of 7,143 points, adopted to solve the Śrī Chakra yantra without errors, and a unit radius of 3,000 points.
Given the high precision required, it would be pointless to embed it in the HTML page, this PDF, requires a powerful reader and a suitable screen.
I embed the image below in the e-book; to scroll, you need to set the zoom view to 'Fit Page', but you can magnify it as much as your reader allows.
In any case, for ease of reading, I reproduce here the most important detail enlarged to 3200%.
The scheme of course requires high definition in the paths of the two overlapping cases: the one based on π = 3.14460 is in green, the one based on π = 3.14159 is in red, without the need for letters and numbers since the curves correspond to the descriptions already given, while it would be difficult to see both ends of the paths here, even if marked with letters.
I also provide the code, to allow further tests to anyone who wants to manipulate its few elementary parameters.
The smaller oval on the left, set for the zoom, indicates the differences between the arcs of the two phases; the larger one on the right is the point of arrival at height =1 (marked by the dotted line), where we can see that the red triangle (from 3.1416) gives rise to a square with a larger side, of which the height =1 can no longer constitute the expected curvature factor.
Of course, even the initial measurement in the 1st box could not fulfill this task, but this could only emerge from the discrepancy between the two opposing phases.
The two arcs having as radii the bases of the larger squares are in solid lines, while those from the hypotenuses are in narrow and thicker hatching, to be able to show that the two cases in green (3.14460) overlap.
A normal view will in fact make the two schemes appear equivalent, while one of the two involves giving up the π and its magnificence.
The 8 decimal places of PostScript are more than enough to highlight the differences, but they require the maximum zoom achievable by your reader, around the areas circled by dashed blue ovals.
To obtain arcs as perfect as possible, however, I had to use a manual algorithm based on sin() and cos(), since the PostScript arc operator is not, for reasons already mentioned in various details.
This is immediately confirmed by the exact intersection of the green arc with the side of the square and the triangle (PI 3.14460 - double-circled), which evidently none of the other curves respects.
To exemplify them tout court, I also wanted to include the circles drawn by the PostScript (360° in dashed lines) which, probably based on the current pi, differ from the more truthful arcs of my code (with a continuous line).
The difference between arcs from the native PS and trigonometric arcs, whose amplitude even varies at different angles, can be clearly seen in the two screenshots above enlarged with the same zoom (3287%), which I reproduce here for the reader's convenience.
Applying my simple PS algorithm to the half of ¼circle, with an extreme reduction of the step one could even try again the rudimentary calculation of its length; but it would no longer make sense to do so, we already have the π.
It will become clear to you that in comparison the current π current is a path with no outlet or access.
at the center of the world
Asking whether π derives from Φ or vice versa it's like asking "whether the chicken or the egg came first".
Generating or generated, π should be acquired and understood as a pillar emanation of Universal Intelligence.
If one observes and meditates on the few emblematic figures that follow, do one not perhaps perceive the powerful attractive and expansive synergy of the π together with the golden section, for the fulfillment of the Cosmic Egg?
Believing to have reached the required π, and that adding celebrations to 3.1416 will increasingly contribute to affirming it as the right goal, at this point would be unscientific and misleading.
I have clearly reiterated the geometric reason in several places on the website and in my first essay: 3.1416 is already incorrect from thousandths onwards, and after this test there can be no doubt left.
But I know all too well that there are those who understand (and can't do anything about it), and those who don't want to understand! If I had participated in memorizing one or two thousand digits queuing at 3.14… perhaps I would react the same way.
Nevertheless, this festive attitude not only does it divert attention from a more in-depth scientific application, but it implicitly prevents the grasping of the most important of truths:
π [the effective one] is the source and from its square is the generator of the global equilibrium in the form of Φ ÷ 1 = 1 ÷ 1 + Φ, that is 1 = x + x × x.
chasing the Squaring of the Circle
Did you think it was over? I thought so too, at various stages.
Even if the essence of this final stage is autonomous and self-sufficient enough to leave one free to ignore any previous one - which applies even more so to any other historical attempt to define the π –. I trust that retracing the various stages of my investigation can point to new ways of sharing further hidden aspects of the celestial Project.
Anyone who feels free from the jumble of 'consolidated' suppositions on an artificial and thankless π will still be able to establish its maximum transparency of a perfectly calculable number, further shortening the overall treatment already published.
It will be enough to aim for the direct search for squaring the circle, starting from "phase 2 - " for a decisive synthesis.
Per l'esatta rettifica di AB in eB, l dovrà essere tale per cui risulti Ce =1/l.
To confirm its function as a rectification of ¼π and therefore a curvature coefficient, it will be possible to trace the same arc cb of length =1 in two separate and convergent ways:
from Ce as radius, by the algebraic enlargement of AB/l.
from the projection of l to 1 on a side of length 1/l,
where it will act as l to curve it into cb.
In fact, only in this way will the projection of l onto bf with length =1 by the proportion 1/l give rise to a square whose side bF ÷ bf = 1 ÷ l, from which the arc cb rectified into bf will have as expected length 1=EB, both equivalent to ⅛ of circumference of length 8, like the perimeter of the square circumscribed to the unit circle
This new arc of radius Cb will be equivalent to the arc of radius Ce with only one value of l: that of the real ¾π, such as to reproduce the function of l as a curvature coefficient at any scale, maintaining in our case Ce = Cb.
It is the bijective combination that opens the safe of the π
The contrivance allows us to determine with the equation described whether the presumed length of l is correct, or what it should be.
Any other value, in fact, more or less approximate, but not exactly ¾π, will make the two radii and their respective arcs different, distancing them from each other, as well as from corresponding to ⅛ of circumference.
Straightedge and compass?
Now it's easy, since Cf = φ. The procedure for obtaining the golden rectangle 1|1.618 from the square CBED is well known: from the center m of CB, an arc with radius mE is drawn to meet the extension of CB at p. Cp is φ (1.618). For the Pi-drawer an arc of radius Cp will reach at f the extension of DE, i.e. at height 1.
Without having to go over the implications already discussed, it will be sufficient to join C to f to cut EB at the point e, which defines eB as a precise rectification of AB = ¾π, our unit of measurement, as well as of squaring of the circumference of radius Ce. with EB × 8.
To make it even shorter and more direct:
a hypotenuse Ce = 1/l, as a radius traces a semi-arc cb from the diagonal CE to the extension of CB, so that Cb is the base of a square to scale 1/l, capable of reproducing on its side bF the rectification fb of its curvature =1 in bc, necessarily reflecting the function performed by l on EB, since fb ÷ Fb = eB ÷ EB, and repeating a cb defined from the beginning by Ce as an increase of AB to 1.
Two valves of the same shell, which contains the most precious Pearl
the last wonder
To conclude as I began, touching this area, the last wonder among the secrets of the circle emerges:
the sine of the angle formed by l is equal to the measure of its tangent squared: Φ ÷ l = √Φ ÷ 1, a tangent which, as the first external projection of the circle, determines determines a circle in perfect quadrature, even for a descriptive outlook.
It was enough to bring ⅛ of a unit circumference to length = 1.
"the ratio between two contiguous sides of a square of side = 1 with the corresponding arc of a circle of radius = 1, virtually inscribed in it… is in my opinion the most suitable to mark the step towards the π"
Thereafter three months were oriented to the celebration of Pi Day 2026; but those who were officially informed preferred not to notice.
Communication forwarded to the Exploratorium Museum's media@ service, as well as received and ignored by the Sito ufficiale del PiGreco Day organizzato dal Ministero dell'Istruzione e del Merito, at the University of Turin.
preface
Starting from the radius [which in this context will always have length 
That being said, from ½π equivalent to ¼ of a circle 
So the next step is to project, with sufficiently low accuracy - a little more than the length of the chord of the unit of measurement 
The basic concept is relatively simple:
To this end, 
inally, the formula is perfect and absolute: an equation
The scheme of course requires high definition in the paths of the two overlapping cases: the one based on π = 3.14460 is in green, the one based on π = 3.14159 is in red, without the need for letters and numbers since the curves correspond to the descriptions already given, while it would be difficult to see both ends of the paths here, even if marked with letters.
If one observes and meditates on the few emblematic figures that follow, do one not perhaps perceive the powerful attractive and expansive synergy of the π together with the golden section, for the fulfillment of the Cosmic Egg?
In fact, only in this way will the projection of 
Now it's easy, since 
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