\ The wonderful π - Squaring the circle

2x2 = 3.14


The Rosetta Stone' detail – © The Trustees of the British Museum
Before numbers, math is based on concepts.
The Geometry – Architecture of the Universe – modalizes these concepts with a precision that
no computing apparatus can achieve but in an abstract and symbolic way.

Myth and Heresy

SQUARING THE CIRCLE USING ONLY A COM­PASS AND A STRAIGHT­EDGE not only is it not im­pos­si­ble, but it is a process inherent in the same 'fig­ur­a­tive' geometry in a natural way, since the π it is the cor­nerstone of the laws of harmony and equi­lib­rium that reg­u­late it.
It was just a matter of discovering the key, geometric balance needle between state of mass and energy state.

In spite of what F. Lin­demann ex­hibited a­bout the un­solv­ability of the prob­lem, his suc­cess was and still is due sole­ly to the fact that the cur­rent cal­cu­la­tion of π it has al­ways been en­trusted to het­er­o­ge­ne­ous cri­te­ria to the laws ­/ na­ture of the cir­cle.

The in­sti­tut­ed π de­rives in fact in the most ad­vanced of cas­es, from re­con­di­tioning the cir­cle to a pol­y­gon, di­lat­ing with­in the cir­cum­fer­ence pro­gres­sive po­lyg­o­nal ac­cen­tu­a­tions, se­quenced by for­mu­las that do not make it ge­o­met­ri­cal­ly de­duc­i­ble.


Put sim­ply, the sup­posed tran­scend­ence of the π it is due to the way in which it was for­mu­lat­ed, not to its al­ge­bra­ic pa­ram­e­ter which has al­ways been ac­ces­si­ble, even if nev­er ac­quired up to now.
True transcendence, even if one has never observed it, is that of the circle itself.

It must first of all be recognized that geometry is not second to math­e­mat­ics, or a dependence on it as we usually believe; if anything it is the op­po­site, since the Creative Intent does not generate numbers but forms, and the numerical expressions, abstract and stripped of any ex­is­ten­tial au­ton­o­my, only serve to make that geometry, which composes them ac­cord­ing to the Universal order, from the atom to the galaxy, legible, com­pre­hen­si­ble and usable.
Numbers are only the alphabet of geometric language, and draw their sa­cred potential from the intrinsic meaning of the figures and the re­la­tions that they represent and express. Even the four mathematical operations that gov­ern them in formulas emulate the cardinal dynamics in space, op­pos­ing and complementary vectors, without which the values would only represent quan­tities suspended in the void.
Do we prefer to think that universal Consciousness manifested itself by re­or­dering chaos through mathematical formulas like ours, or by creating per­fect­ly synergistic space-time forms?

Be that as it may, far from proving that squaring the circle with a ruler and compass is im­pos­si­ble, this highlights that the current calculation of π It is not exact and absolute: it is just realistic, but not real.
Even if this was not Lindemann's intention, he thereby armored the π from any advance on its truth; but because of the force that feeds life, as we know, even a small sprout can pierce the rock and flourish in the light, until nothing remains unfinished.

insertion of June 15, 2024
the flower that pierced the rock
And here it is that sprout, right here, even if it risks making obsolete much of the writing that follows, and to which I place it before for its pre­cious, if not decisive relevance.
The fact is that after the long wandering in the garden of geometry, grasp­ing and exposing some of the most elementary and significant forms, evidently it had to and could only be geometry that directed me to an em­blem­at­ic answer to the above question, leading me back to the paradigm of the Great Pyramid of Giza, from which this whole study began.
The pyramid will indicate alone the verification of the accuracy of the π guard­ed by it, against that considered absolute by our mathematics.
With a flat and linear demonstration of two circles with identical radius, but drawn with different algorithms: a basic one with rapid execution, pre­sum­a­bly pertaining to the π 3.14159, the other with the use of Cartesian sine and cosine coordinates, which create the circumference virtually point to point, the defect of the first is visible, indisputable and explained as far as possible; naturally only at enlargements capable of highlighting thou­sandth differences in the diagram; but what matters most, their dif­fer­ent in­ci­dence with an unexpected meeting point of circle, square and tri­an­gle in the virtual scheme of the pyramid, should leave no room to argue that it is right π = 3.14159 instead of π = 3.14460.
Which goes beyond the impossibility of coming to terms with any hy­poth­e­sis by measuring the stones in the present, thanks to a geometric mes­sage imposing itself.
Anyone who remains unconvinced can always continue reading leaving a­side the attribution of π, to explore the special properties of the number 3.14460 modulated for each case described by that supreme golden tri­an­gle which is a world, endowed with such wonders that one never im­ag­ined existed enclosed in a single figure.

Everything is finite, infinite is only 'the Whole'
An unlikely longing that of a science calling itself as exact, based on the as­sump­tion of seeing the difference, if not the structural diversity, between line segment and arc of a circle. to disappear, albeit to unlikely infinitesimal limits.
It is the blatant mistake that indispensable research has 'had' to indulge; but at some point it will need to give up, beginning by recognizing that the real problem has not really been squaring the circle, but the identity of the au­then­tic π.
A π digital in fact, i.e. digitally faceted according to the criterion of a con­ven­tion­al unfeasible infinite, will never be able to contain the curvature!
Only an organic π, the Pi at the root of Phi Φ it can do this, boasting all the req­ui­sites; which if it were sound I could define as analogue; and that does not mean it isn't...

After more than one year, I have achieved an unequivocal demonstration of this unfailing perfection. Since my study of the golden ratio and the π has been greatly amplified since then in various directions, and it is un­like­ly that it will be read all together, I invite you to verify straightway on this last page the existence of a natural π in essential geometry.

I am striving to be convincing, since I realize that the academic world could hardly follow a purely intuitive path - despite the famous "cogito ergo sum".
On the other hand, if we are faced with a hypothesis which by its very nature cannot be demonstrated, it is a fact that the opposite cannot be demonstrated either, since between the two very close definitions of π, one necessarily artificial and the other likely native, or congenital to numerical harmony (as the discussion developed so far should highlight), the first will not be able to assume such a priv­i­lege, given the recognized contradiction implicit in its for­mu­la­tion.
Precisely for this reason, focusing on this discovery and comparison could only take place from an external and independent source.
It will now be needed to become aware of this and state whether to en­trench in the scholastic compromise, or turn to this 'divine wonder' for that supreme case that grants no alternative.
Or finally being able to measure which one is the right one.

If the π has far ex­ceeded the boast of ce­leb­ri­ty, this is not so much due to its un­dis­put­ed sci­en­tif­ic pri­or­i­ty, much less to be­ing prov­en to be per­fect, as to the fact that it still con­sti­tutes the great­est chal­lenge, i­de­al­ly un­re­solved but there­fore elected as a ban­ner that flaunts end­less dec­i­mals quite use­less for the sole pur­pose, more or less con­scious, of con­cealing the in­ev­i­ta­ble com­pro­mise.
The Golden Section Φ in fact – of whose properties the essential notation is:
Φ × (1 + Φ) = 1  ||  1 / φ = φ - 1 , for the alternate values: 5/2 ± .5 – is the only proportional divisor that arises as a needle of the balance be­tween the sum [+] n/Φ = n+Φn and the sub­trac­tion [-] n/(n+Φn) = Φ of the dividend n, and which we will dis­cov­er closely related to π, which is its square root, regulating attraction/repulsion be­tween particles, the grav­i­ta­tion­al balance and the undulatory motion – it deserves just as much; and in­stead it is eve­ry­where re­duced to a pseudo spiral, mostly attributed with questionable cor­re­spond­ence to works by art­ists who did not even know it, even sur­passed, often sub­sumed into the a­nom­a­lous Fibonacci sequence con­ceived only a few cen­tu­ries ago in a fair tournament.
Rest assured that the widespread exaltation of the Fibonacci sequence is merely dissimulating what has not yet been understood about the depth of the golden ratio in creation. The only desirable outcome is that this ru­di­men­ta­ry ladder can bring peopple closer to the mathematical truth of Φ.

It will be the golden ratio itself that will show us the most contiguous so­lu­tion, which can be deduced in the last phase of the work on this site, but which deserves to be anticipated at this point. If only because this re­search of mine is so long and spread across interactive pages for ever­-in­creas­ing arguments, that very few will be willing to follow it to the end.
Let be given a circle with area Φq, that for any grounded theory of π it has a diameter approximating 1,000.
We want to establish its squaring, which from a semantic, or even literal, point of view, is achieved in a surprisingly simple and direct two-way: bringing it to the square or returning it to its square root.
The first case means not only multiplying it by itself, but transforming its area into that of the value Φq of a square polygon, whose side will be ex novo Φ = 0.78615 and the perimeter Φ × 4 = 3.14460…
Now, if we calculate the radius of the circle given by the formula π × r² = Φ,
for r²=Φ /3.1416, the radius will result r=0.25024 then = 0.50024 .
for r²=Φ /3.1446, the radius will result r=0.25000 i.e. = 0.500000.
Continuing the interesting comparative analysis from a semantic point of view, we observe how extremely easily the squaring of the circle can be defined in order to obtain the equivalent area of the two figures.
In practice, just as the area of the circle is brought to the square to reach the square polygon with the same perimeter π, it can be used in the opposite direction as the identical area of a square polygon, whose square root is the measure of the side, i.e.4Φ = 0.88665; nothing is easier, natural and explanatory.
The ratio between the two opposite results, on the recurrent balance of the square and its root, both in linguistic and geometric terms is exactly 4φ = 1, 12784, and so it must be, according to that limit of the golden reason why Φ × φ = 1, while any greater or lesser value of the starting area will not be able to respect it by progressively diverging from.

All this harmonic congruity decays by millesimal deviations, if the π is fixed at 3.1416: tout court, the final ratio is 1.2732(² = 1.62114 and not 1.618).

If the procedure is then applied to a greater area, let's say = 0.88665 , the circle will have a radius = 0.5310 and circumference = 3.339563, the pe­rimeter of the square with the same area will be 3.7665; with π = 3.1416, radius = 0.5312 and circumference 3.337963, value that moves away even more so from 3.7665.
This should confirm that the Golden Section is, so to speak, the com­bi­na­tion of the safe, resulting in the squaring of the circle inherent, or in­te­grat­ed into the Divine Proportion; as long as you acknowledge the π as its square root multiplied by four, which expresses precisely the virtual syn­the­sis of quadrature of the circle, from its U­nique center~vertex of or­i­gin from which universal energy expands and modalizes through the four vectors mak­ing up earthly existence.

It will therefore be a question of choosing whether to prefer the vanities of an inevitably artificial, and therefore not absolutely perfect, historical re­search, or training in the inevitable perfection of Vital Intelligence, even if it goes beyond any possibility of direct demonstration.

This work of mine will illustrate the initial foundations of these statements, which only require common sense, introducing the radical and de­fin­i­tive so­lu­tion of the enigma π, attributable – much more than to for­mu­las and in­fi­nite cal­cu­la­tions that have been wasted for cen­tu­ries in de­bates and ne­ga­tory dem­on­stra­tions – to the golden properties of that extraordinary triangle, yes! a tri­an­gle to begin with, analyzed and described in my pre­vi­ous pres­en­ta­tion of the "The unknown Great Golden Triangle and the Great Pyr­a­mid of Giza", which dates back to 2003 to one of my publications [ISBN 978-­88-­904390­-5-6].
To conclude on the fly, a summary tout court is condensed at the end of the page; but by clicking on the title you can download the 35-pages PDF; or enter to read the prompt English translation in 35 Web pages.


Perhaps the Great Architect needed nothing more
What do "ruler and compass" mean? Simply e O, a segment and a potential circle of variable dimensions, which can be drawn anywhere as long as starting from points (in addition to the first) ideally defined by the progress of the geometric system.

At the human executive level, 'ideally' it implies that the theoretical ac­cu­ra­cy of the layout may require not so much a steady hand, as that no place­ment of the strokes can be obtained by measurement or square or pro­trac­tor, or ar­bi­trat­ed, such as: a tangent or a perpendicular, or an an­gle, however obvious.
Each point from which a path proceeds, in addition to the starting point, must be able to result from an algebraic equation that reflects the design.

Before delving into the profound relation between circle and square, re­vealed by a triangular formation of the Golden Section, it is worth ex­am­in­ing some simple premises, which introduce a symbolic relationship be­tween the tools, concrete and abstract, necessary and sufficient to frame the fun­da­men­tal geometric structures.
Tools that manual geometry makes usable, in the two which are the es­sen­tial terms of reference:
  1. the ruler, suitable for drawing straight lines, or better segments with a beginning and an end, mostly determined by the meeting points of other lines.
    They represent a concrete and static continuity between elementary units, ideally called points, adjacent and not measurable but only draw­a­ble.
  2. the compass, to trace curved lines, circles or arcs of circumferences that define, according to the same principle, constant and dynamic dis­tances, relative to extremities or meeting points of other lines.
    Its aperture can therefore never be autonomously reused for repetitive purposes.
Just as the ruler itself expresses concreteness, materially recognizable as an already existing line, the compass is abstraction consisting of two points that express only their interspace, not usable to reproduce meas­ure­ments, and which could never join by a rectilinear motion.
In practice, while the ruler outlines itself, enabling a measurement from any starting point without the need for the compass, the latter must vir­tual­ly refer to a defined center, and in various cases to a measure called ra­di­us, virtually represented or representative a segment, without which, in­di­vi­d­u­al­ly, it would not be attributable to any significant expression.
A circle therefore cannot start a path, since neither its center, nor any point of the circumference could be assumed to continue, not resulting from a path that identifies it.
It is interesting to meditate on the fact that in order to recognize to the circle the intrinsic property of defining a segment at its first appearance: a sort of installation, or first stone personified by its ray, on which to build, we should be able to access its center; but the center is unreachable as such; thence the only first point that can be fixed and used is the start or end of a line.
Whereas would this radius be already defined by a segment – or the com­pass would define it with its opening, having cen­ter on one end of the line – its two ex­tremes will allow the circle as if to double repeating the center on each of the two, their intersection giving rise to the most ed­i­fy­ing de­vel­opments.

I propose to point out briefly how the circle by its projection is the matrix of each polygon not only in the manual construction, but in the abstract ge­o­met­ric projection as well. No incompatibility precludes squaring the circle but, as we will see, the inhomogeneity of the two formats is re­solved in the inscrutable link that connects the circle to each of its ‘creatures’.

Triangle From Circle The operation for which, having de­ter­mined a segment AB, we will be able to construct on this segment the e­qui­lat­e­ral triangle, the first among the pol­y­gons, appears primary and in­dic­a­tive. Two circles equal and specular with the centers on the ends A and B will define the intersection point C, equidistant from A and B and potential vertex of the triangle ABC, but also of a rhombus, or of the flat section of a tetrahedron.
I will purposely plot whole circles even if simple arcs of circumference would be enough to highlight a certain symbolism, which an attentive mind will be able to grasp.
If two circles will suffice for the triangle, to obtain a square – starting from a fixed base – the process will not be as immediate and may require some concentration.
In this case, in order to induce an epistemological profile to a research that I find emblematic of the direct and exclusive relationship between circle and square, to maintain the step already started, we would like to be all the more exclusive to repeat the same circumference as a mold, having for radius the base segment, as if to say with a fixed angle com­pass.
This will assign it a univocal valency, excluding any artifice or side game of probable alternative paths, leaving to a rule as natural as possible, all its explanatory transparency, and with the least number of clear pas­sages, whereas the use of circles with variable radius can only represent more or less simple constructs of practical geometry.
While undoubtedly dealing with the most significant postulation, it appears to be extraneous to the multifarious expositions spread on the Web; per­haps be­cause of a rapport now officially banned.
I will therefore not give up submitting it to the reader, as I have submitted it to myself, sure that he will find this little trial stimulating, since I have not seen any examples or even indirect answers in numerous sites that il­lus­trate the use of ruler and compass.
He will find the solution here, but not before concentrating on your own with pencil and paper.


The reflections that follow, precisely as they result from a more care­ful con­sid­er­a­tion gained with the study that I present, will be able to play a useful introductory function, by virtue of a more literary and descriptive than math­e­mat­i­cal character.
The proposed study is in fact the result of reflection, much more than cal­cu­la­tions which nature seems to have bestowed without any effort.
It will be up to the reader's judgment to determine which has prec­e­dence and great­er importance.

Beyond the mere geometric assumption, necessarily limited because of human and instrumental conception, if the square with the four di­rec­tions it presents can be considered an emblem of space, matter and static mass, the circle leads back to energy, to the wave vibration or sound, to time, which takes place and develops through its four phases cyclical.

The square needs all its extremes, or a cross as by the four cardinal points, which virtually defines it.
Creating a circle is far more simple: just dropping a drop, its center, on a still water surface, to see not just one circle, but one sequence of rapidly expanding rings with gradually increasing diameters.

Even if the photographic snapshots only capture a moment of the wave impulses that alternate from the center, it is not difficult to re­late their dy­nam­ics to recurring patterns, which naturally depend on the weight, am­pli­tude and correspondence of the liquid [veil] and other factors, but al­ways giving rise and form to distinct concentric fields, of in­creas­ing pro­por­tions, until they get lost in single waves side by side with the ex­haus­tion of the initial thrust.
Just search for “drop of water image” on the net to compare them in quantity.
The circle expresses at the same time equidistance from a source [con­trac­tion] and isometric continuity [expansion].
Its potential, which can virtually be contained even in the center a­lone, man­i­fests itself on an undulating level, to be translated into ac­tion with­in the framework of its phases.
The geometry of the square 'crystallizes' its continuous cir­cu­lar al­ter­na­tion, pro­ject­ing it into distinct and separable areas and di­rec­tions.
These are short 'literary' notes that we would like to consider ab­stract, but not ab­struse, intended to stimulate attention on the nature and po­ten­tial mean­ing of the circle, as a latent expression on which it de­serves to meditate, in order to better assimilate its essence and root deep and func­tion­al, which can undoubtedly be defined as tran­scen­dent.

The or­i­gin [cen­ter] and the man­i­fest wrap­ping of the whole [cir­cum­fer­ence], can be clear­ly rec­og­nized first of all in the more com­plex and rep­re­sent­a­tive ge­o­met­ric sys­tem of world es­o­ter­ic sym­bol­ism (hand­ed down as Śrī Chak­ra yantra, whose com­plete con­struc­tion I have solved years ago with ab­so­lute ac­cu­ra­cy for the first time in our his­to­ry), since it leads back to a cos­mic con­struct u­ni­ver­sal­ly rec­og­nized as such.

Its planimetry extends with surprising features on the series of 8 con­cen­tric cir­cles, whose diameters are in golden ratio progression between them.

A schematic view refers to
a PDF extract of the original text

The same ones ob­served in this fig­ure, ap­plied to the big mas­ter tri­an­gle of the man­da­la, cor­re­spond­ing to the sec­tion pro­file of the great pyr­a­mid of Giza, ac­cord­ing to an ordered ver­ti­cal pro­jec­tion that I pub­lished in the said 2003, al­low­ing us to hy­poth­e­size a se­quence of vi­bra­tory fields with pre­cise res­o­nance with­in the struc­ture; some­thing that a­bout sev­en years lat­er seems to have found out a con­fir­ma­tion at least from a spec­tral a­nal­y­sis, thanks to the in­stal­la­tion of sci­en­tif­ic in­stru­ments (muon de­tec­tors).

Note that this vertical arrangement with the magical tangentiality and synchronicity of the circles in the golden ratio can give body to only one type of triangle: that great golden triangle which I will speak of with ef­fu­sive­ness.
The square circle argument in turn will find further developments, up to a final article on the φ and π in the Great Pyramid, since this homepage is now just introductory.

Aside from being the basis of every orbital relationship, from a­tom­ic to as­tro­phys­ics, the most mys­te­ri­ous mode of af­fir­ma­tion of the cir­cle is un­doubt­ed­ly the spi­ral, a bi­di­rec­tion­al sym­bi­o­sis of the two val­ues of cen­ter and pe­riph­ery, or­i­gin and des­ti­ny, which con­nects the in­fin­i­tes­i­mal to infinity.

We will find these first ingredients as necessary if not suf­fi­cient for the nat­u­ral res­o­lu­tion of the squar­ing of the cir­cle, pre­sented in this work, which shows the cor­rect def­i­ni­tion and gen­e­sis of the π, the only true un­known and key to the mil­len­ni­al enigma.

2x2 = 3.14


© The Watch Publisher 2021

in a circle of diameter 1, π is the Root Square[d of its golden section
by centuries-old convention multiplied ×4 = 3,144605511029693144

Nonetheless, a further, singular to say the least analysis was de­vel­oped as an ap­pen­dix at As­troTime. Not only the Birth­day-Chart of the tra­di­tion­al pi day is be­ing built – cel­e­brat­ed since 1988 as ' the In­ter­na­tion­al Day of Math­e­mat­ics' – with an a­maz­ing as­tro­log­i­cal dis­cov­ery, but even the ge­net­ic struc­ture of the fa­mous and un­known con­stant π is re­solved, which de­spite the most ka­lei­do­scop­ic tar­get prac­tice im­ag­i­na­ble, con­tin­ues to gov­ern the phys­i­cal world and not only.
As I see it, following a loving and persistent meditation exercise, I want to give a further demonstration of the degree of kinship of π with the Golden Ratio, a confirmation that overcomes any objection.
A proof unknown to the global mathematical tradition, being based on the true π.

February 2024 -A proof that cannot be refused


Antonio Alessi © The Watch Publisher, 2003-24

The published e-book is being rendered here page after page, filtering the Google's English translation, to ease the same to other languages. [fixed broken links, 23/02/2022]

To browse, please tap the up and down arrows
at the top and bottom left of the pages. Thank you