\ The wonderful π - Squaring the circle

2x2 = 3.14

π = SQUARING THE CIRCLE

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.

DISCIPLINE OF π AND NATURAL SQUARING OF THE CIRCLE - © 2021 The Watch Publisher

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.

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 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...
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.

This work of mine will illustrate the 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.
PREMISE

THE LANGUAGE OF COM­PASS AND A STRAIGHT­EDGE

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 SQUARE AND THE CIRCLE

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).

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


DISCIPLINE OF THE π AND NATURAL SQUARE OF THE CIRCLE

© 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.

ENTER

Antonio Alessi © The Watch Publisher, 2003-22

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