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.
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.
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 organicπ, 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) = Φof 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 √Φ_{q}, 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 Φ_{q} 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.^{4} √Φ = 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 ^{4}√φ = 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 asits 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.
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:
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.
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’.
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 question is:
Outline a square on side AB with a fixed radius compass.
If you clicked unwittingly, you can still wait and try it yourself…
otherwise proceed.
In retrospect (Jan. 2023), I can only add that when I intended to open this 'parenthesis', I had no idea of the extraordinary path that would have awaited me, to the point of creating another specific domain dedicated to the golden spiral; I leave the same surprise to those who want to delve into the discovery of things of which they would not have suspected the existence, and which would be too much to anticipate.
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.
Its planimetry extends with surprising features on the series of 8 concentric circles, whose diameters are in golden ratio progression between them.
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.
2_{x}2 = 3.14
DISCIPLINE OF THE π AND NATURAL SQUARE OF THE CIRCLE
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]
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