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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 balance that regulate it.
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.
If the π has far exceeded the boast of celebrity, this is not so much due to its undisputed scientific priority, as to the fact that it still constitutes the greatest challenge, ideally unresolved but therefore elected as a banner that flaunts endless decimals for the sole purpose, more or less conscious, of concealing the inevitable compromise.
This paper will outline the fundamentals of such statements, investigating possible gaps and yet introducing the radical and intrinsic solution of the π enigma.Preliminary observations
Even if the squaring of the circle has risen to the proverbial limelight as a metaphor of the impossible, as it is now a must to believe and make believe, the initial goal was not to be the direct conversion of circumference and circle into the corresponding squares perimeter and area.
If at the beginning of this unlikely research I wondered, questioning the opportunity of such a claim at the various cultures, over the course of this experience I realize how natural the cause was: anyone who has advanced in the search for the solution formula of the calculation of a circle, which imposes a π, it is in fact unbeaten in the need for a verification, to be carried out only through the intermediary of a square shape, flat, symmetrical and free from irrational interventions.
When we say area =, we aim at a number of squares with a unitary side, as the only matrix capable of expressing it with indisputable conceptual precision; even when the basic unit carried an irrational value.
The actual aim has always been to be able to define that constant – hereinafter referred to as pigreek from the first letter of the Greek word 'perimeter', or another ad personam – that would enable the calculation of circumference and area; a problem that has always been too uncomfortable to emphasize as such.
Putting the spotlight on squaring the circle, which would have had only a reporting and verification purpose and that has never found a solution, would have instead strengthened the pedestal of the research in place, to the point of putting its dictates out of the question; a typical reaction in the face of all sorts of riddles not entirely unraveled.
After the first historical attempts, squaring the circle had in fact become the challenge that would have gripped illustrious scholars in vain in the centuries to come, giving impetus to the most disparate experiments and formulas not exempt from unusual expedients, with no other outcome than to be mocked and crumbled shortly, albeit in the shadow of a compromise of transcendence claimed as such; but as I said, this is only a human component, not properly scientific. 
