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To be sure, the very formula that calculates the area of the circle as πr², in its referring to the simple square of the radius (which is CBED, ¼ of the whole) denotes that π is assumed with an implicitly quadruple value, such as to extend the calculation on the quadrant to the whole circle; but conceptually it is πr² that has to quad­ru­ple, not the π itself.
Such virtually improper formulation corroborates the concept sustained here, not at all improvised but which on the contrary leads back to the most correct in­ter­pre­ta­tion.

The π: 4th dimension and gravitational interaction

Prolonged reflections, aimed at integrating the premise line of thought, lead me in­sist­ent­ly to the conception of the π as well as as a mediator between square and circle, as a fulcrum of gravity geometry:
and here we are in the fourth dimension, with a symbol π which in my opinion should be thought of as an expression of cosmic Intelligence of a radial rather than circular way (which is a conception suggested by the flawed polygonal representation of the cir­cum­fer­ence); a sort of potential that, likewise a radar, is valid – even in its constant mode – to mark the time-rhythm in the gravitational curvature.

Going from the two Cartesian axes into a three-dimensional sys­tem in which we can dis­cretize any kind of figure, we ob­serve that to trace a cir­cum­fer­ence in 2D, or to outline a sphere in 3D, a 4th factor is needed, re­presented by that radius, a seg­ment or stretch of wire that an­chored to one end, allows to draw in any direction an in­def­i­nite number of points all e­qui­dis­tant from the fixed head, which be­comes the center of a flat or three­-di­men­sion­al whole. Once the length of that wire has been determined, it guarantees the equipollence of all the points visited in space with respect to the center.

It is sufficient to think of a continuous motion originating from this process, to translate the influence of that thread into a gravitational effect; nor is there any other possibility of tracing a real circle than the use of a compass, as an in­stru­ment that guarantees a continuous and homogeneous anchoring; since any dif­fer­ent definition based on 2 or 3 axes would only give rise to a polygonal fig­ure, it being impossible to achieve what commonly – and too compulsively – we fan­ta­size as an infinite number of points ('indefinite' would be appropriate).

This is the basic concept on which it is necessary to focus, in order to go beyond a perspective, albeit historical, now too rudimentary and in any case untrue.