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The GAP

What needs to be analyzed could in itself be quite simple, if not elementary; but as often happens with simple things, difficult to expose; not to say that it requires such a dose of courage, that only this majestic figure requires me to field, with all the humility of the case.
Experience teaches me that the less certain topics are resolved or understood, the more high-sounding, complicated accents or the privilege of a few, tran­scen­dent a bit like theology are elaborated…

I will therefore try to highlight it in a spontaneous way, since I do not need to report special formulas and calculations, which would probably end up having to be summarized in intuition.
I focused the analysis on the perimeters by default, but the same goes for the areas they enclose.

Let's start from the two extreme points of a side -av- of polygon inscribed or circumscribed to the circumference: theirs is a purely linear distance on a straight line, which represents the length of the segment; there can be no cur­va­ture or chord even in the presence of oval and symmetrical units.

Things change with the presence of a third point, the extreme of the next side -vc-, since together with the first it automatically determines the center of a cir­cum­fer­ence and its potential presence.

12-sided poligons
64-sided poligons (zoom to the next figure)
Now we can translate in -rs- the sum of their lengths, obviously greater than the chord -ac-; but it is still a linear distance, where the distance of a from v ob­served in function of the circle is a curved distance, more or less greater than the polygonal lines; without counting the curvature within the two sides.

The fact is that it ever shall be as far as the exhaustive process may apply; by doubling without limits the number of sides of an inscribed polygon, or other, the curvature factor will never run out, nor will it cease to sidestep con­stric­tions.

According to the concept expressed in the previous figure III, in which ruler and compass are compared, the ruler draws bidirectional straight lines, the compass continuous curves of which each point is at a constant distance from the center.

It is therefore evident that, however short the length of each side may be, even in isometry, only two of its points will be equidistant from the center in respect of the curvature, the others not; yet it seems that only the decreasing length of their sides matters, relying on that fideistic tend to infinity.