The only case [im] possible in which the sum of |
n consecutive cases would correspond to the length of the arc they form, would be that one in which the units in question were indeed sides long only two 'points', a purely dialectical but unlikely supposition, which I suppose unattainable even by the infinitesimal calculus.
In practice, I argue that even when a set has reached its exhaustive approximation, in one sense or another, the same factor, transcendent or not, will always insist on justifying the curvature, for the realistic principle that a perimeter or area they only make sense if they are made up of tangible, if not also usable, units.
From what has been ascertained, it can likely be deduced that even at the limit of geometric counterfeiting, the microscopic arc of circumference in place will always be greater than the underlying chord, just as the whole circle will be greater than the polygon it contains, at least for three thousandths.
Compared to the current focus of pi, if we really want to maintain the usual mode, we get a – let's say – a '
Coefficient of Curvilinear Constancy'
1,000955… from whose presence it is given to deduce the threshold beyond which the calculation is lost in the abstract of infinity, eluding the faculty of a “real” measurement.
The natural prowess of this value denounces the limit of the theoretical framework, comparing the 'official' π with the effective
π, just detected and which else cannot to be;
and the most exciting aspect is that it is precisely ruler and compass that take it out of the oven, ready for use.
I would not exclude that if our ancestors had known this triangle and its aforementioned properties, Golden Section in primis, they would not have hesitated to use the
π that it contains and manifests, thence move on to squaring the circle with all the ease that will follow shortly, saving many torments; but this does not demonstrate what only an instrumental survey, precise beyond the .003 would confirm, assuming this need is felt.
Probably only a cylindrical vessel and a cube of proper proportions, filled with the same quantity of stable liquid, could reserve that satisfaction. …
Self-criticism is science
However, this analysis suggests that the
π measurement may differ from the calculated one, since this cannot be deduced from a linear dimensioning without requiring a precise and dedicated rectification, precisely because that direct superposition of perimeters and areas cannot take place in an elastic way; although this is all that has been compulsively pursued since the very beginning.
After all we are only a few thousand years old, and we like to think that they are who knows how many; (nay, we are going through the worst phase of our history) .
Do you really think that
π could be an entity disconnected from the most seductive harmonies of space and time? only the fulcrum of an autonomous carousel, entrusted entirely to a range of speculative arrangements, even if so far justified?
the maximum vehicle of structural equilibrium together with the hinge of the Golden Section, to be reduced rounded by a heap of fragments, as the unique chance for the methodology?