The other side of the coin

It could be defined as the 'round off of the square', although the purpose remains to define circumference and circle. I wanted to expose it to re-propose on bal­ance the extreme linearity of the procedure as well as the charm of discovery.

Once the key of derivation of the π has been identified, reversing the path leads us to appreciate a sort of message, which brings us closer to the acquisition of all that has been described so far, starting from the Square Root of the Golden Section.

Instead of striving to mirror the Ca circle in a myriad of inscribed segments, we build around it a simple square with side = 1 (as the diameter of the circle it can con­tain), the most direct and familiar of shapes, since it is from it that we will di­rect­ly derive the perimeter and the circular area inscribed.

To this end, in fact, we should only draw as we know the first inner circle Cb with a diameter in golden proportion Φ.

A line tangent to it, reduced to a chord at the two points where it in­ter­sects the larger circle, will be equivalent to that hinge which is Φ, virtually providing the meas­ure­ment of the cir­cum­fer­ence.

The parameter is sub­stan­tial, not only in geometry and math­e­mat­ics but on a phys­i­cal and quantum level. It is the soul of what we call pi.
Eg. four strings in quad­ran­gu­lar sym­me­try add to­geth­er the length of the cir­cum­fer­ence while nec­es­sar­i­ly cros­sing;
and the sym­bol­ism of the fig­ure high­lights an ir­re­place­a­ble ra­tio that helps even more to con­ceive the rel­e­vance of the square root and the gold­en sec­tion, in or­der to ex­trap­o­late the π.

Opening your mind to the na­tive π – not to its circular ver­sion in­tend­ed by a road paved with pol­y­gons – the a­fore­men­tioned intermediary factor between Unity and Φ, becomes equally the conversion factor between square and circle: to carry it out in every sense just multiply the π:
× the perimeter of the circumscribed square to calculate the circumference,
   and / or
× its area to obtain that of the circle!
Have you ever seen greater efficiency?