Independent of the practical importance of a systematic calculation of π the algorithm of Archimedes is a big step forward in the basics of mathematics. It is the first documented perception of convergent infinite series and their limit value (limit).
Already in about 450 BC the presocratic philosopher Zenon of Elea theorized deeply about infinite divisibility of space and time. In this context he perplexed his contemporaries with famous Paradoxa. In the example of Achilles and the turtle the fast racer Achilles competes with a turtle that has an advance at the start. While he reaches its starting point, the turtle has again gained an advance. One can continue this reasoning ad infinitum, resulting in the conclusion that Achilles can never reach the turtle.
Interpreted in today´s mathematical understanding, the issue of the paradox is: can a series of infinite steps, none of which is zero, lead to a finite sum (the point or the time of reaching the turtle). At the time of Zenon number theory did not include such a possibility.
Archimedes introduces the transcendental circle number as finite limit value of an infinite series, and he invents a straightforward algorithm for its calculation. The members of this series are the products of the side length of a triangle and of the number of triangles in the circle. The limit value is the product of the limit of the side length (→0) and the limit of the number of triangles (→∞).
In the case of Archimedes and the circle, the mathematical formula of the series members is rather complicated, as square root of sqare root of square root...with always the same argument − which renders it beautifully symmetric.
In Zenon´s Achilles Paradox the formula is much simpler, as the limiting value of a geometric series.
Archimedes´ algorithm includes its own proof by using 2 series that obviously converge to the same limit, an upper sum (external polygon) that is always larger than the limit and a lower sum (internal polygon) that is always smaller than the limit − for a finite order of the polygons.
The relation of the members of upper and lower sum has a simple geometric meaning, with the half angle as the decisive parameter.