On May 5, 4:58 am, JT <jonas.thornv...@gmail.com> wrote: > Is of course Pi but Pi is irrational and have a unfinished decimal > expansion. How much work has been done in geometry to find a none > irrational substitute to Pi by replace the radius with some other > property of the circle that do not lead to a irrational expression? > > Well my quetsion maybe even not correct stated, but if there was a > relation between some aspect of a polygon and the circumrefrence that > had closed decimal expansion, would that be used instead of the radius > and Pi to calculate the circumreference and area of the circle? > > So is it possible that there could be another expression to calculate > circles circumreference and area that is not an approximation, by > using rounded numbers?
Methods of exhaustion (infinite regress, reductio ad absurdum) have been around for thousands of years basically since Archimedes, and arguably from before then in the early Eastern. The ratio of circumference to diameter pi can be denoted with denoted with the regular polygons that inscribe or circumscribe the circle as the number of sides increases. This is from that properties of regular polygons centered on the origin have that vertices are farthest from the origin and midpoints of sides are closest to the origin, and that (regular) polygons inscribed in the circle have their vertices on the circle and polygons circumscribed about the circle have the midpoints of their sides on the circle. As the number of sides increases, the difference between the distance from origin to vertex and distance from origin to midpoint decreases, toward zero. As the inscribing and circumscribing become indistinguishable, basically that gets to defining the unit circle as where d_v and d_m meet as n->oo.
Then for d_vertex = 1 and d_midpoint = 1, starting with squares that has side length = and side lengths = s_n_v = root(1/2) and s_n_m = 2. For n-gons with n >= 4 this basically describes the ratio of side lengths d_n_v/d_n_m, that goes to one as n->oo.
There are lots of identities for pi in real numbers that are of angle of polygons and circumference, and as well into notions like Wallis' identity for pi, then as to Euler's formula via complex roots of unity.
The ratio of circumference to diameter, of the circle then, is that the disc is the shape with the most area on the plane, for a given maximum distance from the origin. Then the perimeter of that shape, is the minimum among shapes, for the disc, for a given area.
Then, generally referred to as a constant in Euclidean geometry, pi is also reflective of natural minimae/maximae and centers/edges. So, there are lots of formulae for pi, but, given varying regular constructions of shape and distance, the formulae that resolve with particular parameterizations to pi = 3.1415926... may not be generally identical algebraically.