```Date: May 5, 2013 1:43 PM
Author: ross.finlayson@gmail.com
Subject: Re: The relation between circumreference and radius of circle

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) havebeen around for thousands of years basically since Archimedes, andarguably from before then in the early Eastern.  The ratio ofcircumference to diameter pi can be denoted with denoted with theregular polygons that inscribe or circumscribe the circle as thenumber of sides increases.  This is from that properties of regularpolygons centered on the origin have that vertices are farthest fromthe origin and midpoints of sides are closest to the origin, and that(regular) polygons inscribed in the circle have their vertices on thecircle and polygons circumscribed about the circle have the midpointsof their sides on the circle.  As the number of sides increases, thedifference between the distance from origin to vertex and distancefrom origin to midpoint decreases, toward zero.  As the inscribing andcircumscribing become indistinguishable, basically that gets todefining the unit circle as where d_v and d_m meet as n->oo.https://en.wikipedia.org/wiki/Method_of_exhaustionThen for d_vertex = 1 and d_midpoint = 1, starting with squares thathas 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 sidelengths 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 angleof polygons and circumference, and as well into notions like Wallis'identity for pi, then as to Euler's formula via complex roots ofunity.http://www.gusmorino.com/pag3/pi/wallis/index.htmlBailey, Bourwein, and Plouffe discovered a digit extraction algorithmfor pi, that given an index into the (fixed-radix) expansion of pi,gives the value without computing the previous values.http://pi314.net/eng/plouffe.phpThere's a movie.http://www.imdb.com/title/tt0138704/?ref_=fn_al_tt_1The ratio of circumference to diameter, of the circle then, is thatthe disc is the shape with the most area on the plane, for a givenmaximum 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 isalso reflective of natural minimae/maximae and centers/edges.  So,there are lots of formulae for pi, but, given varying regularconstructions of shape and distance, the formulae that resolve withparticular parameterizations to pi = 3.1415926... may not be generallyidentical algebraically.Regards,Ross Finlayson
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