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) 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.

https://en.wikipedia.org/wiki/Method_of_exhaustion

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.

http://www.gusmorino.com/pag3/pi/wallis/index.html

Bailey, Bourwein, and Plouffe discovered a digit extraction algorithm

for 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.php

There's a movie.

http://www.imdb.com/title/tt0138704/?ref_=fn_al_tt_1

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.

Regards,

Ross Finlayson