Calculating Pi...          

There are a number of ways of calculating Pi.

Archimedes stated that if you inscribe a regular polygon, and you also ascribe a regular polygon, both with 3*2n-1 then you can define an increasing sequence, b1, b2, b3.. and another a1, a2, a3.. both with limits of Pi. Using trig we can define each of these terms as;

an = K  tan(p/K),  bn = K sin(p/K),

Archimedes calculated the a6 and b6 terms by purely geometrical means, and showed that b6 < p < a6. Using polygons of up to 96 sides he defined Pi as 3.1418. Other people used this technique to find very large numbers of decimal places. (See the history of Pi section).

p/4 = 1 - 1/3 + 1/5 - 1/7 + ....

This is not a lot of use as it takes 10000 terms to find 4 correct decimal places.

This is a refinement of Gregory's formula and was developed by Machin in 1706.

You can try computer simulation of this method which is fun if you like that kind of thing at:

http://www.angelfire.com/wa/hurben/buff.html

Tenuous link:            American Pie

 

Picture taken from http://www.geocities.com/Hollywood/Studio/9480/americanpie2.htm