[ RadSafe ] Pi

[Conrad Sherman]

Archimedes (and also various other early mathematicians such as Eudoxos) is credited with taking one of the biggest steps forward towards more accurate approximations. He came up with the first theoretical method of approximating pi, all based around regular polygons and circles.

The premise is simple; draw a circle, then draw a regular polygon (for example an octagon) inside the circle, so that the vertices of the octagon touch the circle. Then draw another octagon, but this time so that the sides of the octagon touch the circle. If done correctly you should see a octagon, inside a circle, inside another octagon. He discovered that as you increase the number of sides on the polygons, the area between the perimeters of the polygons and the circumference of the circle gets ever smaller. Eventually in theory the area gets so small that it becomes "exhausted", so the larger the number of sides on the polygon, the closer the polygon's perimeter becomes (in size) to the circumference of the circle. This fact can then be used to find a range of values, inside which must lie.

This was a revolution in the way of thinking at the time, and with slight variations, continued to be used for many, many years to come, by mathematicians such as Liu Hui and Ludolph van Ceulen. Van Ceulen found a value of pi correct to 32 digits with this method, a feat in patience and perservance if nothing else. 


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