The magic and mystery of "pi" ()

Synopsis.
The number "pi" has been a fascinating object for thousands of years. Intimately connected with a circle, it is not an easy object to get hold of completely rigourously. In fact the two main theorems associated to it--the formulas for the area and circumference of a circle of radius pi--are usually simply assumed to be true, on the basis of some rather loose geometrical arguments in high school which are rarely carefully spelt out.

Here we give an introduction to some historically important formulas for pi, going back to Archimedes, Tsu Chung-Chi, Madhava, Viete, Wallis, Newton, Euler, Gauss and Legendre, Ramanujan, the Chudnovsky brothers and S. Plouffe, and culminating in the modern record of ten trillion digits of Yee and Kondo. And I also throw in a formula of my own, obtained from applying Rational Trigonometry to Archimedes' inscribed regular polygons.

It should be emphasized that the formulas here presented are not ones that can easily be rigorously justified, relying as they do on a prior theory of real numbers and often Euclidean geometry. The lecture ends with some speculations about the future role that "pi" might play in our understanding of the continuum--a huge problem which is not properly appreciated today.
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