Finding π in unexpected places
(Patrick Dorey)

Description

The number π is usually defined as the ratio of a circle's circumference to its diameter, but it appears in many other, sometimes surprising, places in mathematics.

Perhaps the most famous example is Buffon's needle, which dates back to the 18^th century. Drop a needle onto a grid of parallel lines. What is the probability that the needle lies on one of the lines? The answer involves π, so in principle you could determine the value of π by dropping a needle (very many) times onto a floor made of parallel strips of wood, and counting the fraction of times that it falls on a join between two strips.

Other examples are much more recent. In 1991, Dave Boll found the number π while investigating the Mandelbrot set, one of the most famous fractals. He posted his observations on an online discussion group; ten years later, Aaron Klebanoff found and published a proof.

Finally, in a 2003 article called "Playing pool with π", Galperin showed that the digits of π can be found by counting collisions in a simple system consisting of two billiard balls of unequal masses, bouncing against a wall. Although the basic idea is simple, this problem has links with some subtle (and unsolved) problems in number theory.

The aim of this project is to explore some of these more unexpected appearances of π, starting with the above three. There is a lot of scope for computer-based experiments, and room for you to extend the study with further examples of your own.

Resources

• A good place to start for Buffon's needle is this Wikipedia article. Follow the links and you will find out about a generalisation called Buffon's noodle.
• Dave Boll posted his observation here; Klebanoff's proof was published in the journal Fractals, Vol 9, 393-402. There's some more discussion here and a nice description in a numberphile video here.
• For a five-minute explanation of the billiard ball problem, see this video. Galperin's paper is here, and a couple of further articles on the topic from Mathematical Intelligencer are here and here.