This is a very cool, short, geometric proof that the square root of 2 cannot be rational number. The high level summary: If the square root of 2 were rational, it would be possible to construct a 45-45-90 triangle where the sides all have integer lengths. It can be proven that from any such triangle you can construct a smaller triangle with the same properties. This could go on forever, which would be impossible, since there is a lower bound to the positive integers.

A new proof that the square root of 2 is irrational

Found via Daring Fireball.