What do you get when you take something combinatorial (the free group on two generators), cram it any which way into something geometric (SO(3) acting on the sphere), and take for granted the ability to pick apart the resulting mess (the axiom of choice)?

Banach-Tarski is what. We'll prove that one can "double the ball," i.e. there is a bijection B\to B\cup B that is defined piecewise by isometries on a finite partition of B.

Free bonus "joke" - Q: What's an anagram of "Banach-Tarski"? A: "Banach-Tarski, Banach-Tarski"