Banach-Tarski

A while ago, someone told me of an interesting paradox of Banach and Tarski which is worth relating.

Consider a ball in space of dimension 3 or greater. One can partition the ball into a finite set of subsets that can be reassembled to form two new balls identical to the original ball. In other words, one can cut up a solid ball and reassemble the pieces into two duplicates, with volumes equal to the original. The volume is effectively doubled with no operation other than cutting--no stretching, twisting or deforming of any kind. Moreover, one can move the pieces in such a way that they do not run into each other.

Intuitively (because I don't understand it mathematically), the key to this process is performing cuts of an infinitely convoluted nature. Clearly this is physically impossible, so we will not be multiplying any balls in reality, but it is entirely possible geometerically.

See Banach-Tarski Paradox for a slightly more rigorous explanation.