The Banach-Tarski paradox is a curious theorem from group theory. I was quite taken by it when I first learned of it, and wrote up these self-contained notes (PDF).

Banach-Tarski shows how to conjure things out of thin air …but not with magic, just with the axiom of choice. You also need an infinitely divisible object, but only on short loan, because after you cut it up and reassemble it, you’ll have two!. Basically, it gives a method for taking a ball, cutting it up into many tiny pieces, and then putting the pieces back together to make two balls just identical the one you started with.