The Banach-Tarski ‘paradox’ is a curious theorem from group theory.
It 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 identical to the one you started with.
I was quite taken by it when I first learned of it, and wrote up these self-contained notes (PDF).