The Banach-Tarski Paradox

This is one of my favourite mathematical topics. There are many ways to state it, but I will give the simplest and most general:

In theory, it is possible to take any object (usually a solid ball) and cut it into a finite number of pieces, apply a rigid transformation (so no stretching is allowed) to each piece, and reassemble them into any other object of any size, or multiple objects of any size!

So you could take a walnut, cut into a few pieces, and reassemble it into a copy of the planet Earth! Of course this only works for a theoretical walnut in which there is no discrete atomic structure, but still the fact that it could be done even in theory is astonishing!

(For more detailed information, try the sites here, here, and here. )

This also leads to a very interesting question in physics. Since normal objects are only excluded because they have discrete structure on the small scale (atoms and molecules), what allows volume to be defined in spacetime? If spacetime is continuous as most scientist believe, then it is possible to apply the Banach-Tarski paradox to a piece of empty space and change its volume arbitrarily. The only way out would be if spacetime itself is discrete on some scale (and in fact this has been proposed in numerous different models of quantum gravity). But does the BT paradox actually require discrete spacetime?