An oversimplified explanation of the mind-numbingly paradoxical Banach-Tarski theorem states that a solid sphere can be cut into non-overlapping pieces (geometricians would say that such a ball has been "decomposed"), and reassembled in a new arrangement, such that the end result is two identical copies of the original sphere — conservation of mass be damned.
Confused? BoingBoing's Maggie Koerth-Baker gives a good rundown:
Imagine a ball. Now imagine cutting that ball up into a finite number of pieces. Six, maybe. Or five. The Banach-Tarski paradox proposes that you could take those pieces and, without stretching or expanding them in any way, use them to form two balls identical to the first. Basically, you've just created mass out of nothing. That is, to put it mildly, not supposed to be able to happen. Thus, the part about the paradox.
Still confused? Me too. But don't let that stop you from appreciating the video up top. In it, members of the University of Copenhagen mathematics department exploit the Banach-Tarski paradox, decomposing and reassembling a multitude of oranges to the (slightly modified) tune of Duck Sauce's "Barbara Streisand." On a related note, this video will make you realize how much more you wish you partied with mathematicians.
[Spotted on BoingBoing]