Layman's Guide to the Banach-Tarski Paradox
The BTP posits that it's possible to take a solid sphere, cut it into pieces, rearrange the pieces using nothing more than rotation and translation, and then re-assemble them into two identical copies of the original sphere. "In other words," <I>www.kiro5hin.org</I> explains, "you've doubled the volume of the original sphere."