Layman's Guide to the Banach-Tarski Paradox. The Banach-Tarski Paradox is well-known among mathematicians, particularly among set theorists.1 The paradox states that it is possible to take a solid sphere (a "ball"), cut it up into a finite number of pieces, rearrange them using only rotations and translations, and re-assemble them into two identical copies of the original sphere. In other words, you've doubled the volume of the original sphere. "Impossible!" I hear you say. "That violates physical laws!" Well, that is what many mathematicians said when they first heard this paradox. But I'd like to point out in this article why this may not be as impossible as one might think at first. 1It revolves around the decades-old debate of whether the Axiom of Choice should be admitted or rejected. More on this in the epilogue. [kuro5hin.org]