As found from a commenter on Hacker News, the Banach-Tarski Paradox is described as thus:

The Banachâ€“Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into several non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are complicated: they are not usual solids but infinite scatterings of points. In a paper published in 1924, Stefan Banach and Alfred Tarski gave a construction of such a “paradoxical decomposition”, based on earlier paradoxical decompositions of a unit interval and of a sphere due to Giuseppe Vitali and Felix Hausdorff, and discussed a number of related questions concerning decompositions of subsets of Euclidean spaces in various dimensions.