Portal:Mathematics/Featured picture/2008 09

The Banach–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 or possibly with changing their shape. However, the pieces themselves are extremely complicated: they are usually not solids but infinite scatterings of points.