CentralNotice Banach–Tarski paradox From Wikipedia, the free encyclopedia Jump to: navigation , search Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original? The Banach–Tarski paradox is a theorem in set-theoretic geometry , which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets , which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces. [ 1 ] A stronger form of the theorem implies ...