BanachTarski

Banach–Tarski paradox is a result in set-theoretic geometry and mathematical logic stating that a solid ball in three-dimensional Euclidean space can be partitioned into a finite number of disjoint pieces that, using only rotations and translations, can be reassembled to form two identical copies of the original ball. The theorem is counterintuitive because it implies a single object can be decomposed into parts that can be rearranged to yield more than one copy of the original.

The result was proven in 1924 by Stefan Banach and Alfred Tarski. Its proof relies on the Axiom of Choice and the existence of non-measurable sets. The standard construction uses a paradoxical decomposition arising from the action of a free group on two generators within the rotation group SO(3). This construction shows that volume is not a property that can be assigned to all subsets in a way that is preserved under isometries.

The paradox does not correspond to anything physically realizable. It uses highly non-measurable, abstract pieces and infinite precision arguments that cannot be implemented in the physical world. Nevertheless, Banach–Tarski has profound implications for the foundations of mathematics, geometric measure theory, and group theory. It demonstrates that certain intuitive notions of volume and dissection fail in the presence of the Axiom of Choice and highlights the distinction between finite, constructive objects and abstract set-theoretic constructions.

In higher dimensions, the phenomena change: in four dimensions, a two-piece paradox is possible, and in higher dimensions, the number of pieces required can vary.