Banach–Tarski paradox

Theorem in set-theoretic geometry

The Banach–Tarski paradox is a theorem in set-theoretic geometry that states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets that can 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 original shape. The pieces themselves are not "solids" in the traditional sense, but infinite scatterings of points.

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