The Banach-Tarski paradox splits one ball into two

A solid ball can be cut into pieces and rebuilt as two identical balls. Not a trick or an approximation. It's a proven theorem — the catch is in what “pieces” means. Banach and Tarski proved in 1924 that a solid three-dimensional ball can be split into pieces that recombine into two identical copies of the original, using only rotations and translations. The pieces have no well-defined volume — their existence depends on the axiom of choice. They aren't chunks or slices but infinitely complex, dust-like clouds of points. Most mathematicians don't treat it as a flaw, but as a demonstration that maths can depart from physical intuition without contradicting itself.

The Banach-Tarski paradox splits one ball into two

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