The Banach-Tarski theorem (1924) states: a solid ball can be decomposed into finitely many non-measurable pieces and reassembled—using only rigid rotations—into two balls identical to the original. The key is the free subgroup F₂ ⊂ SO(3).
The construction uses two rotations a (72°) and b (arccos(1/3) ≈ 109.47°) that generate a free group F₂ in SO(3). This group acts on the sphere without fixed points on a dense set, enabling the paradoxical decomposition. No physical object can be so divided — the pieces are not Lebesgue measurable.