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.