You Can Cut a Ball Into Five Pieces and Reassemble Two of It

Stefan Banach and Alfred Tarski published the result in 1924. Take a solid ball. Cut it into a finite number of pieces — five is enough — and using only rotations and translations, reassemble those pieces into two solid balls, each the same size as the original. No stretching, no overlap, no extra material added. The volume just doubles.

This sounds like it can't be true and isn't, in any physical sense. The pieces aren't shapes you could carve with a knife. They're sets of points so jagged and dispersed that they don't have a measurable volume — they're "non-measurable," in the technical sense. The whole construction depends on the Axiom of Choice, the rule of set theory that lets you select one element from each of an infinite collection of sets, even when no procedure for choosing exists.

That axiom is what does the heavy lifting. Without Choice (and with only the weaker axioms ZF + Dependent Choice), you can prove the paradox is not a theorem. With it, you get to assemble these unreal pieces, and once you have them, doubling a ball is a routine consequence of how rotations of three-dimensional space behave.

The paradox is usually presented as a reason to doubt the Axiom of Choice. Most working mathematicians keep Choice anyway, because the alternative throws out enormous amounts of useful analysis along with the weird pieces. Banach-Tarski lives on as the rare result that shows mathematics happily separating itself from physical intuition. A ball can have a definite volume; sums of certain parts of it cannot. The paradox is that nothing has gone wrong — the universe of sets is just bigger and stranger than the universe of solid objects.