The Theorem That Lets You Double a Sphere
Cut a solid ball into five pieces, reassemble them into two identical balls. The math says yes.
In 1924, Stefan Banach and Alfred Tarski published a six-page paper in Fundamenta Mathematicae showing that a solid ball in three-dimensional space can be cut into a finite number of disjoint pieces and reassembled, by rigid motions alone, into two solid balls each the same size as the original.
No stretching. No duplication. Just translate and rotate. The current record for the minimum number of pieces is five, established by Raphael Robinson in 1947.
The trick is that the 'pieces' are not the kind of thing a knife can produce. They are non-measurable sets, point clouds so pathologically scattered that no consistent notion of volume can be assigned to them. The construction leans on the axiom of choice to pick one representative point from each of uncountably many equivalence classes.
The paradox does not work in the plane. Banach himself proved that in two dimensions, any equidecomposition preserves area. The reason is group-theoretic: the group of rigid motions in the plane is amenable, while in three dimensions the rotation group SO(3) contains a free subgroup on two generators, which is what makes the duplication possible.
For some mathematicians the result is a good reason to be suspicious of the axiom of choice. For others it is the price of a clean theory of infinite sets, and the moral is just that 'volume' is not a thing every subset of space gets to have.
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