BanachTarskiParadoxicalDecomposition

This is the famous "doubling the ball" paradox, whereby using the AxiomOfChoice it is possible to take the unit sphere, cut it up into finitely many pieces and using only rotation and translation, and reform the pieces into two balls that are identical to the original.

At least this is the way it is commonly understood. In fact, however, I believe the paradox is even stronger than that. It states:

Any two bounded subsets (of 3-space) with non-empty interior, are equi-decomposable by translations and rotations. This can be illustrated by saying that a marble can be cut up into finitely many pieces and reassembled into a planet. Note that in the decomposition, the pieces are not measurable, and so they will not have "reasonable" boundaries.

This is yet another example of a DecompositionParadox?, one of the first of which was discovered by Hausdorff who noticed that it is possible to "chop up" the unit interval into countably many pieces which (by translation only) can be reassembled into the interval of length 2.

What these paradoxes point out is that the notions of measure and volume are often more complicated than we intuitively expect.

--RaviDesai