Banach-Tarski Paradox

First stated in 1924, this theorem demonstrates that it is possible to dissect a Ball into six pieces which can be reassembled by rigid motions to form two balls of the same size as the original. The number of pieces was subsequently reduced to five. However, the pieces are extremely complicated. A generalization of this theorem is that any two bodies in $\Bbb{R}^3$ which do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other (they are Equidecomposable).

References

Stromberg, K. ``The Banach-Tarski Paradox.'' Amer. Math. Monthly 86, 3, 1979.

Wagon, S. The Banach-Tarski Paradox. New York: Cambridge University Press, 1993.