**Obtaining Infinitely Many Balls From One**

Using the Banach–Tarski paradox, it is possible to obtain *k* copies of a ball in the Euclidean *n*-space from one, for any integers *n* ≥ 3 and *k* ≥ 1, i.e. a ball can be cut into *k* pieces so that each of them is equidecomposable to a ball of the same size as the original. Using the fact that the free group *F*_{2} of rank 2 admits a free subgroup of countably infinite rank, a similar proof yields that the unit sphere *S**n*−1 can be partitioned into countably infinitely many pieces, each of which is equidecomposable (with two pieces) to the *S**n*−1 using rotations. By using analytic properties of the rotation group SO(*n*), which is a connected analytic Lie group, one can further prove that the sphere *S**n*−1 can be partitioned into as many pieces as there are real numbers (that is, pieces), so that each piece is equidecomposable with two pieces to *S**n*−1 using rotations. These results then extend to the unit ball deprived of the origin. A 2010 article by Vitaly Churkin gives a new proof of the continuous version of the Banach–Tarski paradox.

Read more about this topic: Banach–Tarski Paradox

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