Banach–Tarski paradox
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 of rank 2 admits a free subgroup of countably infinite rank, a similar proof yields that the unit sphere S can be partitioned into countably infinitely many pieces, each of which is equidecomposable (with two pieces) to the S 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 can be partitioned into as many pieces as there are real numbers (that is, 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} pieces), so that each piece is equidecomposable with two pieces to S using rotations. These results then extend to the unit ball deprived of the origin. A 2010 article by Valeriy Churkin gives a new proof of the continuous version of the Banach–Tarski paradox.
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