# Banach–Tarski Paradox - Obtaining Infinitely Many Balls From One

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 F2 of rank 2 admits a free subgroup of countably infinite rank, a similar proof yields that the unit sphere Sn−1 can be partitioned into countably infinitely many pieces, each of which is equidecomposable (with two pieces) to the Sn−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 Sn−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 Sn−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.

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Science is Christian, not when it condemns itself to the letter of things, but when, in the infinitely little, it discovers as many mysteries and as much depth and power as in the infinitely great.
Edgar Quinet (1803–1875)