Von Neumann paradox

In mathematics, the von Neumann paradox, named after John von Neumann, is the idea that one can break a planar figure such as the unit square into sets of points and subject each set to an area-preserving affine transformation such that the result is two planar figures of the same size as the original. This was proved in 1929 by John von Neumann, assuming the axiom of choice. It is based on the earlier Banach–Tarski paradox, which is in turn based on the Hausdorff paradox.

Banach and Tarski had proved that, using isometric transformations, the result of taking apart and reassembling a two-dimensional figure would necessarily have the same area as the original. This would make creating two unit squares out of one impossible. But von Neumann realized that the trick of such so-called paradoxical decompositions was the use of a group of transformations that include as a subgroup a free group with two generators. The group of area-preserving transformations (whether the special linear group or the special affine group) contains such subgroups, and this opens the possibility of performing paradoxical decompositions using them.

The following is an informal description of the method found by von Neumann. Assume that we have a free group H of area-preserving linear transformations generated by two transformations, σ and τ, which are not far from the identity element. Being a free group means that all the its elements can be expressed uniquely in the form $\sigma ^{u_{1}}\tau ^{v_{1}}\sigma ^{u_{2}}\tau ^{v_{2}}\cdots \sigma ^{u_{n}}\tau ^{v_{n}}$ for some n, where the $u$ s and $v$ s are all non-zero integers, except possibly the first $u$ and the last $v$ . We can divide this group into two parts: those that start on the left with σ to some non-zero power (we call this set A) and those that start with τ to some power (that is, $u_{1}$ is zero—we call this set B, and it includes the identity).

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