The Von Neumann Paradox in The Euclidean Plane
In the Euclidean plane, two figures that are equidecomposable with respect to the group of Euclidean motions are necessarily of the same area, therefore, a paradoxical decomposition of a square or disk of Banach–Tarski type that uses only Euclidean congruences is impossible. A conceptual explanation of the distinction between the planar and higher-dimensional cases was given by John von Neumann: unlike the group SO(3) of rotations in three dimensions, the group E(2) of Euclidean motions of the plane is solvable, which implies the existence of a finitely-additive measure on E(2) and R2 which is invariant under translations and rotations, and rules out paradoxical decompositions of non-negligible sets. Von Neumann then posed the following question: can such a paradoxical decomposition be constructed if one allowed a larger group of equivalences?
It is clear that if one permits similarities, any two squares in the plane become equivalent even without further subdivision. This motivates restricting one's attention to the group SA2 of area-preserving affine transformations. Since the area is preserved, any paradoxical decomposition of a square with respect to this group would be counterintuitive for the same reasons as the Banach–Tarski decomposition of a ball. In fact, the group SA2 contains as a subgroup the special linear group SL(2,R), which in its turn contains the free group F2 with two generators as a subgroup. This makes it plausible that the proof of Banach–Tarski paradox can be imitated in the plane. The main difficulty here lies in the fact that the unit square is not invariant under the action of the linear group SL(2, R), hence one cannot simply transfer a paradoxical decomposition from the group to the square, as in the third step of the above proof of the Banach–Tarski paradox. Moreover, the fixed points of the group present difficulties (for example, the origin is fixed under all linear transformations). This is why von Neumann used the larger group SA2 including the translations, and he constructed a paradoxical decomposition of the unit square with respect to the enlarged group (in 1929). Applying the Banach–Tarski method, the paradox for the square can be strengthened as follows:
- Any two bounded subsets of the Euclidean plane with non-empty interiors are equidecomposable with respect to the area-preserving affine maps.
As von Neumann notes,
- "Infolgedessen gibt es bereits in der Ebene kein nichtnegatives additives Maß (wo das Einheitsquadrat das Maß 1 hat), dass gegenüber allen Abbildungen von A2 invariant wäre."
- "In accordance with this, already in the plane there is no nonnegative additive measure (for which the unit square has a measure of 1), which is invariant with respect to all transformations belonging to A2 ."
To explain this a bit more, the question of whether a finitely additive measure exists, that is preserved under certain transformations, depends on what transformations are allowed. The Banach measure of sets in the plane, which is preserved by translations and rotations, is not preserved by non-isometric transformations even when they do preserve the area of polygons. The points of the plane (other than the origin) can be divided into two dense sets which we may call A and B. If the A points of a given polygon are transformed by a certain area-preserving transformation and the B points by another, both sets can become subsets of the A points in two new polygons. The new polygons have the same area as the old polygon, but the two transformed sets cannot have the same measure as before (since they contain only part of the A points), and therefore there is no measure that "works".
The class of groups isolated by von Neumann in the course of study of Banach–Tarski phenomenon turned out to be very important for many areas of mathematics: these are amenable groups, or groups with an invariant mean, and include all finite and all solvable groups. Generally speaking, paradoxical decompositions arise when the group used for equivalences in the definition of equidecomposability is not amenable.
Read more about this topic: Banach–Tarski Paradox
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