The

theorem (or "

paradox") applies to the

surface of a

sphere too (i.e. as well as to the solid sphere, described above).

A sphere can be cut into three equal pieces, such that two of them put together are the same size as the third.

*Take a moment to let that sink in...*

More formally, there exists a partition of the sphere into sets A, B, and C, such that every point of the sphere is in exactly one of the three sets. Each of three sets is congruent to any of the other two, i.e. is the same size and shape.

And the union of A and B (just moving them around, no squashing or bending) is also the same size and shape as C. *Twilight Zone music...*

Now I can't find confirmation of that claim. All the websites I've looked at keep talking balls.

The result about solid balls is an extension of the Banach-Tarski theorem for spheres. Actually it's easier if you delete the centre point. Then the pricked ball can be cut into four pieces and reassembled into two identical pricked balls.

The theorem was proved independently in 1924 by Stefan Banach (1892-1945) and Alfred Tarski and published jointly in 1926. It shows that Lebesgue measure cannot be extended universally to spaces of dimension 3 or higher. But in 1926 Lindeman proved that the line and plane are safe: no bounded subset of the plane can have such a paradoxical decomposition.

In fact the results don't just apply to balls and spheres but to any bounded subset of the space.

Apparently Banach and Tarski felt about this the way Einstein did about the EPR paradox: they intended it to show the absurdity of the Axiom of Choice (AC). But other mathematicians went "Cool!" and could taunt physicists like Feynman with it.

Besides, you can construct similarly bizarre results without AC, such as this: You can take finitely many subsets from an arbitrarily small region and rearrange them to fill (be dense in) an arbitarily big region.

ariels and

AxelBoldt have both questioned the phrasing of this; so I might be misrepresenting something, but I can't tell what. I archived their messages until I would have time to look at hard problems and decide what here needs fixing. But it's been ages since I did any real

maths and I think my brain is too old for its subtleties now, so I'm just going to state their comments:

AxelBoldt said the theorem applies to bounded sets in 3 dimensions *with non-empty interior*. Also, there are paradoxical decompositions in 1 and 2 dimensions; Hausdorff constructed them to show that not all sets are measurable. AxelBoldt then recommends (gasp) looking it up on Wikipedia.

ariels questions the use of the word 'dense' in my last paragraph. You should be able to do it without denseness. You don't need choice for denseness; e.g. divide the points of the cube into 4 sets, according to rationality of the X and Y coordinate s. Then you can rearrange them easily to be dense in the 2x2x2 cube. The density thing is definitely out. ariels doubts you even need to say something like "co-meagre" (i.e. complement of first category or maybe measure zero).