On Sat, 13 Mar 1999, Joth Tupper wrote:
> I am beginning to feel sorry for starting these threads, especially as they
> are off topic. ...
Well, off topic it is, but I would like to add a few remarks for the
benefit of the curious who might not know about the theorem yet.
...
> Take two solid 3 dimensional spheres S1 and S2 of unequal volume (i.e.,
> different radii). The volumes can be as different as you wish -- Newman
> suggests taking one the size of a pea and the other the size of the sun.
> You can divide S1 into finitely many (n) disjoint pieces A1,A2,...,An and
> S2 into finitely many disjoint pieces B1,B2,...,Bn such that (after
> suitable rearranging, if needed), A1 is congruent to B1, A2 is congruent
> to B2, and so on until An is congruent to Bn. [Note that the word
> "congruent" applies -- not merely "similar".]
...
The way I prefer to state the theorem is that the pieces Ai are moved,
i.e., rotated and translated; Bi is just Ai in a different position.
Also, you do not need to use spheres: any bounded subsets of R3 with
non-empty interior will do: you may cut any small stone into finitely many
pieces and rearrange them (as if they were pieces of a 3D jig-saw puzzle)
in order to produce a real-size replica of your favorite sculpture.
The same theorem (with practically the same proof) applies to any Rn, n>2.
For R2, however, this does *not* work: you can not cut a measurable subset
of the plane and rearrange the pieces in order to obtain a measurable set
of different area. However, you *can* cut a round disk into finitely many
parts and rearrange them to obtain a square of equal area.
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