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I suspect it is impossible to split a (any) 3d solid into two, such that each of the pieces is identical in shape (but not volume) to the original. How can I prove this?

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    On a related note: Kimmo Eriksson proves in [The American Mathematical Monthly Vol. 103, No. 5 (May, 1996), pp. 393-400] that a convex polygon is splittable in two properly congruent pieces iff it has rotational symmetry.2010-09-07
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    A couple of questions. 1) What exactly does split mean? 2) Why is this tagged topology?2010-09-07
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    Is this connected to [the Banach-Tarski paradox](http://en.wikipedia.org/wiki/Banach-Tarski)?2010-09-07
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    Was thinking of a 3d analogue to Pythagoras' Theorem.2010-09-07
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    Isaac, I don't think so, because that paradox involves dividing a sphere into non-spherical pieces. But I am also curious if there are any solutions involving pathological shapes or division methods.2010-09-08

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