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?
Can we slice an object into two pieces similar to the original?
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geometry
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1On 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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0A couple of questions. 1) What exactly does split mean? 2) Why is this tagged topology? – 2010-09-07
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1Is this connected to [the Banach-Tarski paradox](http://en.wikipedia.org/wiki/Banach-Tarski)? – 2010-09-07
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0Was thinking of a 3d analogue to Pythagoras' Theorem. – 2010-09-07
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0Isaac, 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
2 Answers
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You can certainly take a rectangular box, $2^{1/3} \times 2^{2/3} \times 2$ and slice it into two boxes of size $1 \times 2^{1/3} \times 2^{2/3}$.
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0I'm confused, doesn't this contradict the theorem in the answer Mariano Suárez-Alvarez wrote? – 2010-09-07
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0Not impossible then. – 2010-09-07
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2@muad: Boxes aren't strictly convex. – 2010-09-07
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0@whuber, thank you - I understand now – 2010-09-07
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1Also can we make the pieces different sizes and connect them sideways, like a Golden Rectangle? How many different ways are there to do this in n-space? – 2010-09-08
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It seems that Puppe and others proved that this is impossible for any strictly convex solid. See [B. L van den Waerden, Aufgabe Nr 51, Elem. Math. 4 (1949) 18, 140]
The reference comes from Unsolved problems in geometry by Hallard T. Croft, K. J. Falconer and Richard K. Guy.
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3Is it also true that no strictly convex solid can be divided into _any_ two strictly convex pieces, whether or not they are identical with each other or either with the original? – 2010-09-08
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0@Dab, I would imagine that that is true (for at a point where the two pieces touch which is not at the boundary of the original body at most one of them will be strictly convex) – 2010-09-08