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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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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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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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