Can we slice an object into two pieces similar to the original?

Question

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?

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. — Mariano Suárez-Álvarez, Sep 07 '10 at 17:16
A couple of questions. 1) What exactly does split mean? 2) Why is this tagged topology? — Aryabhata, Sep 07 '10 at 17:25
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. — Dan Brumleve, Sep 08 '10 at 01:13

score 9 · Answer 1 · answered Sep 07 '10 at 18:09

9

You can certainly take a rectangular box, $2^{1/3} \times 2^{2/3} \times 2$ and slice it into two boxes of size $1 \times 2^{1/3} \times 2^{2/3}$.

answered Sep 07 '10 at 18:09

David E Speyer

61,724

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I'm confused, doesn't this contradict the theorem in the answer Mariano Suárez-Alvarez wrote? – Sep 07 '10 at 18:47
Not impossible then. – Guillermo Phillips Sep 07 '10 at 18:59
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@muad: Boxes aren't strictly convex. – whuber Sep 07 '10 at 19:51
@whuber, thank you - I understand now – Sep 07 '10 at 20:24
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Also 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? – Dan Brumleve Sep 08 '10 at 01:37

score 4 · Answer 2 · answered Sep 07 '10 at 17:22

4

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.

answered Sep 07 '10 at 17:22

Mariano Suárez-Álvarez

135,076

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Is 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? – Dan Brumleve Sep 08 '10 at 01:06
@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) – Mariano Suárez-Álvarez Sep 08 '10 at 09:51

Can we slice an object into two pieces similar to the original?

2 Answers2