Convex and discrete geometry Polytopes and polyhedra Problem

Hilbert's third problem

Problem

Let $P$ and $Q$ be two polyhedra in $\R^3$ of equal volume. Can $P$ always be cut into finitely many polyhedral pieces that can be reassembled by isometries into $Q$ ?

Remarks

The answer is no. Max Dehn proved in 1900 that the regular tetrahedron and the cube of the same volume cannot be cut into each other in this way. His tool was the Dehn invariant, which does not change under cutting and reassembling: it is $0$ for the cube and nonzero for the regular tetrahedron. Jean-Pierre Sydler proved in 1965 that two polyhedra are scissors-congruent if and only if they have the same volume and the same Dehn invariant. In the plane the analogous question has a positive answer, by the Wallace–Bolyai–Gerwien theorem.

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Tag	`81035XQ5HE`
URI	`m2://81035XQ5HE`
Kind	problem
Writing	W0 stub
MSC	52B45, 51M20, 51M25
Status	This copy of the site has no build data. Run `m2 status 81035XQ5HE`.
About	Polyhedron
Uses	Polyhedron, Dehn invariant
Same as	Wikidata Q2667025 Wikipedia Hilbert's third problem

Hilbert's third problem

Problem

Remarks

Reading path

Further reading

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