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The volume as a metric invariant of polyhedra. (English) Zbl 0922.52006

While almost all polygons (with rigid edges and hinged vertices) are flexible, most polyhedra (with rigid faces and hinged edges) are rigid. Indeed, it was not until 1978 that Connelly found the first example of a flexible polyhedron. This, and subsequent examples, did not change its volume when flexed – again, in contrast to flexible polygons. Connelly’s conjecture that this was always the case was proved by I. Kh. Sabitov in 1996; the present paper is a modified English version of that paper.
The proof is extremely elegant, and conceptually straightforward, although the algebra involved is highly nontrivial. Essentially, it is shown that the volume of a polyhedron with a specified combinatorial structure satisfies a certain polynomial identity, whose coefficients depend on the lengths of its edges and face diagonals. As the set of roots of this polynomial is finite, the volume must be constant.
The paper closes with an application to the question of realizability of simplicial 2-complexes with prescribed edge lengths, and the interesting observation that a polyhedron with algebraic edge lengths must necessarily have algebraic volume.
Reviewer: R.Dawson (Halifax)

MSC:

52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
51M25 Length, area and volume in real or complex geometry
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