Dehn invariant of flexible polyhedra

Abstract

We prove that the Dehn invariant of any flexible polyhedron in Euclidean space of dimension greater than or equal to 3 is constant during the flexion. In dimensions 3 and 4 this implies that any flexible polyhedron remains scissors congruent to itself during the flexion. This proves the Strong Bellows Conjecture posed by Connelly in 1979. It was believed that this conjecture was disproved by Alexandrov and Connelly in 2009. However, we find an error in their counterexample. Further, we show that the Dehn invariant of a flexible polyhedron in either sphere or Lobachevsky space of dimension greater than or equal to 3 is constant during the flexion if and only if this polyhedron satisfies the usual Bellows Conjecture, i.e., its volume is constant during every flexion of it. Using previous results due to the first listed author, we deduce that the Dehn invariant is constant during the flexion for every bounded flexible polyhedron in odd-dimensional Lobachevsky space and for every flexible polyhedron with sufficiently small edge lengths in any space of constant curvature of dimension greater than or equal to 3.

16 pages

References in corpus (4)

• Flexible cross-polytopes in spaces of constant curvature
• Embedded flexible spherical cross-polytopes with non-constant volumes
• The bellows conjecture for small flexible polyhedra in non-Euclidean spaces
• The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces

Cited by in corpus (4)

• Necessary conditions for the extendibility of a first-order flex of a polyhedron to its flex
• A sufficient condition for a polyhedron to be rigid
• A flexible polyhedron without self-intersections in Euclidean 3-space, all of whose dihedral angles change during a flex
• The spectrum of the Laplacian in a domain bounded by a flexible polyhedron in ${\mathbb{R}}^d$ does not always remain unaltered during the flex