Dehn invariant of flexible polyhedra

TL;DR

This paper proves that the Dehn invariant remains constant during flexion of flexible polyhedra in Euclidean, spherical, and Lobachevsky spaces, confirming the Strong Bellows Conjecture and correcting previous misconceptions.

Contribution

It establishes the invariance of the Dehn invariant during flexion in various spaces, proving the Strong Bellows Conjecture and addressing prior errors in counterexamples.

Findings

Dehn invariant is constant during flexion in Euclidean space for dimensions ≥3.

Dehn invariant is constant in spherical and Lobachevsky spaces if volume remains constant.

Confirmed the Strong Bellows Conjecture and corrected previous counterexamples.

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.