Dehn Invariant and Scissors Congruence of Flexible Polyhedra

Abstract

We prove that the Dehn invariant of any flexible polyhedron in n-dimensional Euclidean space, where n ≥ 3, is constant during the flexion. For n = 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 S n or Lobachevsky space Λ n , where n ≥ 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 3.2010 Mathematics Subject Classification. 52C25, 52B45 (Primary), 51M25, 32D99 (Secondary).

Necessary conditions for the extendibility of a first-order flex of a polyhedron to its flex

“…In [16], among other results, A.A. Gaifullin and L.S. Ignashchenko proved that the Dehn invariants of any flexible oriented polyhedron with non-degenerate triangular faces in R 3 do not alter during a flex.…”

“…However, the real flowering of the theory of flexible polyhedra began in the mid-1970s when Robert Connelly constructed a sphere-homeomorphic self-intersection free flexible polyhedron [7]. Now we already know that any flexible polyhedron in Euclidean 3-space preserves the total mean curvature (see [1]), enclosed volume (there are especially many articles devoted to this issue, so we point out several of them in chronological order: [23], [24], [25], [10], [26], [27], [14], and [13]), and Dehn invariants (see [16]). The reader can find more details about the theory of flexible polyhedra in the above mentioned articles, as well as in the following review articles, which we list in chronological order: [21], [8], [29], [28], and [15].…”

“…Namely, in Section 3 we derive the desired equations using the fact that the Dehn invariants remain unaltered during the flex, a fact that was established for the first time by A.A. Gaifullin and L.S. Ignashchenko [16]. In Section 4, we derive another set of desired equations using the notion of the rigidity matrix.…”

A sufficient condition for a polyhedron to be rigid

“…(d) during the flex, they necessarily keep unaltered the Dehn invariants, see [21]; (e) the notion of a flexible polyhedron can be introduced in all spaces of constant curvature of dimension 3 and above (see [36], [37], [16], [17], [19], as well as in pseudo-Euclidean spaces of dimension 3 and above (see [3]); moreover, it is known that in many of these spaces flexible polyhedra do exist and possess properties similar to properties (a)-(d) (see [24], [14], [15], [18], [20]).…”

“…Our arguments essentially use one auxiliary statement, proved by A. A. Gaifullin and L. S. Ignashchenko in [21], see Lemma 1 in Section 2.…”