Flexible polyhedra and their volumes

Abstract

We discuss some recent results on flexible polyhedra and the bellows conjecture, which claims that the volume of any flexible polyhedron is constant during the flexion. Also, we survey main methods and several open problems in this area.

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

“…Another proof, which is also limited to the case d = 3, was published in 1997 in [8] (see also an expository paper [30]). For d 4, the Bellows Conjecture was proved by A.A. Gaifullin in 2014 in [12] and [13] (see also an expository paper [17]).…”

“…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].…”

Zero‐sum cycles in flexible polyhedra

“…Gluck [13] then showed that “almost all” simply connected polyhedra are rigid as well. The research then focused on finding necessary and sufficient conditions for the flexibility of polyhedra, and it is still an active research field, as witnessed by, among others, recent works by Gaifullin [8] and Alexandrov [2, 3]. In the latter, Alexandrov proved that the edge lengths of a flexible (orientable) polyhedron with triangular faces must satisfy a $$\mathbb {Q}$$‐linear relation.…”

Zero-sum cycles in flexible polyhedra

“…Gluck [Glu75] then showed that "almost all" simply connected polyhedra are rigid as well. The research then focused on finding necessary and sufficient conditions for the flexibility of polyhedra, and it is still an active research field, as witnessed by, among others, recent works by Gaifullin [Gai18] and Alexandrov [Ale19,Ale20].…”