Rigidity of nonconvex polyhedra with respect to edge lengths and dihedral angles

TL;DR

This paper proves that nonconvex polyhedra with certain conditions are uniquely determined by their edge lengths and dihedral angles across Euclidean, hyperbolic, and spherical geometries, expanding classical rigidity results.

Contribution

It introduces new conditions ensuring the rigidity of nonconvex polyhedra and provides a unified proof across different geometries, along with counterexamples and conjectures.

Findings

Polyhedra are uniquely determined by edge lengths and dihedral angles under specified conditions.

Counterexamples show what happens when conditions are violated.

Rigidity results are extended to various families of polyhedra.

Abstract

We prove that every three-dimensional polyhedron is uniquely determined by its dihedral angles and edge lengths, even if nonconvex or self-intersecting, under two plausible sufficient conditions: (i) the polyhedron has only convex faces and (ii) it does not have partially-flat vertices, and under an additional technical requirement that (iii) any triple of vertices is not collinear. The proof is consistently valid for Euclidean, hyperbolic and spherical geometry, which takes a completely different approach from the argument of the Cauchy rigidity theorem. Various counterexamples are provided that arise when these conditions are violated, and self-contained proofs are presented whenever possible. As a corollary, the rigidity of several families of polyhedra is also established. Finally, we propose two conjectures: the first suggests that Condition (iii) can be removed, and the second concerns the rigidity of spherical nonconvex polygons.