Continuous deformations of polyhedra that do not alter the dihedral angles

TL;DR

This paper investigates nonconvex polyhedral surfaces in hyperbolic and spherical spaces that can undergo continuous deformations without changing dihedral angles, revealing invariance of volume and existence of tilings with similar properties.

Contribution

It demonstrates the existence of nontrivial deformations preserving dihedral angles in hyperbolic and spherical polyhedra, and explores their geometric properties and tiling implications.

Findings

Volume remains constant during dihedral angle-preserving deformations.

Surface area and curvatures can vary even when dihedral angles are preserved.

Existence of tilings with nontrivial dihedral angle-preserving deformations.

Abstract

We prove that, both in the hyperbolic and spherical 3-spaces, there exist nonconvex compact boundary-free polyhedral surfaces without selfintersections which admit nontrivial continuous deformations preserving all dihedral angles and study properties of such polyhedral surfaces. In particular, we prove that the volume of the domain, bounded by such a polyhedral surface, is necessarily constant during such a deformation while, for some families of polyhedral surfaces, the surface area, the total mean curvature, and the Gauss curvature of some vertices are nonconstant during deformations that preserve the dihedral angles. Moreover, we prove that, in the both spaces, there exist tilings that possess nontrivial deformations preserving the dihedral angles of every tile in the course of deformation.