On a class of inflexible polyhedra
A polyhedron \(P\) is called inflexible if it admits no continuous deformation that changes at least one of its dihedral angles while preserving the congruency of each of its faces. A pyramid is a polyhedron with a vertex (called a main vertex, of which there may be more than one) that is joined by edges to each of the remaining vertices. In this paper, the author proves that a particular class of pyramids in three-dimensional space of constant curvature (with the appropriate conditions on spherical spaces) is inflexible, with extensions to pyramids in higher dimensional constant curvature spaces. In particular, in dimension three, the author considers the following class (referred to as pyramids of class A) of all geometric realizations in \(\mathbb{R}^{3}\) of some given pyramidal triangulation: those that, over each edge of the pyramid, admit a nondegenerate triangle with vertex at a main vertex of the pyramid (the existence of at least one main vertex with this property is assumed). The following theorems are proved: Theorem 1. Every simplicial pyramid of class A in three-dimensional constant curvature space is inflexible (provided that no two edge lengths equal \(\pi/2\) in the case of positive curvature \(K=1\)); Theorem 2. Every pyramid immersed in three-dimensional constant curvature space is inflexible (with the same positive curvature provision of Theorem 1). In higher dimensions, analogous results are proved for the class of pyramids, also referred to as pyramids of class A, satisfying the following: (1) the pyramids are the images of mappings into \(\mathbb{R}^{n}\) of a pyramidal simplicial \((n-1)\)-dimensional pseudomanifold; and (2) the \((n-1)\)-dimensional simplex formed from a main vertex and any \((n-2)\)-dimensional face not incident to that vertex is nondegenerate.
scientific article; zbMATH DE number 6386974