Infinitesimally rigid polyhedra. II. Modified spherical frameworks

Abstract

In the first paper, Alexandrov’s Theorem was studied, and extended, to show that convex polyhedra form statically rigid frameworks in space, when built with plane-rigid faces. This second paper studies two modifications of these polyhedral frameworks: (i) block polyhedral frameworks, with some discs as open holes, other discs as space-rigid blocks, and the remaining faces plane-rigid; and (ii) extended polyhedral frameworks, with individually added bars (shafts) and selected edges removed. Inductive methods are developed to show the static rigidity of particular patterns of holes and blocks and of extensions, in general realizations of the polyhedron. The methods are based on proof techniques for Steinitz’s Theorem, and a related coordinatization of the proper realizations of a 3 3 -connected spherical polyhedron. Sample results show that: (a) a single k k -gonal block and a k k -gonal hole yield static rigidity if and only if the block and hole are k k -connected in a vertex sense; and (b) a 4 4 -connected triangulated sphere, with one added bar, is a statically rigid circuit (removing any one bar leaves a minimal statically rigid framework). The results are also interpreted as a description of which dihedral angles in a triangulated sphere will flex when one bar is removed.