Infinitesimally rigid polyhedra. II. Modified spherical frameworks

Abstract

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-connected spherical polyhedron. Sample results show that: (a) a single fc-gonal block and a fc-gonal hole yield static rigidity if and only if the block and hole are fc-connected in a vertex sense; and (b) a 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.

On the number of realizations of certain Henneberg graphs arising in protein conformation

“…The complete rigidity matrixR refers to the case where G is the full clique on V . The rigidity matrix is used to give a stronger definition of rigidity (namely that of infinitesimal rigidity) which is used in statics [47,48]. Intuitively, infinitesimally rigid structures are resilient to collapse when certain forces are applied to them.…”

Tensegrities and Global Rigidity

“…There is a good body of work devoted to the detection and understanding of local rigidity. One can see some good surveys in [31,35,36].…”

“…See [34,1,37] for example. See [31,38,35,36,32] for the first-order theory, and see [17] for the more general approach that combines the first-order theory and the stress matrix approach that is developed here. In computational geometry, there was the carpenter's rule conjecture, inspired by a problem in robot arm manipulation.…”

The Rigidity of Spherical Frameworks: Swapping Blocks and Holes

“…In [5,17], we introduced a construction process for block and hole polyhedra. This construction permitted us to extract the generic static rigidity properties of the framework of the block and hole polyhedra from the generic behaviour of an underlying 'base' block and hole polyhedron.…”

“…In this paper, a number of the construction details from [5] and [17] will not be relevant. Thus, we give a simplified definition of a block and hole polyhedra, starting with the definition of an abstract spherical polyhedron.…”