Frameworks, Symmetry and Rigidity

TL;DR

This paper develops symmetry equations for various framework systems in R^d, leading to generalized counting rules and necessary conditions for isostaticity, expanding the theoretical understanding of framework rigidity and symmetry.

Contribution

It introduces symmetry equations for the rigidity matrix and Jacobian of diverse frameworks, generalizing Fowler-Guest formulas and counting rules.

Findings

Symmetry equations for the rigidity matrix in R^d are derived.

Generalized Fowler-Guest character formulas are established.

Necessary conditions for isostaticity in asymmetric frameworks are identified.

Abstract

Symmetry equations are obtained for the rigidity matrix of a bar-joint framework in R^d. These form the basis for a short proof of the Fowler-Guest symmetry group generalisation of the Calladine-Maxwell counting rules. Similar symmetry equations are obtained for the Jacobian of diverse framework systems, including constrained point-line systems that appear in CAD, body-pin frameworks, hybrid systems of distance constrained objects and infinite bar-joint frameworks. This leads to generalised forms of the Fowler-Guest character formula together with counting rules in terms of counts of symmetry-fixed elements. Necessary conditions for isostaticity are obtained for asymmetric frameworks, both when symmetries are present in subframeworks and when symmetries occur in partition-derived frameworks.