When is a symmetric pin-jointed framework isostatic?

TL;DR

This paper explores symmetry-based necessary conditions for 2D and 3D frameworks to be isostatic, revealing restrictions on structural components and identifying only six possible 2D point groups for isostatic frameworks.

Contribution

It introduces symmetry-based necessary conditions for isostatic frameworks and proposes conjectures for sufficient conditions in both 2D and 3D cases.

Findings

Only six point groups can host 2D isostatic frameworks.

Symmetry imposes restrictions on unshifted structural components.

Initial results and conjectures for sufficiency are presented.

Abstract

Maxwell's rule from 1864 gives a necessary condition for a framework to be isostatic in 2D or in 3D. Given a framework with point group symmetry, group representation theory is exploited to provide further necessary conditions. This paper shows how, for an isostatic framework, these conditions imply very simply stated restrictions on the numbers of those structural components that are unshifted by the symmetry operations of the framework. In particular, it turns out that an isostatic framework in 2D can belong to one of only six point groups. Some conjectures and initial results are presented that would give sufficient conditions (in both 2D and 3D) for a framework that is realized generically for a given symmetry group to be an isostatic framework.