The orbit rigidity matrix of a symmetric framework

TL;DR

This paper introduces the orbit matrix, a new tool for analyzing symmetric frameworks, which helps predict symmetry-preserving mechanisms and offers insights into their rigidity and flexibility across various geometries.

Contribution

The paper presents the orbit matrix as an analog to the rigidity matrix, connecting symmetry analysis with infinitesimal flexibility and stress in frameworks, and extends the understanding of symmetric mechanisms.

Findings

Orbit matrix provides a matrix representation for symmetric infinitesimal flexes and stresses.

Analysis of the orbit matrix predicts symmetry-preserving finite mechanisms.

Framework extends to spherical, hyperbolic, and other geometries with shared symmetry groups.

Abstract

A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connects these two areas and provides a matrix representation for fully symmetric infinitesimal flexes, and fully symmetric stresses of symmetric frameworks. The orbit matrix is a true analog of the standard rigidity matrix for general frameworks, and its analysis gives important insights into questions about the flexibility and rigidity of classes of symmetric frameworks, in all dimensions. With this narrower focus on fully symmetric infinitesimal motions, comes the power to predict symmetry-preserving finite mechanisms - giving a simplified analysis which covers a wide range of the known mechanisms, and generalizes the classes of known mechanisms. This initial exploration of the properties of the orbit matrix also opens up a number of new questions and possible extensions of the previous results, including transfer of symmetry based results from Euclidean space to spherical, hyperbolic, and some other metrics with shared symmetry groups and underlying projective geometry.