Crystal frameworks, symmetry and affinely periodic flexes

TL;DR

This paper develops symmetry equations for rigidity matrices of crystal frameworks, deriving formulas that connect symmetry properties with flexibility and stress characteristics, advancing understanding of periodic structures.

Contribution

It introduces symmetry equations for rigidity matrices and derives symmetry-adapted Maxwell-Calladine and Fowler-Guest formulas for crystal frameworks.

Findings

Derived symmetry equations for various infinitesimal flexes.

Established symmetry-adapted Maxwell-Calladine counting formulas.

Connected character lists of subrepresentations with framework properties.

Abstract

Symmetry equations are obtained for the rigidity matrices associated with various forms of infinitesimal flexibility for an idealised bond-node crystal framework $\C$ in $\bR^d$. These equations are used to derive symmetry-adapted Maxwell-Calladine counting formulae for periodic self-stresses and affinely periodic infinitesimal mechanisms. The symmetry equations also lead to general Fowler-Guest formulae connecting the character lists of subrepresentations of the crystallographic space and point groups which are associated with bonds, nodes, stresses, flexes and rigid motions. A new derivation is also given for the Borcea-Streinu rigidity matrix and the correspondence between its nullspace and the space of affinely periodic infinitesimal flexes.