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Crystal frameworks, symmetry and affinely periodic flexes

Research output: Contribution to journalJournal article

Published

Journal publication date31/07/2014
JournalNew York Journal of Mathematics
Volume20
Number of pages28
Pages665-693
Early online date31/07/14
Original languageEnglish

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 Rd. 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.