Crystal frameworks, symmetry and affinely periodic flexes

Power, Stephen (2014) Crystal frameworks, symmetry and affinely periodic flexes. New York Journal of Mathematics, 20. pp. 665-693. ISSN 1076-9803

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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.