New York Journal of Mathematics
Volume 20 (2014) 665-693


S. C. Power

Crystal frameworks, symmetry and affinely periodic flexes

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Published: July 31, 2014
Keywords: Periodic bar-joint framework, rigidity matrix, symmetry
Subject: 52C75, 46T20

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.


Partly supported by EPSRC grant EP/J008648/1

Author information

Department of Mathematics and Statistics, Lancaster University, Lancaster, United Kingdom LA1 4YF