Crystal frameworks, symmetry and affinely periodic flexes

Details Crystal frameworks, symmetry and affinely periodic flexes Crystal frameworks, symmetry and affinely periodic flexes

2011-03-09

arxiv.org/abs/1103.1914v3

Mathematics

Combinatorics

Glide symmetry case proof corrected, extra example given. 27 pages, 4 figures

Scientific paper

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

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