Periodic frameworks and flexibility

Abstract

We formulate a concise deformation theory for periodic bar-and-joint frameworks in Rd and illustrate our algebraic-geometric approach on frameworks related to various crystalline structures. Particular attention is given to periodic frameworks modelled on silica, zeolites and perovskites. For frameworks akin to tectosilicates, which are made of one-skeleta of d-dimensional simplices, with each vertex common to exactly two simplices, we prove the existence of a space of periodicity-preserving infinitesimal flexes of dimension at least (d2). However, these infinitesimal flexes need not come from genuine flexibility, as shown by rigid examples. The changes implicated in passing from a given lattice of periods to a sublattice of periods are illustrated with frameworks modelled on perovskites. © 2010 The Royal Society.