The Almost Periodic Rigidity of Crystallographic Bar-Joint Frameworks

A crystallographic bar-joint framework, C in R, is shown to be almost periodically infinitesimally rigid if and only if it is strictly periodically infinitesimally rigid and the rigid unit mode (RUM) spectrum, Ω(C), is a singleton. Moreover, the almost periodic infinitesimal flexes of C are characterised in terms of a matrix-valued function, ΦC(z), on the d-torus, T, determined by a full rank translation symmetry group and an associated motif of joints and bars.