We formulate a concise deformation theory for periodic bar-and-joint frameworks in R d 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 d 2 . 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.

Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with n vertices. We show that, modulo planar rigid motions, this number is at most 2n−4 n−2 ≈ 4 n . We also exhibit several families which realize lower bounds of the order of 2 n , 2.21 n and 2.28 n .For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley-Menger variety CM 2,n (C) ⊂ P ( n 2 )−1 (C) over the complex numbers C. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with 2n − 4 hyperplanes yields at most deg(CM 2,n ) zero-dimensional components, and one finds this degree to be D 2,n = 1 2 2n−4 n−2 . The lower bounds are related to inductive constructions of minimally rigid graphs via Henneberg sequences.The same approach works in higher dimensions. In particular, we show that it leads to an upper bound of 2D 3,n = (2 n−3 /(n − 2)) 2n−6 n−3 for the number of spatial embeddings with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to rigid motions.Our technique can also be adapted to the non-Euclidean case.

We formulate a mathematical theory of auxetic behaviour based on one-parameter deformations of periodic frameworks. Our approach is purely geome- tric, relies on the evolution of the periodicity lattice and works in any dimension. We demonstrate its usefulness by predicting or recognizing, without experiment, computer simulations or numerical approximations, the auxetic capabilities of several well-known structures available in the literature. We propose new principles of auxetic design and rely on the stronger notion of expansive behaviour to provide an infinite supply of planar auxetic mechanisms and several new three-dimensional structures.

We prove a rigidity theorem of Maxwell–Laman type for periodic frameworks in arbitrary dimension.

Rigid frameworks in some Euclidian space are embedded graphs having a unique local realization (up to Euclidian motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with n vertices. We show that, modulo planar rigid motions, this number is at most 2n−4 n−2 ≈ 4 n . We also exhibit several families which realize lower bounds of the order of 2 n , 2.21 n and 2.88 n .For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley-Menger variety CM 2,n (C) ⊂ P ( n 2 )−1 (C) over the complex numbers C. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with 2n − 4 hyperplanes yields at most deg(CM 2,n ) zero-dimensional components, and one finds this degree to be D 2,n = 1 2 2n−4 n−2 . The lower bounds are related to inductive constructions of minimally rigid graphs via Henneberg sequences.The same approach works in higher dimensions. In particular we show that it leads to an upper bound of 2D 3,n = 2 n−3 n−2 n−6 n−3 for the number of spatial embeddings with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to rigid motions.