The Rigidity of Infinite Graphs

Abstract

A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in the normed spaces (R d , · q ), for d ≥ 2 and 1 < q < ∞. Generalisations are obtained for the Laman and Henneberg combinatorial characterisations of generic infinitesimal rigidity for finite graphs in (R 2 , · 2). Also Tay's multi-graph characterisation of the rigidity of generic finite body-bar frameworks in (R d , · 2) is generalised to the non-Euclidean norms and to countably infinite graphs. For all dimensions and norms it is shown that a generically rigid countable simple graph is the direct limit G = lim − → G k of an inclusion tower of finite graphs G1 ⊆ G2 ⊆ . . . for which the inclusions satisfy a relative rigidity property. For d ≥ 3 a countable graph which is rigid for generic placements in R d may fail the stronger property of sequential rigidity, while for d = 2 the equivalence with sequential rigidity is obtained from the generalised Laman characterisations. Applications are given to the flexibility of non-Euclidean convex polyhedra and to the infinitesimal and continuous rigidity of compact infinitely-faceted simplicial polytopes.

Crystal flex bases and the RUM spectrum

“…Consider the crystal framework which is obtained from by adding bonds of length 1 between polar joints whenever this is possible. It is straightforward to see that is sequentially infinitesimally rigid [12, 14] and so is infinitesimally rigid in the strongest possible sense. Note that every 2D subframework parallel to the xy -plane is a copy of the fully triangulated framework .…”

Non-Euclidean braced grids

“…We note that this phenomenon is not possible for an infinite generic bar-joint framework (G, p) in R 2 . Such a framework is infinitesimally rigid if and only if (G, p) is sequentially infinitesimally rigid (Kitson and Power [12]). This means that there exists an increasing chain of infinitesimally rigid finite subframeworks (G n , p) with G equal to the union of the G n .…”

Coboundary operators for infinite frameworks

“…It is evident that G is a spanning tree of G . Moreover, (G , q) is infinitesimally rigid, since it is sequentially infinitesimally rigid (see [12]). Indeed, let G n be the subgraph of G that is determined by the restriction of (G , q) on the half-plane H n := {(x, y) : x ≤ n}.…”

“…These generalise respectively a directed graph and an incidence matrix commonly used in graph theory and the notion of a bar-joint framework and a rigidity matrix used in Euclidean space rigidity theory. Also, the k-frame matrices of Whiteley [18] and the rigidity matrices of bar-joint frameworks in non-Euclidean spaces [12] are particular instances of coboundary matrices. We show that, under suitable conditions, coboundary matrices give rise to bounded operators and we either provide a formula for the operator norm, or provide upper and lower bounds for the operator norm.…”

“…In particular, C(G, q) can be used to determine the infinitesimal rigidity and hence rigidity properties of a bar-joint framework in X. (See for example [12] which considers the special case where X is a finite dimensional p -space).…”