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Some aspects of a mathematical theory of rigidity and flexibility are developed for general infinite frameworks and two main results are obtained. In the first sufficient conditions, of a uniform local nature, are obtained for the existence of a proper flex of an infinite framework. In the second it is shown how continuous paths in the plane may be… (More)

Symmetry equations are obtained for the rigidity matrix of a bar-joint framework in R d. These form the basis for a short proof of the Fowler-Guest symmetry group generalisation of the Calladine-Maxwell counting rules. Similar symmetry equations are obtained for the Jacobian of diverse framework systems, including constrained point-line systems that appear… (More)

A theorem of Laman gives a combinatorial character-isation of the graphs that admit a realisation as a minimally rigid generic bar-joint framework in R 2. A more general theory is developed for frameworks in R 3 whose vertices are constrained to move on a two-dimensional smooth submanifold M. Furthermore, when M is a union of concentric spheres, or a union… (More)

A theory of flexibility and rigidity is developed for general infinite bar-joint frameworks (G, p). Determinations of nondeformability through vanishing flexibility are obtained as well as sufficient conditions for deformability. Forms of infinitesimal flexibility are defined in terms of the operator theory of the associated infinite rigidity matrix R(G,… (More)

We show that planar embeddable 3-connected Laman graphs are generically non-soluble. A Laman graph represents a configuration of points on the Euclidean plane with just enough distance specifications between them to ensure rigidity. Formally, a Laman graph is a maximally independent graph, that is, one that satisfies the vertex-edge count 2v − 3 = e… (More)

The graphs G = (V, E) with |E| = 2|V | − that satisfy |E | ≤ 2|V | − for any subgraph G = (V , E) (and for = 1, 2, 3) are the (2,)-tight graphs. The Henneberg–Laman theorem characterizes (2, 3)-tight graphs inductively in terms of two simple moves, known as the Henneberg moves. Recently, this has been extended, via the addition of a graph extension move, to… (More)

We show that planar embeddable 3-connected CAD graphs are generically non-soluble. A CAD graph represents a configuration of points on the Euclidean plane with just enough distance dimensions between them to ensure rigidity. Formally, a CAD graph is a maximally independent graph, that is, one that satisfies the vertex-edge count 2v − 3 = e together with a… (More)

Symmetry equations are obtained for the rigidity matrix of a bar-joint framework in R d. This leads to a short proof of the Fowler-Guest symmetry group generalisation of the Calladine-Maxwell counting rules. Similar symmetry equations are obtained for the Jacobian of diverse framework systems, including the constrained point-line systems that appear in CAD,… (More)