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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)

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)

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)

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)

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