Rigidity

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Rigid polyhedron

A polyhedron is rigid if it cannot be continuously deformed into another configuration. A rigid polyhedron may have two or more stable forms which cannot be continuously deformed into each other without bending or tearing (Wells 1991).A polyhedron that can change form from one stable configuration to another with only a slight transient nondestructive elastic stretch is called a multistable polyhedron (Goldberg 1978).A non-rigid polyhedron may be "shaky" (infinitesimally movable) or flexible. An example of a concave flexible polyhedron with 18 triangular faces was given by Connelly (1978), and a flexible polyhedron with only 14 triangular faces was subsequently found by Steffen (Mackenzie 1998).Jessen's orthogonal icosahedronis an example of a shaky polyhedron.

Rigid graph

The word "rigid" has two different meaning when applied to a graph. Firstly, a rigid graph may refer to a graph having a graph automorphism group containing a single element.A framework (or graph) is rigid iff continuous motion of the points of the configuration maintaining the bar constraints comes from a family of motions of all Euclidean space which are distance-preserving. A graph that is not rigid is said to be flexible (Maehara 1992).For example, the cycle graph is rigid, while is flexible. An embedding of the bipartite graph in the plane is rigid unless its six vertices lie on a conic (Bolker and Roth 1980, Maehara 1992).A graph is (generically) -rigid if, for almost all (i.e., an open dense set of) configurations of , the framework is rigid in .Cauchy (1813) proved the rigidity theorem, one of the first results in rigidity theory. Although rigidity problems were of immense interest to engineers, intensive mathematical study of..

Laman's theorem

Let a graph have exactly graph edges, where is the number of graph vertices in . Then is "generically" rigid in iff for every subgraph of having graph vertices and graph edges.

Just rigid

A framework is called "just rigid" if it is rigid, but ceases to be so when any single bar is removed. Lamb (1928, pp. 93-94) proved that a necessary (but not sufficient) condition that a graph be just rigid is thatwhere is the number of edges (bars) and is the node of vertices (i.e., pivots; Coxeter and Greitzer 1967, p. 56).

Flexible polyhedron

Although the rigidity theorem states that if the faces of a convex polyhedron are made of metal plates and the polyhedron edges are replaced by hinges, the polyhedron would be rigid, concave polyhedra need not be rigid. A nonrigid polyhedron may be "shaky" (infinitesimally movable) or flexible (continuously movable; Wells 1991).In 1897, Bricard constructed several self-intersecting flexible octahedra (Cromwell 1997, p. 239). Connelly (1978) found the first example of a true flexible polyhedron, consisting of 18 triangular faces (Cromwell 1997, pp. 242-244). Mason discovered a 34-sided flexible polyhedron constructed by erecting a pyramid on each face of a cube adjoined square antiprism (Cromwell 1997). Kuiper and Deligne modified Connelly's polyhedron to create a flexible polyhedron having 18 faces and 11 vertices (Cromwell 1997, p. 245), and Steffen found a flexible polyhedron with only 14 triangular..

Braced polygon

The braced square problem asks: given a hinged square composed of four equal rods (indicated by the thick lines above), how many more hinged rods must be added in the same plane (with no two rods crossing) so that the original square is rigid in the plane. The best solution known (left figure above), uses a total of 27 rods, where , , and are collinear (Gardner 1964; Gardner 1984; Wells 1991). If rods are allowed to cross, the best known solution requires 19 rods (right figure above).Friedman has also considered the minimum number of rods needed to construct rigid regular -gons. In 1963, T. H. O'Beirne found solutions for the pentagon (64 rods), octagon (105 rods), and dodecagon (45 rods). His 64-rod solution for the pentagon is illustrated above (Frederickson 2002, p. 71). The best solutions known with overlapping permitted for polygons with , 4, ... sides require 3, 19, 31, 11, 79, 31, 51, ... rods (A. Khodulyov; cited in Friedman..

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