Rigidity of Graphs and Frameworks

The first reference to the rigidity of frameworks in the mathematical literature occurs in a problem posed by Euler in 1776. Consider a polyhedron P in 3-space. We view P as a ‘ panel-and-hinge framework’ in which the faces are 2-dimensional panels and the edges are 1-dimensional hinges. The panels are free to move continuously in 3space, subject to the constraints that the shapes of the panels and the adjacencies between them are preserved, and that the relative motion between pairs of adjacent panels is a rotation about their common hinge. The polyhedron P is rigid if every such motion results in a polyhedron which is congruent to P . Euler’s conjecture was that every polyhedron is rigid. The conjecture was verified for the case when P is convex by Cauchy in 1813. Gluck showed in 1975 that it is true when P is ‘generic’ i.e. there are no algebraic dependencies between the coordinates of the vertices of P . Connelly finally disproved the conjecture in 1982 by constructing a polyhedron which is not rigid. I will describe results and open problems concerning the rigidity of various other types of frameworks. I will be mostly concerned with the generic case for which the problem of characterizing rigidity reduces to a problem which depends only on the incidence graph of the framework. ∗School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, England. e-mail: [email protected].