Notes on the Rigidity of Graphs

The first reference to the rigidity of frameworks in the mathematical literature occurs in a problem posed by Euler in 1776, see [8]. 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 3-space, subject to the constraints that the shapes of the panels and the adjacencies between pairs of panels 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 [3] in 1813. Indeed Cauchy proved an even stronger result. Suppose P1 and P2 are two convex polyhedra. If there is a bijection between the faces of P1 and P2 which preserves both the shapes of faces and the adjacencies between pairs of faces, then P1 and P2 are congruent. Cauchy’s strengthening of Euler’s conjecture is not true for all polyhedra, however. Consider the icosahedron, P1. We can reflect one of the vertices of P1 in the plane containing it’s five neighbouring vertices to obtain a non-convex polyhedron P2 with the same faces and adjacencies between faces as P1. Clearly P1 and P2 are not congruent. This example is not a counterexample to Euler’s original conjecture since the reflection is not a continous motion from P1 to P2. Gluck [9] showed in 1975 that Euler’s conjecture is true when P is a ‘generic’ polyhedron i.e. there are no algebraic dependencies between the coordinates of the vertices of P . It follows that ‘almost all’ polyhedra are