Rigidity of certain polyhedra in $ {\bold R}^3 $

Abstract.

We extend the Cauchy theorem stating rigidity of convex polyhedra in ${\bold R}^3$. We do not require that the polyhedron be convex nor embedded, only that the realization of the polyhedron in ${\bold R}^3$ be linear and isometric on each face. We also extend the topology of the surfaces to include the projective plane in addition to the sphere. Our approach is to choose a convenient normal to each face in such a way that as we go around the star of a vertex the chosen normals are the vertices of a convex polygon on the unit sphere. When we can make such a choice at each vertex we obtain rigidity. For example, we can prove that the heptahedron is rigid.