Zero-sum cycles: a necessary condition for the flexibility of polyhedra

Abstract

It is known that all convex polyhedra are rigid as well as almost all simply connected ones. Hence, it is a non-trivial task to provide a flexible polyhedron. However, flexible polyhedra exist, for instance Bricard’s octahedra or Steffen’s polyhedron. We provide a necessary condition on the flexibility of a polyhedron, namely, if a polyhedron flexes, then there is a cycle of edges with a sign assignment such that the signed sum of the edge lengths is zero. Our result is a generalization of the analogous statement for suspensions. This is a joint work with Matteo Gallet, Georg Grasegger and Josef Schicho.