A counterexample to the rigidity conjecture for polyhedra

Abstract

Are closed surfaces rigid? The conjecture that in fact they all are r i g i d a t least for polyhedra has been with us a long time. We propose here to give a counterexample. This is a closed polyhedral surface (topologically a sphere), embedded in three-space, which flexes. (See Gluck [5] for definitions and some references for the history of the problem.) Certain ambiguities arising from definition io of the eleventh book of Euclid's Elements have led many to conjecture the rigidity of closed surfaces. In 1813 Cauchy [2] proved that strictly convex surfaces were rigid, and this result is the basic tool for many other rigidity theorems. Recently Gluck [5] has shown that almost all simply connected closed surfaces are rigid. On the other hand we have shown [3] that there are immersed surfaces which flex. The ideas in [3] are part of the motivation behind the example described here. The first step is to find an example of an immersed flexible sphere that is not only immersed but has just two singular points in its image. Locally the singular points look like two dihedral surfaces that intersect at just one point in their edges. The next step is to alter the polyhedron only in the neighborhood of these singular points in such a way that the dihedral surfaces flex as before, but one dihedral surface is crinkled such that near the intersection point it is pushed in. When this is done the resulting polyhedron still flexes, but the singular points have been erased; no new ones have been created, so it is embedded.