2/16/2012

Ileana Streinu
Smith College

Rigidity and origami

Cauchy's famous rigidity theorem for 3D convex polyhedra has been extended in various directions by Dehn, Weyl, A.D.Alexandrov, Gluck and Connelly. These results imply that a disk-like polyhedral surface with simplicial faces is, generically, flexible, if the boundary has at least 4 vertices. What about surfaces with rigid but not necessarily simplicial faces? A natural, albeit extreme family is given by flat-faced origamis. Around 1995, Robert Lang, a well-known origamist, proposed a method for designing a crease pattern on a flat piece of paper such that it has an isometric flat-folded realization with an underlying, predetermined metric tree structure. Important mathematical properties of this algorithm remain elusive to this day. In this talk I will show that Lang's beautiful method leads, most often, to a crease pattern that cannot be continuously deformed to the desired flat-folded shape if its faces are to be kept rigid. Most surprisingly, sometimes the initial crease pattern is simply rigid: the (real) configuration space of such a structure may be disconnected, with one of the components being an isolated point. This is joint work with my student John Bowers, who also implemented a very nice program to visualize what is going on.