Folding and Unfolding in Computational Geometry

Abstract

I will present several recent results about geometric folding and unfolding problems. For example, place several rigid rods down on the table, and hinge them together at their ends to form a chain. Is it always possible to fold the hinges in such a way that the linkage is straightened into a line? The rods cannot change in length or cross each other, and must remain on the table. While at first glance it may seem intuitive that any chain can be ``unraveled'' into a straight line, this ``carpenter's rule problem'' has proved difficult and remained unsolved for over 25 years. This problem was solved in a recent burst of interest in folding and unfolding, a branch of discrete and computational geometry. In the past few years, several intriguing and surprising results have been proved about various contexts of folding and unfolding: reconfiguring linkages, paper folding (origami), unfolding polyhedra into nonoverlapping ``nets,'' and gluing polygons into polyhedra. Many of these problems have applications in areas including manufacturing, robotics, and protein folding.