When Can a Polygon Fold to a Polytope

We show that the decision question posed in the title can be answered with an algorithm of time and space complexity O(n), for a polygon of n vertices. We use a theorem of Aleksandrov that says that if the edges of the polygon can be matched in length so that the resulting complex is homeomorphic to a sphere, and such that the \complete angle" at each vertex is no more than 2 , then the implied folding corresponds to a unique convex polytope. We check the Aleksandrov conditions via dynamic programming. The algorithm has been implemented and tested.