Bumpy pyramid folding

Abstract

We investigate folding problems for a class of petal (or star-like) polygons P, which have an n-polygonal base B surrounded by a sequence of triangles for which adjacent pairs of sides have equal length. We give linear time algorithms using constant precision to determine if P can fold to a pyramid with flat base B, and to determine a triangulation of B (crease pattern) that allows folding into a convex (triangulated) polyhedron. By Alexandrov's Theorem, the crease pattern is unique if it exists, but the general algorithm known for this theorem is pseudo-polynomial, with running time O(n456.5); ours is the first efficient algorithm for Alexandrov's Theorem for a special class of polyhedra. We also give a polynomial time algorithm that finds the crease pattern to produce the maximum volume triangulated polyhedron.