The zipper foldings of the diamond

Abstract

In this paper, we classify and compute the convex foldings of a particular rhombus that are obtained via a zipper folding along the boundary of the shape. In the process, we explore computational aspects of this problem; in particular, we outline several useful techniques for computing both the edge set of the final polyhedron and its three-dimensional coordinates. We partition the set of possible zipper starting points into subintervals representing equivalence classes induced by these edge sets. In addition, we explore nonconvex foldings of this shape which are obtained by using a zipper starting point outside of the interval corresponding to a set of edges where the polygon folds to a convex polyhedron; surprisingly, this results in multiple families of nonconvex and easily computable polyhedra.