The zipper foldings of the diamond

Abstract

A zipper folding of a polygon P given a source point x ∈ ∂P is the polyhedron generated by identifying all points in ∂P equidistant from x, measured along the perimeter of P , in essence “zipping” the boundary of the polygon. A theorem of Alexandrov shows that as long as every glued point has nonnegative curvature, then any zipper folding of a convex polygon leads to unique convex polyhedron (where a doubly covered polygon is considered a “flat” polyhedron). Alexandrov’s theorem is existential, but a more recent constructive proof by Bobenko and Izmestiev allows for the explicit construction of the polyhedron by solving a certain differential equation [2]. An implementation of the constructive algorithm has been coded by Stefan Sechelmann, which will output the folded convex polyhedron given a input triangulation of the polygon with gluing instructions. However, it is difficult to extract the creases and adjacencies from the initial polygon in their final output polyhedron. We seek a more combinatorial approach to computing this information. Previous work has also looked at determining all the combinatorially different polyhedra obtained via foldings, primarily for regular convex polygons as well as a few other shapes such as the Latin cross [3]. In this paper, we classify and compute the convex foldings of a diamond shape which are obtained via zipper foldings. Our primary goal was to seek a simpler combinatorial approach to testing for the correct set of folds, or crease pattern. As was observed by Alexandrov and noted in [1], there are a finite number of possible crease patterns. However, in our experience, verifying or discounting a crease pattern has been surprisingly difficult in more