Zipper Unfoldings of Polyhedral Complexes

Abstract

We explore which polyhedra and polyhedral complexes can be formed by folding up a planar polygonal region and fastening it with one zipper. We call the reverse process a zipper unfolding. A zipper unfolding of a polyhedron is a path cut that unfolds the polyhedron to a planar polygon; in the case of edge cuts, these are Hamiltonian unfoldings as introduced by Shephard in 1975. We show that all Platonic and Archimedean solids have Hamiltonian unfoldings. We give examples of polyhedral complexes that are, and are not, zipper [edge] unfoldable. The positive examples include a polyhedral torus, and two tetrahedra joined at an edge or at a face.