Flat Zipper-Unfolding Pairs for Platonic Solids

We show that four of the five Platonic solids’ surfaces may be cut open with a Hamiltonian path along edges and unfolded to a polygonal net each of which can “zipper-refold” to a flat doubly covered parallelogram, forming a rather compact representation of the surface. Thus these regular polyhedra have particular flat “zipper pairs.” No such zipper pair exists for a dodecahedron, whose Hamiltonian unfoldings are “zip-rigid.” This report is primarily an inventory of the possibilities, and raises more questions than it answers.

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Joseph O'Rourke
• Keywords: CORR 2010, Education, Hamiltonian Path, Polygonal Net, five Platonic