  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.
