Flat Zipper-Unfolding Pairs for Platonic Solids
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2010
- DOI:
- 10.48550/arXiv.1010.2450
- arXiv:
- arXiv:1010.2450
- Bibcode:
- 2010arXiv1010.2450O
- Keywords:
-
- Computer Science - Computational Geometry;
- Computer Science - Discrete Mathematics;
- 52B10;
- F.2.2
- E-Print:
- 15 pages, 14 figures, 8 references. v2: Added one new figure. v3: Replaced Fig. 13 to remove a duplicate unfolding, reducing from 21 to 20 the distinct unfoldings. v4: Replaced Fig. 13 again, 18 distinct unfoldings