Zipper unfoldings of convex polyhedra
A zipper unfolding of a convex polyhedron $P$ cuts the surface of $P$ along a single path and unfolds it into the plane as one non-overlapping polygon. Since the cut must reach every vertex, the path is a Hamiltonian path; its segments need not run along edges of $P$.
Demaine, Demaine, Lubiw, Shallit and Shallit introduced zipper unfoldings. If the cuts are required to follow edges of $P$, a polyhedron whose $1$-skeleton has no Hamiltonian path, such as the rhombic dodecahedron, has no zipper edge-unfolding; Brown, among others, studied polyhedra of this kind.
Problem. Does every convex polyhedron have a zipper unfolding?
Source: Demaine, Demaine, Lubiw, Shallit and Shallit (2010), Open Problem 2; Problem 77 of The Open Problems Project (Demaine, Mitchell and O'Rourke).
Imported from topp.openproblem.net.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CO
- Keywords
- convex polyhedra, Hamiltonian path, The Open Problems Project, zipper unfolding
References
@inproceedings{demaine2010,
author = {Demaine, Erik and Demaine, Martin and Lubiw, Anna and Shallit, Arlo and Shallit, Jonah},
title = {Zipper unfoldings of polyhedral complexes},
booktitle = {Proc. 22nd Canad. Conf. Comput. Geom.},
year = {2010},
pages = {219--222},
}
@article{brown1961,
author = {Brown, Thomas},
title = {Simple paths on convex polyhedra},
journal = {Pacific J. Math.},
volume = {11},
number = {4},
year = {1961},
pages = {1211--1241},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01108, "Zipper unfoldings of convex polyhedra", clopen.solutions.