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Problem 77: Zipper Unfoldings of Convex Polyhedra
- Does every convex polyhedron P have a zipper unfolding?
A zipper unfolding cuts open P via a single
path, necessarily a Hamiltonian path (to span all vertices),
and unfolds the surface to a non-overlapping polygon in the plane.
The segments of the path need not lie along edges of P.
- Posed as Open Problem 2 in [DDL+10], which
introduced the term "zipper unfolding."
- Partial and Related Results
- With the restriction that the cuts follow edges,
any P without a Hamiltonian path in its 1-skeleton
has no zipper edge-unfolding, e.g., a rhombic dodecahedron.
(Such polyhedra have been studied, e.g., in [Bro61].)
- Related Open Problems
- Problem 9.
- Entry Revision History
- J. O'Rourke, 7 Feb. 2012.
Simple paths on convex polyhedra.
Pacific J. Math., 11(4):1211-1241, 1961.
Erik Demaine, Martin Demaine, Anna Lubiw, Arlo Shallit, and Jonah Shallit.
Zipper unfoldings of polyhedral complexes.
In Proc. 22nd Canad. Conf. Comput. Geom., pages 219-222,
The Open Problems Project - September 19, 2017