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- Statement
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.- Origin
Posed as Open Problem 2 in [DDL+10], which introduced the term “zipper unfolding.”

- Status/Conjectures
Open.

- 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
- Categories
polyhedra

- Entry Revision History
J. O’Rourke, 7 Feb. 2012.

- [Bro61]
Thomas Brown. Simple paths on convex polyhedra.

*Pacific J. Math.*, 11(4):1211–1241, 1961.- [DDL+10]
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, August 2010.