Problem 77: Zipper Unfoldings of Convex Polyhedra

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

Problem 9.

Categories

polyhedra

Entry Revision History

J. O'Rourke, 7 Feb. 2012.

Bibliography

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.