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- Statement
Is there any genus-zero orthogonal polyhedron P built by gluing together cubes face-to-face that cannot be edge-unfolded, where all cube edges on the surface of P are considered edges available for cutting? These orthogonal polyhedra are sometimes known as

*polycubes*, 3D versions of 2D*polyominoes*.- Origin
George Hart and Joseph O’Rourke, 2004.

- Status/Conjectures
Open.

- Motivation
More general problems seem even more difficult.

- Partial and Related Results
This is a special case of a more general problem, which is equally open. The goal, as in Problem 9, is to cut the surface and unfold without overlap. An

*edge unfolding*only permits cutting along edges of the polyhedron. A*grid unfolding*adds extra edges to the surface by intersecting the polyhedron with planes parallel to coordinate planes through every vertex, and so is easier to edge-unfold. Easier still is the posed problem: The orthogonal polyhedron is built from cubes, and all cube edges are available for cutting. Is there any such polyhedron that cannot be edge-unfolded? Such an example would narrow the options, but it may be that every orthogonal polyhedron can be grid-unfolded. (An easy box-on-box example [BDD+98] shows that without some surface refinement [DO05], not all orthogonal polyhedra can be edge-unfolded.) The posed question is among the most specific whose answer would make progress.Only a few narrow subclasses of orthogonal polyhedra are known to have grid-unfolding algorithms: orthotubes, orthostacks of orthogonally convex slabs, and orthogonal terrains. See [O'R08].

- Related Open Problems
Problem 9: Edge-Unfolding Convex Polyhedra.

Problem 42: Vertex-Unfolding Polyhedra.

Problem 43: General Unfolding of Nonconvex Polyhedra.- Appearances
- Categories
folding and unfolding; polyhedra

- Entry Revision History
J. O’Rourke, 14 Jul 2006, 16 Jul 2007.

- [BDD+98]
Therese Biedl, Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Joseph O’Rourke, Mark Overmars, Steve Robbins, and Sue Whitesides. Unfolding some classes of orthogonal polyhedra. In

*Proc. 10th Canad. Conf. Comput. Geom.*, pages 70–71, 1998. Full version in*Elec. Proc.*: http://cgm.cs.mcgill.ca/cccg98/proceedings/cccg98-biedl-unfolding.ps.gz.- [DO07]
Erik D. Demaine and Joseph O’Rourke.

*Geometric Folding Algorithms: Linkages, Origami, Polyhedra*. Cambridge University Press, July 2007. http://www.gfalop.org.- [DO05]
Erik D. Demaine and Joseph O’Rourke. Open problems from CCCG 2004. In

*Proc. 17th Canad. Conf. Comput. Geom.*, pages 303–306, 2005.- [O'R08]
Joseph O’Rourke. Unfolding orthogonal polyhedra. In J.E. Goodman, J. Pach, and R. Pollack, editors,

*Proc. Snowbird Conference Discrete and Computational Geometry: Twenty Years Later*, pages 307–317. American Mathematical Society, 2008.