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**Previous:** Problem 42: Vertex-Unfolding Polyhedra

- Statement
Can every closed polyhedron be cut along its surface and unfolded into one piece in the plane without overlap? Such an unfolding is called a

*general unfolding*to distinguish from*edge-unfoldings*(see Problem 9) and*vertex-unfoldings*(see Problem 42).- Origin
Perhaps [BDE+03].

- Status/Conjectures
Open.

- Partial and Related Results
It is known that every convex polyhedron has a general unfolding. In fact, there are three general methods for unfolding convex polyhedra: the star unfolding [AO91], [AAOS97], the source unfolding [MMP87], and unfolding via quasigeodesics [IOV07].

On the nonconvex side, Bern et al. [BDE+03] show a general unfolding for a nonconvex simplicial polyhedron (whose faces are all triangles) that has no edge unfolding, establishing that general unfoldings are more powerful than edge unfoldings. (This was known earlier [BDD+98] but with an example using nonconvex faces.)

It is now known that all orthogonal polyhedra (those with all edges parallel to coordinate axes) have a general unfolding [DFO07], although the resulting single piece can be exponentially thin and long. See [O'R08] for a survey of progress on orthogonal polyhedra.

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

Problem 42: Vertex-Unfolding Polyhedra.- Appearances
- Categories
folding and unfolding; polyhedra

- Entry Revision History
E. Demaine, 7 Aug. 2002; J. O’Rourke, 24 Jul 2008.

- [AAOS97]
Pankaj K. Agarwal, Boris Aronov, Joseph O’Rourke, and Catherine A. Schevon. Star unfolding of a polytope with applications.

*SIAM J. Comput.*, 26:1689–1713, 1997.- [AO91]
Boris Aronov and Joseph O’Rourke. Nonoverlap of the star unfolding. In

*Proc. 7th Annu. ACM Sympos. Comput. Geom.*, pages 105–114, 1991.- [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.- [BDE+03]
Marshall Bern, Erik D. Demaine, David Eppstein, Eric Kuo, Andrea Mantler, and Jack Snoeyink. Ununfoldable polyhedra with convex faces.

*Comput. Geom. Theory Appl.*, 24(2):51–62, 2003.- [DFO07]
Mirela Damian, Robin Flatland, and Joseph O’Rourke. Epsilon-unfolding orthogonal polyhedra.

*Graphs and Combinatorics*, 23[Suppl]:179–194, 2007. Akiyama-Chvátal Festschrift.- [DO07]
Erik D. Demaine and Joseph O’Rourke.

*Geometric Folding Algorithms: Linkages, Origami, Polyhedra*. Cambridge University Press, July 2007. http://www.gfalop.org.- [IOV07]
Jin-ichi Itoh, Joseph O’Rourke, and Costin Vîlcu. Unfolding convex polyhedra via quasigeodesics. Technical Report 085, Smith College, July 2007. arXiv:0707.4258v2 [cs.CG].

- [MMP87]
Joseph S. B. Mitchell, David M. Mount, and Christos H. Papadimitriou. The discrete geodesic problem.

*SIAM J. Comput.*, 16:647–668, 1987.- [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.