General unfoldings of nonconvex polyhedra

Decide whether every polyhedron can be cut open, with cuts anywhere on its surface, and flattened into one non-overlapping piece.

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1	# General unfoldings of nonconvex polyhedra
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3	If the cuts may run anywhere on the surface, not only along edges, can every polyhedron be cut open and flattened into a single non-overlapping piece? Every convex polyhedron can, and so can every orthogonal polyhedron of genus zero, but no method is known for polyhedra in general.
4
5	## Problem
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7	A polyhedron here is a closed surface in $\mathbb{R}^3$ made of finitely many flat polygons, glued edge to edge with each edge shared by exactly two faces. A general unfolding cuts the surface along a finite set of polygonal paths, which need not follow edges, so that what remains is connected and can be developed isometrically into the plane as one simple polygon: different points of the surface go to different points of the plane, apart from the two sides of a cut. The cuts must pass through every vertex whose angles do not sum to $2\pi$.
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9	Question. Does every polyhedron have a general unfolding? The question is open already for polyhedra homeomorphic to a sphere.
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11	## Background
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13	Unfoldings that cut only along edges are not enough: Bern, Demaine, Eppstein, Kuo, Mantler and Snoeyink (2003) found nonconvex polyhedra, one of them with only triangular faces, that have no edge unfolding but do have general unfoldings [1]. The Open Problems Project tentatively traces the general question to that paper. It is Open Problem 22.3 in Demaine and O'Rourke's book on folding [2], and O'Rourke's 2019 survey lists it among the six unsettled cases in its table of unfolding questions [3]. Nonconvex polyhedra are also where most applications of unfolding lie [3].
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15	## What is known
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17	- Convex polyhedra. Every convex polyhedron has a general unfolding, for example the star unfolding, which cuts along shortest paths from a generic point to every vertex [5], or the source unfolding [4].
18	- Orthogonal polyhedra. Damian, Flatland and O'Rourke (2007) showed that every orthogonal polyhedron (all faces meeting at right angles) of genus zero has a general unfolding, with cuts parallel to the edges [6]. Their unfolding may contain strips as thin as $2^{-\Omega(n)}$ for $n$ vertices. Damian, Demaine and Flatland (2014) reduced the cuts to the planes of the vertex grid plus $\Theta(n^2)$ extra coordinate planes between consecutive grid planes ([Graphs Combin. 30, 125–140](https://doi.org/10.1007/s00373-012-1257-9)).
19	- No counterexample is known: no polyhedron has been shown to lack a general unfolding.
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21	## What would count as progress
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23	- An algorithm, with a proof, that produces a general unfolding for every polyhedron in a broad new class, for example all genus-zero polyhedra whose faces are triangles.
24	- Software that outputs a certified unfolding (cut paths and the planar layout, checked for overlap in exact or interval arithmetic) for any given polyhedron, tested on hard examples from the literature.
25	- A polyhedron together with a proof that it has no general unfolding.
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27	## References
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29	1. M. Bern, E. D. Demaine, D. Eppstein, E. Kuo, A. Mantler and J. Snoeyink, Ununfoldable polyhedra with convex faces, Comput. Geom. 24 (2003), 51–62. [arXiv:cs/9908003](https://arxiv.org/abs/cs/9908003)
30	2. E. D. Demaine and J. O'Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, 2007. [doi:10.1017/CBO9780511735172](https://doi.org/10.1017/CBO9780511735172)
31	3. J. O'Rourke, Unfolding polyhedra, Proc. 31st Canadian Conf. Comput. Geom. (2019), 85–86. [arXiv:1908.07152](https://arxiv.org/abs/1908.07152)
32	4. J. S. B. Mitchell, D. M. Mount and C. H. Papadimitriou, The discrete geodesic problem, SIAM J. Comput. 16 (1987), 647–668. [doi:10.1137/0216045](https://doi.org/10.1137/0216045)
33	5. B. Aronov and J. O'Rourke, Nonoverlap of the star unfolding, Discrete Comput. Geom. 8 (1992), 219–250. [doi:10.1007/BF02293047](https://doi.org/10.1007/BF02293047)
34	6. M. Damian, R. Flatland and J. O'Rourke, Epsilon-unfolding orthogonal polyhedra, Graphs Combin. 23 (2007), 179–194. [arXiv:cs/0602095](https://arxiv.org/abs/cs/0602095)
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36	## Source
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38	Imported from [The Open Problems Project, Problem 43](https://topp.openproblem.net/p43), curated by Erik D. Demaine, Joseph S. B. Mitchell and Joseph O'Rourke. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.