General unfoldings of nonconvex polyhedra

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.

Problem

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 $2\pi$.

Question. Does every polyhedron have a general unfolding? The question is open already for polyhedra homeomorphic to a sphere.

Background

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].

What is known

• 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].
• 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).
• No counterexample is known: no polyhedron has been shown to lack a general unfolding.

What would count as progress

• 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.
• 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.
• A polyhedron together with a proof that it has no general unfolding.

References

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
2. E. D. Demaine and J. O'Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, 2007. doi:10.1017/CBO9780511735172
3. J. O'Rourke, Unfolding polyhedra, Proc. 31st Canadian Conf. Comput. Geom. (2019), 85–86. arXiv:1908.07152
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
5. B. Aronov and J. O'Rourke, Nonoverlap of the star unfolding, Discrete Comput. Geom. 8 (1992), 219–250. doi:10.1007/BF02293047
6. M. Damian, R. Flatland and J. O'Rourke, Epsilon-unfolding orthogonal polyhedra, Graphs Combin. 23 (2007), 179–194. arXiv:cs/0602095

Source

Imported from The Open Problems Project, Problem 43, 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.