Dürer's problem: edge-unfolding convex polyhedra

Can the surface of every convex polyhedron be cut along some of its edges and laid flat in one piece without overlap? Albrecht Dürer drew such nets in 1525, and every convex polyhedron anyone has tried has one, yet there is neither a proof nor a counterexample.

Problem

Let $P$ be a convex polyhedron in $\mathbb{R}^3$. An edge unfolding of $P$ cuts the surface along a set $T$ of edges of $P$ and develops what remains isometrically into the plane, face by face. For the result to be one connected piece that lies flat, $T$ must be a spanning tree of the graph of vertices and edges of $P$: it has to reach every vertex and may not contain a cycle. The unfolding is a net if the faces, laid out in the plane, have pairwise disjoint interiors, so that together they form a single simple polygon.

Question. Does every convex polyhedron have a spanning tree of edges whose unfolding is a net?

Background

Dürer's Unterweysung der Messung (1525), known in English as The Painter's Manual, contains the earliest known drawings of polyhedra unfolded into nets. The question was stated mathematically by Shephard in 1975 [1], and it is Problem 9 in the original list of thirty problems of The Open Problems Project. Convexity matters: there are nonconvex polyhedra, even ones whose faces are all triangles, with no edge unfolding at all [2]. If the cuts may cross faces, every convex polyhedron can be unfolded (for example by the star or source unfolding), so the whole difficulty lies in cutting only along edges.

What is known

• No combinatorial obstruction. Ghomi (2014) showed that every convex polyhedron has a net after a suitable affine stretch, which answers a question of Croft, Falconer and Guy [3]. Computer searches have found nets for countless convex polyhedra, although a random edge unfolding of a generic polyhedron overlaps with probability that appears to tend to 1 as the number of vertices grows (Ghomi, 2018).
• An intrinsic counterexample. Barvinok and Ghomi (2020) built a convex polyhedron with a pseudo-edge graph (geodesic edges forming a 3-connected graph with convex faces) along which no unfolding is a net [4]. So the statement fails if edges are only defined intrinsically.
• Special classes. Nested prismatoids have nets (Radons, 2024) [5], and so do polar zonohedra (O'Rourke, 2023). Whether every prismatoid, the convex hull of two convex polygons in parallel planes, has a net is still unknown (O'Rourke, 2026).
• Overlap can be severe. For every $t$ there is a convex polyhedron with an edge unfolding in which some point of the plane is covered at least $t$ times (Akitaya et al., 2026) [6].

What would count as progress

• A convex polyhedron, given by exact vertex coordinates, with a proof that none of its edge unfoldings is a net. For a candidate with few vertices this can be confirmed by enumerating spanning trees and testing each layout for overlap in exact arithmetic.
• An algorithm, with a proof, that finds a net for every polyhedron in a new natural class, such as all prismatoids.
• A proof that every convex polyhedron has a net.

References

1. G. C. Shephard, Convex polytopes with convex nets, Math. Proc. Cambridge Philos. Soc. 78 (1975), 389–403. doi:10.1017/S0305004100051860
2. 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
3. M. Ghomi, Affine unfoldings of convex polyhedra, Geom. Topol. 18 (2014), 3055–3090. doi:10.2140/gt.2014.18.3055
4. N. Barvinok and M. Ghomi, Pseudo-edge unfoldings of convex polyhedra, Discrete Comput. Geom. 64 (2020), 671–689. arXiv:1709.04944
5. M. Radons, Edge-unfolding nested prismatoids, Comput. Geom. 116 (2024), 102033. doi:10.1016/j.comgeo.2023.102033
6. H. A. Akitaya, E. D. Demaine, F. Frei, S. Langerman, A. Lubiw and J. O'Rourke, Overlapping unfoldings of cones and convex polyhedra, CCCG 2026. arXiv:2607.09606

Source

Imported from The Open Problems Project, Problem 9, 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.