[#P3094] Dürer’s edge-unfolding problem
Problem. Let \(P\) be the boundary of a convex three-dimensional polytope and let \(G(P)\) be its edge graph. Must there exist a spanning tree \(T\subseteq G(P)\) such that cutting \(P\) along \(T\) and isometrically developing the remaining disk into the plane produces one polygon whose face interiors are pairwise disjoint?
1Context
Known frontier: Every convex polyhedron becomes edge-unfoldable after a suitable affine transformation, and every nested three-prismatoid has a nonoverlapping edge unfolding; recent work also constructs highly overlapping edge unfoldings. Open boundary: Prove a nonoverlapping spanning-tree edge unfolding exists for every original convex three-polytope, or exhibit one polytope for which every such unfolding overlaps.
2Problem setup
Definition 1 (edge unfolding). A planar development obtained by cutting a spanning tree of the original edge graph and rotating faces rigidly about uncut edges.
Definition 2 (nonoverlapping). Distinct face interiors have disjoint planar images; shared boundary points are allowed.
Remark 1. An edge unfolding cuts only original polytope edges. The surviving hinges flatten the connected surface into one planar polygon. The question asks whether every convex polyhedron has at least one cut tree that avoids face-interior overlap.
3What counts as a solution
- For a proof, construct or establish a valid spanning tree for every convex three-polytope and prove injectivity on face interiors.
- For a disproof, give explicit coordinates and faces for one convex polytope and exhaust or structurally exclude every spanning-tree edge unfolding.
1Status
Current status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: Every convex polyhedron becomes edge-unfoldable after a suitable affine transformation, and every nested three-prismatoid has a nonoverlapping edge unfolding; recent work also constructs highly overlapping edge unfoldings. Exact unresolved remainder: Prove a nonoverlapping spanning-tree edge unfolding exists for every original convex three-polytope, or exhibit one polytope for which every such unfolding overlaps.[1][2][3]
1Records
Notes and companion material
Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: Every convex polyhedron becomes edge-unfoldable after a suitable affine transformation, and every nested three-prismatoid has a nonoverlapping edge unfolding; recent work also constructs highly overlapping edge unfoldings. Exact unresolved remainder: Prove a nonoverlapping spanning-tree edge unfolding exists for every original convex three-polytope, or exhibit one polytope for which every such unfolding overlaps.
- Equivalent-formulation queries: Dürer edge unfolding every convex polyhedron open; convex 3-polytope spanning tree nonoverlapping net; Dürer problem affine unfolding pseudo-edge counterexample
- Strongest checked neighboring result: Every convex polyhedron becomes edge-unfoldable after a suitable affine transformation, and every nested three-prismatoid has a nonoverlapping edge unfolding; recent work also constructs highly overlapping edge unfoldings.
- Exact unresolved remainder: Prove a nonoverlapping spanning-tree edge unfolding exists for every original convex three-polytope, or exhibit one polytope for which every such unfolding overlaps.
How the 4 records connect
ProblemDürer’s edge-unfolding problem
2See also
How to cite
TheoremDB contributors, “Dürer’s edge-unfolding problem,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/durers-edge-unfolding-problemThis page as plain text: durers-edge-unfolding-problem.md
This problem includes 4 records joined by 3 typed links, sourced from doi.org[1], current as of August 1, 2026.
1References
- Packet source. Mohammad Ghomi, “Affine unfoldings of convex polyhedra”. Geometry & Topology 18(5) (2014), 3055-3090. DOI 10.2140/gt.2014.18.3055. abstract and main theorem. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves that every convex polyhedron becomes edge-unfoldable after a suitable affine transformation, while leaving the original metric problem open.Also cited at M. Ghomi, “Affine unfoldings of convex polyhedra,” Geometry & Topology 18(5) (2014), 3055–3090. abstract and main theorem.Source used to assess the problem's recorded status.For Dürer’s edge-unfolding problem: This is the dated publication status for the canonical target Dürer’s edge-unfolding problem.Source named by the research packet.
- Manuel Radons, “Edge-unfolding nested prismatoids”. Computational Geometry 116 (2024), 102033. DOI 10.1016/j.comgeo.2023.102033. abstract and main theorem. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves nonoverlapping edge unfoldings for every nested three-prismatoid.Source used to assess the problem's recorded status.For Dürer’s edge-unfolding problem: Proves nonoverlapping edge unfoldings for every nested three-prismatoid.
- MIT CompGeom Group, H. A. Akitaya, E. D. Demaine, F. Frei, S. Langerman, A. Lubiw, and J. O’Rourke, “Overlapping Unfoldings of Cones and Convex Polyhedra,” arXiv:2607.09606 (2026). abstract and introduction. ↗preprint · primary source · arXiv:2607.09606, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Frames Dürer’s problem as unresolved and constructs edge unfoldings with arbitrarily high overlap thickness, which does not settle existence of a nonoverlapping tree.Source used to assess the problem's recorded status.For Dürer’s edge-unfolding problem: Frames Dürer’s problem as unresolved and constructs edge unfoldings with arbitrarily high overlap thickness, which does not settle existence of a nonoverlapping tree.
Original TheoremDB editorial statement and source synthesis; external works are used for citation only.