OP-00516 Open

An overlap-free edge unfolding for every convex polyhedron

Let $P \subset \mathbb{R}^3$ be a convex polytope with nonempty interior and let $G$ be the graph formed by its vertices and edges. Cutting $\partial P$ along a spanning tree $T$ of $G$ and developing the resulting intrinsic Euclidean surface into the plane gives an edge unfolding of $P$, in which each face keeps its shape and size. Ghomi proved that every convex polyhedron has an edge unfolding without overlaps after an affine stretching in almost any direction. Bern, Demaine, Eppstein, Kuo, Mantler and Snoeyink constructed nonconvex polyhedra with convex faces that have no edge unfolding without overlaps.

Problem. Does there always exist a spanning tree $T \subset G$ such that cutting $\partial P$ along $T$ and developing into the plane produces a single net whose face interiors are pairwise disjoint? Cuts may use only original edges; contact along boundaries of unfolded faces is permitted, overlapping interiors are not.

Source: Shephard (1975), who first formulated it, after Dürer's nets of polyhedra (1525); the conjecture that the answer is yes was first published by Grünbaum (1991), according to Ghomi (2014).

Imported from github.com/MColbrook/AIM.

Classification

Primary: MG — Metric Geometry
Secondary: CO
Keywords: AIM Open Applied Problems, convex polyhedra, edge unfoldings, nets

References

@article{shephard1975,
  author  = {Shephard, G. C.},
  title   = {Convex polytopes with convex nets},
  journal = {Math. Proc. Cambridge Philos. Soc.},
  volume  = {78},
  number  = {3},
  year    = {1975},
  pages   = {389--403},
  doi     = {10.1017/S0305004100051860},
}
@book{durer1525,
  author    = {D{\"u}rer, Albrecht},
  title     = {The Painter's Manual},
  note      = {Translated by Walter L. Strauss from Underweysung der Messung (1525)},
  publisher = {Abaris Books},
  address   = {New York},
  year      = {1977},
}
@article{grunbaum1991,
  author  = {Gr{\"u}nbaum, Branko},
  title   = {Nets of polyhedra {II}},
  journal = {Geombinatorics},
  volume  = {1},
  number  = {3},
  year    = {1991},
  pages   = {5--10},
}
@article{ghomi2014,
  author        = {Ghomi, Mohammad},
  title         = {Affine unfoldings of convex polyhedra},
  journal       = {Geom. Topol.},
  volume        = {18},
  number        = {5},
  year          = {2014},
  pages         = {3055--3090},
  doi           = {10.2140/gt.2014.18.3055},
  eprint        = {1305.3231},
  archivePrefix = {arXiv},
}
@article{bern2003,
  author        = {Bern, Marshall and Demaine, Erik D. and Eppstein, David and Kuo, Eric and Mantler, Andrea and Snoeyink, Jack},
  title         = {Ununfoldable polyhedra with convex faces},
  journal       = {Comput. Geom.},
  volume        = {24},
  number        = {2},
  year          = {2003},
  pages         = {51--62},
  doi           = {10.1016/S0925-7721(02)00091-3},
  eprint        = {cs/9908003},
  archivePrefix = {arXiv},
}
@misc{aim-durer-edge-unfolding,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{An overlap-free edge unfolding for every convex polyhedron}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/244-durer-edge-unfolding.md},
  note         = {Entry 244 at commit a153f46, 2026-10-13},
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00516, "An overlap-free edge unfolding for every convex polyhedron", clopen.solutions.