Durer's Conjecture

Definitions and mathematical statement

Let $P\subset\mathbb R^3$ be a bounded convex polytope with nonempty interior. Its surface is the union of its polygonal facets, equipped with the intrinsic length metric inherited face by face from Euclidean space. An edge unfolding cuts this surface only along a selection of original polytope edges; no cut through the interior of a facet, and no subdivision introducing new cut edges, is allowed.

More precisely, choose a tree $T$ in the edge graph of $P$ containing all polytope vertices. Cut the surface along every edge of $T$, retaining the individual facets and the identifications along uncut edges. The resulting surface is a topological disk. Develop that disk in $\mathbb R^2$: choose one facet's position and successively place adjacent facets by isometries so that their shared uncut edges coincide and their interiors lie on opposite sides of the shared line. The uncut facet adjacency graph is a tree, so the placements are determined up to an overall planar isometry.

The conjecture asks whether every convex three-dimensional polytope admits such a tree for which the developed interiors of distinct facets are disjoint. Boundary contacts are permitted; overlap of positive area is forbidden. The resulting connected planar polygonal assembly is an edge net of $P$. Only existence of the final nonoverlapping development is required; the motion taking the original surface to that development need not itself avoid intersections.

Short English statement

Can every convex three-dimensional polytope be cut along its edges and flattened into one connected net without overlapping face interiors?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3101 (OPG-37459), OpenGarden. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Durer's Conjecture. Original problem and discussion. http://www.openproblemgarden.org/op/d_urers_conjecture.