Zipper Unfoldings of Convex Polyhedra

Definitions and mathematical statement

Let $P$ be the boundary of a convex three-dimensional polytope. A zipper cut is a single simple polygonal arc on this surface which visits every polytope vertex. It has no branches and does not revisit a point; between vertices it may cross face interiors and pass through points on edges. It is not required to follow the original edge graph.

Cut the surface along this arc, retaining both boundary copies of each cut segment. A zipper unfolding is an isometric development of the cut surface into one polygonal region in $\mathbb R^2$ with no overlap of distinct interior points. Each face piece remains rigid in the development. The visit to every vertex opens all concentrated positive curvature.

Does every convex polytope admit a zipper unfolding? Both the path and its order of visiting vertices may be chosen. A counterexample must rule out all such surface paths, including those that cross faces.

Requiring the path to use only original edges is a stronger restriction: a polytope whose edge graph has no Hamiltonian path already fails that restricted test. It does not therefore fail the general zipper question. A branching cut tree, even if it produces a valid net, is not a zipper cut.

Short English statement

Can every convex polyhedron be flattened without overlap after one nonbranching cut that visits all its vertices and may cross face interiors?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5500077 (AMR-054-0077), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problems Project - Computational Geometry, Problem 77. Original source indexed by AMR-054-0077. https://topp.openproblem.net/p77.