General Unfoldings of Nonconvex Polyhedra

Definitions and mathematical statement

Let $P\subset\mathbb R^3$ be a compact connected embedded polyhedral surface without boundary, equipped with the intrinsic length metric on its polygonal faces. It may be nonconvex and may have positive genus. A general cut consists of finitely many polygonal arcs on the surface; these arcs may cross face interiors and need not lie on the original edges.

After cutting, take the surface with its separate boundary copies of the cuts. A general unfolding develops this cut surface into the plane isometrically on each remaining face piece, producing one polygonal region with connected interior and no overlap of distinct interiors. Boundary contacts can be allowed, but separate pieces merely touching at points do not supply the required connected interior.

Does every $P$ admit a general unfolding? The cut graph may be chosen freely, subject to finiteness and retention of one piece. No upper bound on the number of cuts or the thinness of the resulting region is imposed.

Allowing cuts through faces distinguishes this question from edge unfolding. Keeping a single piece distinguishes it from arbitrary dissection into separately placed triangles. The question concerns the existence of the final isometric planar layout; it does not require a collision-free folding animation.

Short English statement

Can every closed polyhedral surface, including nonconvex ones, be cut along finitely many surface arcs and flattened into one piece without interior overlap?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5500043 (AMR-054-0043), 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 43. Original source indexed by AMR-054-0043. https://topp.openproblem.net/p43.