Edge-Unfolding Polycubes

Definitions and mathematical statement

A polycube is a finite face-connected union of congruent axis-aligned unit cubes from the cubic lattice. Assume its boundary is an embedded polyhedral two-sphere, so there are no handles, cavities, or nonmanifold boundary junctions. Retain the entire tiling of that boundary by exposed unit-square cube faces.

An allowed cut follows edges of this square tiling, including edges lying inside larger coplanar faces of the geometric solid. An edge unfolding cuts some such edges and develops the remaining surface into one planar piece with connected interior and pairwise disjoint interiors of its unit squares. Uncut shared edges retain their length and incidence. No square may be cut through its interior.

Does every such genus-zero polycube have an edge unfolding, or does a counterexample exist? The unit grid supplied by the cubes is available for cutting in full, but it may not be refined by introducing further subdivisions.

The question is more permissive than cutting only the nonflat edges of the solid, and more restrictive than an arbitrary surface dissection. An unfolding of a finer grid does not answer the specified unit-square version. Only existence of a flat layout is required, with no prescribed physical unfolding motion.

Short English statement

Can every polycube with spherical boundary be unfolded without overlap using only the existing unit-square edges on its surface?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5500064 (AMR-054-0064), 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 64. Original source indexed by AMR-054-0064. https://topp.openproblem.net/p64.