Folding polyominoes into cubes

TL;DR

This work analyzes grid-foldings of edge-connected polyominoes into the unit cube $\mathcal{C}$ with edge creases and angles $\pm 90^\circ$ or $\pm 180^\circ$, allowing multiple face coverings. It develops an algorithmic framework based on consistent mappings to cube faces and ties realisability to unlink recognition in topology, situating CubeFolding in NP. It delivers complete classifications for tree-shaped polyominoes, a full characterization for rectangular polyominoes with two or more unit holes (and no other holes), and a broad analysis of simply connected cases, resolving several open questions. The results connect combinatorial folding with topological decision problems and provide concrete, verifiable criteria for foldability across broad classes of polyominoes, with explicit constructive proofs and computer-assisted enumerations for small bounds.

Abstract

Which polyominoes can be folded into a cube, using only creases along edges of the square lattice underlying the polyomino, with fold angles of $\pm 90^\circ$ and $\pm 180^\circ$, and allowing faces of the cube to be covered multiple times? Prior results studied tree-shaped polyominoes and polyominoes with holes and gave partial classifications for these cases. We show that there is an algorithm deciding whether a given polyomino can be folded into a cube. This algorithm essentially amounts to trying all possible ways of mapping faces of the polyomino to faces of the cube, but (perhaps surprisingly) checking whether such a mapping corresponds to a valid folding is equivalent to the unlink recognition problem from topology. We also give further results on classes of polyominoes which can or cannot be folded into cubes. Our results include (1) a full characterisation of all tree-shaped polyominoes that can be folded into the cube (2) that any rectangular polyomino which contains only one simple hole (out of five different types) does not fold into a cube, (3) a complete characterisation when a rectangular polyomino with two or more unit square holes (but no other holes) can be folded into a cube, and (4) a sufficient condition when a simply-connected polyomino can be folded to a cube. These results answer several open problems of previous work and close the cases of tree-shaped polyominoes and rectangular polyominoes with just one simple hole.

Figures (34)

• Figure 1: The five simple holes: unit square hole, slit of size 1, I-slit of size 2, L-slit of size 2, U-slit of size 3. This Figure is similar to aich19 where these holes are called basic.
• Figure 2: Cutting the polyomino along the red lines gives one connected component which is homeomorphic to a disk, and three connected components each of which is homeomorphic to an annulus.
• Figure 3: Creating a polyomino from a link diagram
• Figure 4: Standard cube net
• Figure 5: Strip of size $1 \times k$ for even $k$

Theorems & Definitions (42)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.4
• proof
• Theorem 3.5
• proof : Proof sketch
• Remark 4.1
• Example 4.2
• Definition 4.3