When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special “basic” holes guarantee foldability.

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Computational Geometry
School of Computer Science

Aichholzer, O. (Oswin), Akitaya, H.A. (Hugo A.), Cheung, K.C. (Kenneth C.), Demaine, E.D. (Erik D.), Demaine, M.L. (Martin L.), Fekete, S.P. (Sándor P.), … Schmidt, C. (Christiane). (2021). Folding polyominoes with holes into a cube. Computational Geometry, 93. doi:10.1016/j.comgeo.2020.101700