Folding Polyominoes with Holes into a Cube

O. Aichholzer, H. A. Akitaya, K. C. Cheung, E. D. Demaine, M. L. Demaine, S. P. Fekete, L. Kleist, I. Kostitsyna, M. Löffler, Z. Masárová, K. Mundilova, and C. Schmidt

Abstract:

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 hole(s) to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special simple holes guarantee foldability.

Reference: O. Aichholzer, H. A. Akitaya, K. C. Cheung, E. D. Demaine, M. L. Demaine, S. P. Fekete, L. Kleist, I. Kostitsyna, M. Löffler, Z. Masárová, K. Mundilova, and C. Schmidt. Folding polyominoes with holes into a cube. In Proc. $31^{th}$ Annual Canadian Conference on Computational Geometry CCCG 2019, pages 164-170, Edmonton, Alberta, Canada, 2019.