Folding Polyominoes into (Poly)Cubes∗ Oswin Aichholzer Institute for Software Technology, Graz University of Technology,
[email protected]. Michael Biro Department of Mathematics and Statistics, Swarthmore College,
[email protected]. Erik D. Demaine, Martin L. Demaine Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology,
[email protected],
[email protected]. David Eppstein Computer Science Department, University of California, Irvine,
[email protected]. S´andorP. Fekete Department of Computer Science, TU Braunschweig, Germany.
[email protected] Adam Hesterberg Department of Mathematics, Massachusetts Institute of Technology,
[email protected] Irina Kostitsyna Computer Science Department, Universit´eLibre de Bruxelles,
[email protected]. arXiv:1712.09317v2 [cs.CG] 25 Mar 2018 Christiane Schmidt Communications and Transport Systems, ITN, Link¨oping University,
[email protected]. ABSTRACT We study the problem of folding a polyomino P into a polycube Q, allowing faces of Q to be covered multiple times. First, we define a variety of folding models according to whether the folds (a) must be along grid lines of P or can divide squares in half (diagonally and/or orthogonally), (b) must be mountain or can be both mountain and valley, (c) can remain flat (forming an angle of 180◦), and (d) must lie on just the ∗A preliminary extended abstract appears in the Proceedings of the 27th Canadian Conference on Computational Geometry [2]. 1 2 polycube surface or can have interior faces as well. Second, we give all the inclusion relations among all models that fold on the grid lines of P . Third, we characterize all polyominoes that can fold into a unit cube, in some models.