Folding Polyominoes into (Poly)Cubes

Oswin Aichholzer, Michael Biro, Erik Demaine, Martin Demaine, David Eppstein, Sándor P. Fekete, Adam Hesterberg, Irina Kostitsyna, Christiane Schmidt

Research output: Contribution to journalArticleResearchpeer-review

Abstract

We study the problem of folding a given 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 $180^), and (d)~whether the folding must lie on just the polycube surface or can have interior faces as well. Second, we give all inclusion relations among all models that fold on the grid lines of~$P$. Third, we characterize all polyominoes that can fold into a unit cube, in some models. Fourth, we give a linear-time dynamic programming algorithm to fold a tree-shaped polyomino into a constant-size polycube, in some models. Finally, we consider the triangular version of the problem, characterizing which polyiamonds fold into a regular tetrahedron.
Original languageEnglish
Pages (from-to)197-226
Number of pages30
JournalInternational Journal of Computational Geometry and Applications
Volume28
Issue number3
DOIs
Publication statusPublished - 2018

Keywords

    Fields of Expertise

    • Information, Communication & Computing

    Cite this

    Aichholzer, O., Biro, M., Demaine, E., Demaine, M., Eppstein, D., Fekete, S. P., ... Schmidt, C. (2018). Folding Polyominoes into (Poly)Cubes. International Journal of Computational Geometry and Applications, 28(3), 197-226. https://doi.org/10.1142/S0218195918500048

    Folding Polyominoes into (Poly)Cubes. / Aichholzer, Oswin; Biro, Michael; Demaine, Erik; Demaine, Martin; Eppstein, David; Fekete, Sándor P.; Hesterberg, Adam; Kostitsyna, Irina; Schmidt, Christiane.

    In: International Journal of Computational Geometry and Applications, Vol. 28, No. 3, 2018, p. 197-226.

    Research output: Contribution to journalArticleResearchpeer-review

    Aichholzer, O, Biro, M, Demaine, E, Demaine, M, Eppstein, D, Fekete, SP, Hesterberg, A, Kostitsyna, I & Schmidt, C 2018, 'Folding Polyominoes into (Poly)Cubes' International Journal of Computational Geometry and Applications, vol. 28, no. 3, pp. 197-226. https://doi.org/10.1142/S0218195918500048
    Aichholzer, Oswin ; Biro, Michael ; Demaine, Erik ; Demaine, Martin ; Eppstein, David ; Fekete, Sándor P. ; Hesterberg, Adam ; Kostitsyna, Irina ; Schmidt, Christiane. / Folding Polyominoes into (Poly)Cubes. In: International Journal of Computational Geometry and Applications. 2018 ; Vol. 28, No. 3. pp. 197-226.
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    AU - Eppstein, David

    AU - Fekete, Sándor P.

    AU - Hesterberg, Adam

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    AB - We study the problem of folding a given 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 $180^), and (d)~whether the folding must lie on just the polycube surface or can have interior faces as well. Second, we give all inclusion relations among all models that fold on the grid lines of~$P$. Third, we characterize all polyominoes that can fold into a unit cube, in some models. Fourth, we give a linear-time dynamic programming algorithm to fold a tree-shaped polyomino into a constant-size polycube, in some models. Finally, we consider the triangular version of the problem, characterizing which polyiamonds fold into a regular tetrahedron.

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