### Abstract

Cube tilings formed by n-dimensional 4ℤn-periodic hypercubes with side 2 and integer coordinates are considered here. By representing the problem of finding such cube tilings within the framework of exact cover and using canonical augmentation, pairwise nonisomorphic 5-dimensional cube tilings are exhaustively enumerated in a constructive manner. There are 899,710,227 isomorphism classes of such tilings, and the total number of tilings is 638,560,878,292,512. It is further shown that starting from a 5-dimensional cube tiling and using a sequence of switching operations, it is possible to generate any other cube tiling.

Original language | English |
---|---|

Pages (from-to) | 1112-1122 |

Number of pages | 11 |

Journal | Discrete and Computational Geometry |

Volume | 50 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2013 |

Externally published | Yes |

### Fingerprint

### Keywords

- Classification
- Cube tilings
- Exact cover
- Switching graph

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

*Discrete and Computational Geometry*,

*50*(4), 1112-1122. https://doi.org/10.1007/s00454-013-9547-4

**Enumerating Cube Tilings.** / Mathew, K. Ashik; Östergård, Patric R J; Popa, Alexandru.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 50, no. 4, pp. 1112-1122. https://doi.org/10.1007/s00454-013-9547-4

}

TY - JOUR

T1 - Enumerating Cube Tilings

AU - Mathew, K. Ashik

AU - Östergård, Patric R J

AU - Popa, Alexandru

PY - 2013

Y1 - 2013

N2 - Cube tilings formed by n-dimensional 4ℤn-periodic hypercubes with side 2 and integer coordinates are considered here. By representing the problem of finding such cube tilings within the framework of exact cover and using canonical augmentation, pairwise nonisomorphic 5-dimensional cube tilings are exhaustively enumerated in a constructive manner. There are 899,710,227 isomorphism classes of such tilings, and the total number of tilings is 638,560,878,292,512. It is further shown that starting from a 5-dimensional cube tiling and using a sequence of switching operations, it is possible to generate any other cube tiling.

AB - Cube tilings formed by n-dimensional 4ℤn-periodic hypercubes with side 2 and integer coordinates are considered here. By representing the problem of finding such cube tilings within the framework of exact cover and using canonical augmentation, pairwise nonisomorphic 5-dimensional cube tilings are exhaustively enumerated in a constructive manner. There are 899,710,227 isomorphism classes of such tilings, and the total number of tilings is 638,560,878,292,512. It is further shown that starting from a 5-dimensional cube tiling and using a sequence of switching operations, it is possible to generate any other cube tiling.

KW - Classification

KW - Cube tilings

KW - Exact cover

KW - Switching graph

UR - http://www.scopus.com/inward/record.url?scp=84890394508&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84890394508&partnerID=8YFLogxK

U2 - 10.1007/s00454-013-9547-4

DO - 10.1007/s00454-013-9547-4

M3 - Article

VL - 50

SP - 1112

EP - 1122

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 4

ER -