Enumerating Cube Tilings

K. Ashik Mathew; Patric R.J. Östergård; Alexandru Popa

doi:10.1007/s00454-013-9547-4

Enumerating Cube Tilings

K. Ashik Mathew, Patric R.J. Östergård, Alexandru Popa

Department of Computer Science

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

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	https://doi.org/10.1007/s00454-013-9547-4
Publication status	Published - Dec 2013

Keywords

Classification
Cube tilings
Exact cover
Switching graph

ASJC Scopus subject areas

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

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AU - Östergård, Patric R.J.

AU - Popa, Alexandru

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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.

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KW - Exact cover

KW - Switching graph

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