Enumerating Cube Tilings

Cube tilings formed by n -dimensional 4 Z n -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 a...

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Bibliographic Details
Published in:Discrete & computational geometry 2013-12, Vol.50 (4), p.1112-1122
Main Authors: Mathew, K. Ashik, Östergård, Patric R. J., Popa, Alexandru
Format: Article
Language:English
Subjects:
Classification
Combinatorics
Computational Mathematics and Numerical Analysis
Euclidean space
Graphs
Integer programming
Mathematics
Mathematics and Statistics
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Summary:Cube tilings formed by n -dimensional 4 Z n -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.
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-013-9547-4