Enumerating Cube Tilings

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

Research output: Contribution to journalArticle

2 Citations (Scopus)


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 languageEnglish
Pages (from-to)1112-1122
Number of pages11
JournalDiscrete and Computational Geometry
Issue number4
Publication statusPublished - Jan 1 2013



  • 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

Cite this

Mathew, K. A., Östergård, P. R. J., & Popa, A. (2013). Enumerating Cube Tilings. Discrete and Computational Geometry, 50(4), 1112-1122. https://doi.org/10.1007/s00454-013-9547-4