# Enumerating Cube Tilings

Discrete & Computational Geometry, Sep 2013

Cube tilings formed by $$n$$-dimensional $$4\mathbb Z ^n$$-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.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00454-013-9547-4.pdf

K. Ashik Mathew, Patric R. J. Östergård, Alexandru Popa. Enumerating Cube Tilings, Discrete & Computational Geometry, 2013, 1112-1122, DOI: 10.1007/s00454-013-9547-4