Keller's cube-tiling conjecture is false in high dimensions

Jeffrey C. Lagarias; Peter W. Shor

Keller's cube-tiling conjecture is false in high dimensions

Metric Geometry 2016-09-06 v1 Combinatorics

Authors: Jeffrey C. Lagarias , Peter W. Shor

Abstract

O. H. Keller conjectured in 1930 that in any tiling of $\Bbb R^n$ by unit $n$ -cubes there exist two of them having a complete facet in common. O. Perron proved this conjecture for $n\le 6$ . We show that for all $n\ge 10$ there exists a tiling of $\Bbb R^n$ by unit $n$ -cubes such that no two $n$ -cubes have a complete facet in common.

Keywords

tilings cubic fourfolds

Cite

@article{arxiv.math/9210222,
  title  = {Keller's cube-tiling conjecture is false in high dimensions},
  author = {Jeffrey C. Lagarias and Peter W. Shor},
  journal= {arXiv preprint arXiv:math/9210222},
  year   = {2016}
}

Comments

5 pages

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