Keller's cube-tiling conjecture is false in high dimensions
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.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- September 1992
- DOI:
- 10.48550/arXiv.math/9210222
- arXiv:
- arXiv:math/9210222
- Bibcode:
- 1992math.....10222L
- Keywords:
-
- Mathematics - Metric Geometry;
- Mathematics - Combinatorics
- E-Print:
- 5 pages