# Keller’s cube-tiling conjecture is false in high dimensions

@article{Lagarias1992KellersCC,
title={Keller’s cube-tiling conjecture is false in high dimensions},
author={Jeffrey C. Lagarias and Peter W. Shor},
journal={Bulletin of the American Mathematical Society},
year={1992},
volume={27},
pages={279-283}
}
• Published 1 March 1992
• Mathematics
• Bulletin of the American Mathematical Society
O. H. Keller conjectured in 1930 that in any tiling of R n by unit n-cubes there exist two of them having a complete facet in common. 0. Perron proved this conjecture for n ≤ 6. We show that for all n ≥ 10 there exists a tiling of R n by unit n-cubes such that no two n-cubes have a complete facet in common
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## References

SHOWING 1-10 OF 12 REFERENCES

A family of translates of a closedn-dimensional cube is called a cube tiling if the union of the cubes is the wholen-space and their interiors are disjoint. According to a famous unsolved conjecture
Inhaltsverzeichnis. Einleitung 231 Vorbemerkungen 233 I. Teil: Eigenschaften der Raumerfüllung in n Dimensionen 233—241 § 1. Der Strahlensatz 233 § 2. Staffeln 234 § 3. Gerade und ungerade
• Mathematics
• 1990
The statement, that in a tiling by translates of ann-dimensional cube there are two cubes having common (n-1)-dimensional faces, is known as Keller's conjecture. We shall prove that there is a

• (Reprint:
• 1907

### AT&T Bell Laboratories

• AT&T Bell Laboratories

### Reprint: 1961 Physica-Verlag, Würzberg.) [see Chapter 2, §4 and Chapter 3, §7. Minkowski's Conjecture appears on p. 28 and its geometric interpretation on p

• Reprint: 1961 Physica-Verlag, Würzberg.) [see Chapter 2, §4 and Chapter 3, §7. Minkowski's Conjecture appears on p. 28 and its geometric interpretation on p
• 1907