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Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(online) ISSN 0273-0979(print)

 

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


Authors: Jeffrey C. Lagarias and Peter W. Shor
Journal: Bull. Amer. Math. Soc. 27 (1992), 279-283
MSC (2000): Primary 52C22; Secondary 05B45
MathSciNet review: 1155280
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Abstract: O. H. Keller conjectured in 1930 that in any tiling of $ {\mathbb{R}^n}$ by unit n-cubes there exist two of them having a complete facet in common. O. Perron proved this conjecture for $ n \leq 6$. We show that for all $ n \geq 10$ there exists a tiling of $ {\mathbb{R}^n}$ by unit n-cubes such that no two n-cubes have a complete facet in common.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0273-0979-1992-00318-X
PII: S 0273-0979(1992)00318-X
Article copyright: © Copyright 1992 American Mathematical Society