# 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} }

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

## 117 Citations

### Keller's Conjecture on the Existence of Columns in Cube Tilings of R^n

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- 2008

It is shown that if n<7, then each tiling of R^n by translates of the unit cube [0,1)^n contains a column; that is, a family of the form {[0,1)^n+(s+ke_i): k \in Z}, where s \in R^n, e_i is an…

### Cube-Tilings of Rn and Nonlinear Codes

- MathematicsDiscret. Comput. Geom.
- 1994

Families of non-lattice tilings of R n by unit cubes are constructed that provide building blocks for the construction of cube-tilings such that no two cubes have a high-dimensional face in common.

### A Formalized Reduction of Keller’s Conjecture

- MathematicsCPP
- 2023

Keller’s conjecture in d dimensions states that there are no faceshare-free tilings of d-dimensional space by translates of a d-dimensional cube. In 2020, Brakensiek et al. resolved this 90-year-old…

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- Mathematics
- 2021

It is shown that every cube tiling of R is layered and the structure of non-layered cube tilings of R is described. It is also shown that in every cube tiling of R a cylinder contains a column.…

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- MathematicsProceedings of the Steklov Institute of Mathematics
- 2015

We consider the sequential random packing of integral translates of cubes [0, N]n into the torus ℤn/2Nℤn. Two particular cases are of special interest: (1) N = 2, which corresponds to a discrete case…

### On the structure of cube tilings and unextendible systems of cubes in low dimensions

- MathematicsEur. J. Comb.
- 2011

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- Mathematics
- 2011

We give a structural description of cube tilings and unextendible cube packings of R3. We also prove that up to dimension 4 each cylinder of a cube tiling contains a column and demonstrate by an…

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- Mathematics
- 2022

We construct a unilateral lattice tiling of R n into hypercubes of two diﬀernet side lengths p or q . This generalizes the Pythagorean tiling in R 2 . We also show that this tiling is unique up to…

### Fuglede's conjecture is false in 5 and higher dimensions

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- 2003

We give an example of a set Ω ⊂ R 5 which is a finite union of unit cubes, such that L 2 (Ω) admits an orthonormal basis of exponentials { 1 |Ω|1/2 e 2πiξj ·x :

### Spectra for cubes in products of finite cyclic groups

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- 2016

We consider "cubes" in products of finite cyclic groups and we study their tiling and spectral properties. (A set in a finite group is called a tile if some of its translates form a partition of the…

## References

SHOWING 1-10 OF 12 REFERENCES

### A reduction of Keller's conjecture

- Mathematics
- 1986

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…

### Über die lückenlose Erfüllung des Raumes mit Würfeln.

- 1930

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…

### A combinatorial approach for Keller's conjecture

- 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…

### Physica-Verlag, Würzberg.) [see Chapter 2, §4 and Chapter 3, §7

- (Reprint:
- 1907

### Über einfache und mehrfache Bedeckung desn-dimensionalen Raumes mit einem Würfelgitter

- Mathematics
- 1942

### 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