Filling space with hypercubes of two sizes -- The pythagorean tiling in higher dimensions

@inproceedings{Fuhrer2022FillingSW,
  title={Filling space with hypercubes of two sizes -- The pythagorean tiling in higher dimensions},
  author={Jakob Fuhrer},
  year={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 symmetries, which proves a variation of a conjecture by B¨olcskei from 2001. For positive integers p and q this tiling also provides a tiling of ( Z / ( p n + q n ) Z ) n . 

Figures from this paper

References

SHOWING 1-10 OF 24 REFERENCES

Filling space with cubes of two sizes

The problem to classify the unilateral and equitransitive tilings of the plane by squares of different sizes has been revived in the last few years ([6], [1], [8]). The analogous problem in

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

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

A Cube Tiling of Dimension Eight with No Facesharing

A cube tiling of eight-dimensional space in which no pair of cubes share a complete common seven-dimensional face is constructed shows that the first dimension in which such a tiling can exist is seven or eight.

Lattice Tilings by Cubes: Whole, Notched and Extended

A new class of simple shapes that admit lattice tilings, the "extended cubes", which are unions of two axis-aligned rectangles that share a vertex and have intersection of odd codimension are exhibited.

Perfectly packing a square by squares of nearly harmonic sidelength

A well known open problem of Meir and Moser asks if the squares of sidelength 1/n for n ≥ 1 can be packed perfectly into a square of area ∑∞ n=1 1 n2 = π 6 . In this paper we show that for any 1/2 <

On the Shannon capacity of a graph

  • L. Lovász
  • Mathematics, Computer Science
    IEEE Trans. Inf. Theory
  • 1979
It is proved that the Shannon zero-error capacity of the pentagon is \sqrt{5} and a well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases.

On Filling Space with Different Integer Cubes

What is known about unit cubes

The purpose of this article is to figure out what is known about the unit cubes and what do the authors want to know about them.

New lower bounds for the Shannon capacity of odd cycles

Improved lower bounds on the Shannon capacity of C_7$$C7 and C_{15}$$C15 are shown by prescribing stabilizers for the independent sets in Cpd and using stochastic search methods.