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Translational Tile

A finite set S subset Z^d is a translational tile of the integer lattice if there is a set T subset Z^d such that the translates S+t, for t in T, partition Z^d.

For fixed d>=1, let t_(n,d) be the number of subsets of the integer box [n]^d, where [n]={1,2,...,n}, that tile Z^d by translations. Benjamini et al. (2026) proved the asymptotic formula

t_(n,d)=(3^(1/3))^(n^d+/-o(n^d)).

See also

Integer Lattice, Tiling, Translate, Translation

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References

Benjamini, I.; Kozma, G.; and Tzalik, E. "The Number of Tiles of Z^d." Elec. J. Combin. 33, P3.28, 2026. https://doi.org/10.37236/14513.

Cite this as:

Weisstein, Eric W. "Translational Tile." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TranslationalTile.html

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