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arXiv · 1710.05506

Multiple Lattice Tilings in Euclidean Spaces

Qi Yang·Chuanming Zong
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Abstract

This paper proves the following results: Besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form a two-, three- or four-fold lattice tiling in the Euclidean plane. If a centrally symmetric octagon can form a lattice multiple tiling, then the multiplicity is at least seven. However, there are decagons which can form five-fold $($or six-fold$)$ lattice tilings. Consequently, whenever $n\ge 3$, there are non-parallelohedral polytopes which can form five-fold lattice tilings in the $n$-dimensional Euclidean space.

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Qi Yang, Chuanming Zong. 2018-03-17. Multiple Lattice Tilings in Euclidean Spaces. https://doi.org/10.4153/s0008439518000103

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