Multiple Lattice Tilings in Euclidean Spaces

Yang, Qi ; Zong, Chuanming

Canadian mathematical bulletin, Tome 62 (2019) no. 4, pp. 923-929

Résumé

In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results. Except for parallelograms and centrally symmetric hexagons, there are no other convex domains that can form two-, three- or four-fold lattice tilings in the Euclidean plane. However, there are both octagons and decagons that can form five-fold lattice tilings. Whenever $n\geqslant 3$, there are non-parallelohedral polytopes that can form five-fold lattice tilings in the $n$-dimensional Euclidean space.

Détail

DOI : 10.4153/S0008439518000103

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