Geodesic
Multiple Lattice Tilings in Euclidean Spaces
Canadian mathematical bulletin, Tome 62 (2019) no. 4, pp. 923-929

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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. 
Yang, Qi; Zong, Chuanming. Multiple Lattice Tilings in Euclidean Spaces. Canadian mathematical bulletin, Tome 62 (2019) no. 4, pp. 923-929. doi: 10.4153/S0008439518000103
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     title = {Multiple {Lattice} {Tilings} in {Euclidean} {Spaces}},
     journal = {Canadian mathematical bulletin},
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     year = {2019},
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