Characterization of the two-dimensional fivefold and sixfold lattice tiles

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. It is known that there is no other convex domain which can form a two, three or fourfold lattice tiling in the Euclidean plane, but there are centrally symmetric convex octagons and decagons which can form fivefold lattice tilings. This paper characterizes all the convex domains which can form five or sixfold lattice tilings of the Euclidean plane. They are parallelograms, centrally symmetric hexagons, three types of centrally symmetric octagons and three types of centrally symmetric decagons.