Multiple Translative Tilings in Euclidean Spaces
arXiv:1711.02514
Abstract
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: Besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form a two-, three- or four-fold translative tiling in the Euclidean plane. However, there are two-dimensional convex domains which is neither a parallelogram nor a centrally symmetric hexagon can form five-fold translative tilings.
12 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1712.01122, arXiv:1710.05506