Characterization of the Two-Dimensional Five-Fold Lattice Tiles

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. It is known that there is no other convex domain which can form a two-, three- or four-fold lattice tiling in the Euclidean plane, but there is a centrally symmetric convex decagon which can form a five-fold lattice tiling. This paper characterizes all the convex domains which can form a five-fold lattice tiling of the Euclidean plane.

20 pages, 14 figures. arXiv admin note: text overlap with arXiv:1711.02514, arXiv:1710.05506