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.

arXiv comment20 pages, 14 figures. arXiv admin note: text overlap with arXiv:1711.02514, arXiv:1710.05506

References in corpus (2)

• Exhaustive search of convex pentagons which tile the plane
• Multiple Translative Tilings in Euclidean Spaces

Cited by in corpus (6)

• Multiple Translative Tilings in Euclidean Spaces
• Characterization of the Two-Dimensional Five-Fold Translative Tiles
• Characterization of the Two-Dimensional Six-Fold Lattice Tiles
• Twofold Translative Tiles in Three-Dimensional Space
• Can You Pave the Plane Nicely with Identical Tiles
• The Three and Fourfold Translative Tiles in Three-Dimensional Space