2017/10/31 by Qi Yang, Chuanming Zong
Computer Science · Engineering · Materials Science · Mathematics · #Cellular Automata and Applications #Combinatorics #Convex body #Convex domain #Convex hull #Euclidean distance #Euclidean geometry #Euclidean space #Geometry #Lattice (music) #Mathematics #Parallelogram #Polytope #Quasicrystal Structures and Properties #Regular polygon #Substitution tiling #graph theory and CDMA systems #math.MG #msc:52C22
paper · pdf · doi:10.4153/s0008439518000103
published as Can. Math. Bull. 62 (2019) 923-929 · 6 pages, 2 figures
arxiv created 2018/03/17 · openalex publication_date 2018/11/16 · arxiv updated 2019/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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. 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.