Multiple Translative Tilings in Euclidean Spaces

2017/11/04 by Qi Yang, Yang, Qi, Chuanming Zong +1 · 1 citation
Computer Science · Engineering · Materials Science · #52C22 #Cellular Automata and Applications #FOS: Mathematics #Metric Geometry (math.MG) #Quasicrystal Structures and Properties #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1711.02514

openalex publication_date 2017/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Characterization of the Two-Dimensional Five-Fold Translative Tiles

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