The Three and Fourfold Translative Tiles in Three-Dimensional Space

2021/09/15 by Han, Mei, Sriamorn, Kirati, Yang, Qi +1
#05B45 #52C17 #52C22 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2109.07116

Abstract

This paper proves the following statement: If a convex body can form a three or fourfold translative tiling in the three-dimensional space, it must be a parallelohedron. In other words, it must be a parallelotope, a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, or a truncated octahedron.

Related

• Twofold Translative Tiles in Three-Dimensional Space
• Characterization of the Three-Dimensional Fivefold Translative Tiles
• Multiple Translative Tilings in Euclidean Spaces
• Characterization of the Two-Dimensional Five-Fold Translative Tiles
• Twofold Translative Tilings with Convex Bodies