Characterization of the Three-Dimensional Fivefold Translative Tiles

2023/07/15 by Han, Mei, Sriamorn, Kirati, Yang, Qi +1
#05B45 #11H31 #52C17 #52C22 #52C23 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2307.07824

Abstract

This paper proves the following statement: If a convex body can form a fivefold translative tiling in 𝔼³, it must be a parallelotope, a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, a truncated octahedron, a cylinder over a particular octagon, or a cylinder over a particular decagon, where the octagon and the decagon are fivefold translative tiles in 𝔼². Furthermore, it presents an example of multiple tiles in 𝔼³ with multiplicity at most 10 which is neither a parallelohedron nor a cylinder.

Related

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