Characterization of the Three-Dimensional Fivefold Translative Tiles

Abstract

This paper proves the following statement: If a convex body can form a fivefold translative tiling in $\mathbb{E}^3$, 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 $\mathbb{E}^2$. Furthermore, it presents an example of multiple tiles in $\mathbb{E}^3$ with multiplicity at most 10 which is neither a parallelohedron nor a cylinder.