Characterization of the Three-Dimensional Fivefold Translative Tiles

TL;DR

This paper characterizes all convex bodies in three-dimensional space that can form fivefold translative tilings, identifying specific polyhedra and cylinders, and provides an example of a non-tiling multiple tile.

Contribution

It proves a complete classification of 3D convex bodies capable of fivefold translative tilings, extending understanding of tiling geometry.

Findings

Only specific polyhedra and cylinders can form fivefold translative tilings in 3D.

An example of a multiple tile with multiplicity at most 10 that is neither a parallelohedron nor a cylinder.

Provides a comprehensive characterization of 3D 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.