Twofold Translative Tiles in Three-Dimensional Space

TL;DR

This paper proves that in three-dimensional space, any convex body capable of forming a twofold translative tiling must be one of the known parallelohedra, thus characterizing such tilings completely.

Contribution

It establishes a complete characterization of convex bodies that can form twofold translative tilings in three dimensions, showing they are necessarily parallelohedra.

Findings

Twofold translative tilings in 3D are only possible with parallelohedra.

Identifies all convex bodies that can form such tilings as specific known polyhedra.

Provides a rigorous proof confirming the classification of these tilings.

Abstract

This paper proves the following statement: {\it If a convex body can form a twofold translative tiling in $\mathbb{E}^3$, 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.