The Three and Fourfold Translative Tiles in Three-Dimensional Space

TL;DR

This paper characterizes three- and fourfold translative tilings in 3D space, proving that only specific convex polyhedra, known as parallelohedra, can form such tilings.

Contribution

It establishes a complete classification of convex bodies capable of three- or fourfold translative tilings in three-dimensional space.

Findings

Only parallelohedra can form three- or fourfold translative tilings in 3D.

Identifies five specific polyhedra that can tile in this manner.

Provides a proof confirming the classification of these tilings.

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.