The Three and Fourfold Translative Tiles in Three-Dimensional Space

This paper proves the following statement: If a convex body can form a three or fourfold translative tiling in 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.