Twofold Translative Tiles in Three-Dimensional Space

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.

References in corpus (5)

• The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices
• Multi-tiling and equidecomposability of polytopes by lattice translates
• Characterization of the Two-Dimensional Five-Fold Lattice Tiles
• Twofold Translative Tilings with Convex Bodies
• Characterization of the Two-Dimensional Six-Fold Lattice Tiles