Tiling space by platonic solids, I

Abstract

There exist precisely 914, 58, and 46 equivariant types of tile-transitive tilings of three-dimensional euclidean space by topological cubes, octahedra, and tetrahedra, that fall into 11, 3, and 9 topological families, respectively. Representatives are described for all topological families. A general method for obtaining results of this kind is introduced.