On tile-k-transitive tilings of the space

General methods for finding tile-k-transitive tilings of the three-dimensional Euclidean space with polyhedral bodies are discussed. Analogous methods for enumerating k-isohedral tilings of a two-dimensional plane of constant curvature have been obtained previously.     

The classification of 2-isohedral tilings of the plane

Two different methods for enumerating k-isohedral tilings are discussed. One is geometric: by splitting and gluing tiles. The other is combinatorial: by enumerating the appropriate Delaney—Dress symbols. Both methods yield 1270 types of proper 2-isohedral tilings of the plane.