A locally finite face-to-face tiling of euclidean d-space by convex polytopes is called combinatorially multihedral\/ if its combinatorial automorphism group has only finitely many orbits on the tiles. The paper describes a local characterization of combinatorially multihedral tilings in terms of centered coronas. This generalizes the Local Theorem for Monotypic Tilings, established in dolsch, which characterizes the case of combinatorial tile-transitivity.
Долбилин et al. (Mon,) studied this question.