Highly symmetric (a, b, c, d) tilings with cubical symmetries

This paper presents a method for constructing tilings of the Euclidean space . Here, an tiling is defined as a tiling with , , and transitivity classes of vertices, edges, faces, and tiles, respectively, under the action of its symmetry group. By employing the process of dualization and subgroup structures of symmetry groups, tilings with varying transitivity properties can be derived from a known tiling. Using the method, one can use the subgroup structure of the symmetry group of a known tiling to construct new tilings. In particular, application of this method to the regular cubical tiling gives rise a new (1,1,1,2) tiling or a quasiregular tiling.