Color Symmetry of Certain Tilings with a Singular Center

"Tilings with singular points, or tilings that are not locally finite, are classified in [1] among tilings that are not ""well-behaved"". In [2], colorings of tilings with a singular center were obtained from certain colorings of regular Euclidean tilings. It was observed that not all such colorings could be transformed into colorings of tilings with a singularity. Moreover, the existence of maximum color indexes was surmised. In this paper, we provide a mathematical basis for the said observations by utilizing conformal maps that distort a regular Euclidean tiling into a tiling with a singular center. That is, we determine conditions so that a coloring of a regular Euclidean tiling can be transformed into a coloring of a tiling with a singular center. In addition, we establish that a maximum number of colors exists. Finally, we give conditions so that the symmetry group of the tiling with a singular center induces a permutation of the colors."

Non-Perfect Colorings of Symmetrical Tilings with Color Groups of Index 4

In this work we present a method that will allow for the construction and enumeration of non-perfect colorings of symmetrical tilings. If G is the symmetry group of an uncolored symmetrical tiling, then a coloring of the symmetrical tiling is non-perfect if its associated color group is a proper subgroup of G. The process will facilitate a systematic construction of non-perfect colorings of a wider class of symmetrical tilings where the stabilizer of a tile in the symmetry group G of the uncolored symmetrical tiling is non-trivial and the set of tiles may not form a transitive set under the action of G. This poster discusses results on how to identify and characterize non-perfect colorings arising from the method with associated color groups of index 4. The approach obtained here provides an avenue to model and characterize various chemical structures with atoms of different proportions, and their symmetries. This is relevant particularly for understanding new and emerging structures, such as structural analogues of carbon nanotubes, where a lot of its physical and electronic properties depend on their symmetry.