Abstract. In this work we study the color symmetries pertaining to colorings of M n ¼ Z½x, where x ¼ exp ð2pi=nÞ for n 2 f5; 8; 12g which yield standard symmetries of quasicrystals. The first part of the paper treats M n as a four dimensional lattice L with symmetry group G and a result is provided on sublattices of L which are invariant under the point group of G. The second part of the paper characterizes the color symmetry groups and color fixing groups corresponding to Bravais colorings of M n using an approach involving ideals.

Abstract. We obtain tilings with a singular point by applying conformal maps on regular tilings of the Euclidean plane, and determine its symmetries. The resulting tilings are then symmetrically colored by applying the same conformal maps on colorings of regular tilings arising from sublattice colorings of the centers of its tiles. In addition, we determine conditions so that the coloring of a tiling with singularity that is obtained in this manner is perfect.

"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."