Symmetries and color symmetries of a family of tilings with a singular point

TL;DR

This paper constructs tilings with singular points using conformal maps on regular Euclidean tilings, analyzes their symmetries, and explores conditions for perfect colorings derived from sublattice colorings.

Contribution

It introduces a method to generate singular point tilings via conformal maps and characterizes their symmetry and coloring properties, including conditions for perfect colorings.

Findings

Tilings with singular points are obtained through conformal transformations.

Symmetries of these tilings are explicitly determined.

Conditions for perfect colorings of the tilings are established.

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.