Patterns with Symmetries of the Wallpaper Group on the Hyperbolic Space

Abstract

Equivariant function with respect to symmetries of the wallpaper group is constructed by trigonometric functions. A proper transformation is established between Euclidean plane and hyperbolic spaces. With the resulting function and transformation, wallpaper patterns on the Poincaré and Klein models are generated by means of dynamic systems. This method can be utilized to produce infinity of beautiful pattern automatically.