Tessellations with symmetries of the wallpaper groups and the modular group in the hyperbolic 3-space from dynamics

Abstract

Automatic generation of colored patterns in the hyperbolic 3-space is considered from a dynamical system’s point of view. The symmetries of various two-dimensional cross-sections of a tessellation are related to the 17 wallpaper groups and the modular group. The computer images generated in this paper reveal the exotic structures of the tessellations in the hyperbolic 3-space. A color scheme is described, which reflects the rate of convergence of various orbits and, at the same time, enhances the artistic appeal of a generated image. This method can be used to create a great variety of aesthetic symmetrical patterns.