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

Automatic generation of colored patterns in the hyperbolic 3-space is consi dered from a dynamical system's point of view. The symmetries of various tw o-dimensional cross-sections of a tessellation are related to the 17 wallpa per groups and the modular group. The computer images generated in this pap er reveal the exotic structures of the tessellations in the hyperbolic 3-sp ace. 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 ge nerated image. This method can be used to create a great variety of aesthet ic symmetrical patterns. (C) 2001 Elsevier Science Ltd. All rights reserved .