Tessellations in three-dimensional hyperbolic space from dynamics and the quaternions

Automatic generation of coloured patterns in the three-dimensional hyperbolic space is considered from a dynamical system’s point of view. Equivariant mappings with the symmetry of the Picard group is constructed. A convergence colour scheme is described, which reveals the convergence rate 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 exotic symmetrical patterns.