Symmetric Tiling Patterns with the Extended Picard Group in Three-Dimensional Space

摘要

Automatic generation of tiling patterns with the symmetry of the extended Picard group in three-dimensional hyperbolic space is considered. We generate the patterns by repeating the fundamental patterns created in the fundamental region to all other equivalent regions. We also produce such a kind of tiling patterns in the unit sphere by conformal mappings. The method provides a novel approach for devising exotic symmetric tiling patterns from a dynamical system’s point of view.