Symmetric Tiling Patterns with the Extended Picard Group in Three-Dimensional Space
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.
KeywordsConformal Mapping Hyperbolic Space Modular Group Symmetric Pattern Fundamental Region
Unable to display preview. Download preview PDF.
- 3.Escher, M.C.: Escher on Escher–Exploring the Infinity. Harry N. Abrams, New York (1989)Google Scholar
- 4.Schattschneider, D.: Visions of Symmetry: Notebooks, Periodic Drawings and Related Works of M.C. Escher. Freeman, New York (1990)Google Scholar
- 5.Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups, 4th edn. Springer, New York (1980)Google Scholar
- 10.Pickover, C.A.: Computers, Pattern, Chaos and Beauty. Alan Sutton Publishing, Stroud Glouscestershire (1990)Google Scholar