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

  • Rui-song Ye
  • Jian Ma
  • Hui-liang Li
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4035)


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.


Conformal Mapping Hyperbolic Space Modular Group Symmetric Pattern Fundamental Region 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Zhu, M., Murray, J.D.: Parameter domains for spots and stripes in mechanical models for biological pattern formation. Journal of Nonlinear Science 5, 317–336 (1995)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Weyl, H.: Symmetry. Princeton University Press, Princeton (1952)MATHGoogle Scholar
  3. 3.
    Escher, M.C.: Escher on Escher–Exploring the Infinity. Harry N. Abrams, New York (1989)Google Scholar
  4. 4.
    Schattschneider, D.: Visions of Symmetry: Notebooks, Periodic Drawings and Related Works of M.C. Escher. Freeman, New York (1990)Google Scholar
  5. 5.
    Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups, 4th edn. Springer, New York (1980)Google Scholar
  6. 6.
    Armstrong, M.A.: Groups and Symmetry. Springer, New York (1988)MATHGoogle Scholar
  7. 7.
    Carter, N., et al.: Chaotic attractors with discrete planar symmetries. Chaos, Solitons and Fractals 9, 2031–2054 (1998)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Dunham, D.: Hyperbolic symmetry. Computers & Mathematics with Applications 12B, 139–153 (1986)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Chung, K.W., Chan, H.S.Y., Wang, B.N.: Hyperbolic symmetries from dynamics. Computers & Mathematics with Applications 31, 33–47 (1996)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Pickover, C.A.: Computers, Pattern, Chaos and Beauty. Alan Sutton Publishing, Stroud Glouscestershire (1990)Google Scholar
  11. 11.
    Field, M., Golubistky, M.: Symmetry in Chaos. Oxford University Press, New York (1992)MATHGoogle Scholar
  12. 12.
    Chung, K.W., Chan, H.S.Y., Wang, B.N.: Tessellations with the modular group from dynamics. Computers & Graphics 21, 523–534 (1997)CrossRefGoogle Scholar
  13. 13.
    Chung, K.W., Chan, H.S.Y., Wang, B.N.: Tessellations in three-dimensional hyperbolic space from dynamics and the quaternions. Chaos, Solitons and Fractals 25, 1181–1197 (2001)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Ye, R., Zou, Y., Lu, J.: Fractal Tiling with the Extended Modular Group. In: Zhang, J., He, J.-H., Fu, Y. (eds.) CIS 2004. LNCS, vol. 3314, pp. 286–291. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. 15.
    Elstrodt, J., Grunewald, F., Mennicke, J.: Groups on hyperbolic space. Springer, Heidelberg (1998)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Rui-song Ye
    • 1
  • Jian Ma
    • 1
  • Hui-liang Li
    • 1
  1. 1.Department of MathematicsShantou UniversityShantou, GuangdongP.R. China

Personalised recommendations