Abstract
Automatic generation of fractal tiling patterns with the symmetry of the extended modular group is considered. Thanks to the unique one to one relationship between points in the upper half complex plane and the corresponding points in the fundamental region of the extended modular group, we can generate fractal tiling with the extended modular group by repeating the fractal pattern created in the fundamental region to the other tiling regions. We also produce such a kind of tiling in the unit disk by conformal mapping fractal tiling in the upper half complex plane. The method provides a novel approach for devising exotic fractal tiling patterns with symmetries.
Supported by Tianyuan Foundation, National Nature Science Foundation of China (No. A0324649).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Escher, M.C.: Escher on Escher–Exploring the Infinity. Harry N. Abrams, New York (1989)
Schattschneider, D.: Visions of Symmetry: Notebooks, Periodic Drawings and Related Works of M.C. Escher. Freeman, New York (1990)
Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups, 4th edn. Springer, New York (1980)
Weyl, H.: Symmetry. Princeton University Press, Princeton (1952)
Armstrong, M.A.: Groups and Symmetry. Springer, New York (1988)
Dunham, D.: Hyperbolic symmetry. Computers & Mathematics with Applications 12B, 139–153 (1986)
Pickover, C.A.: Computers, Pattern, Chaos and Beauty. Alan Sutton Publishing, Stroud Glouscestershire (1990)
Field, M., Golubistky, M.: Symmetry in Chaos. Oxford University Press, New York (1992)
Carter, N., Eagles, R., Grimes, S., Hahn, A., Reiter, C.: Chaotic attractors with discrete planar symmetries. Chaos, Solitons and Fractals 9, 2031–2054 (1998)
Chung, K.W., Chan, H.S.Y., Wang, B.N.: Hyperbolic symmetries from dynamics. Computers & Mathematics with Applications 31, 33–47 (1996)
Chung, K.W., Chan, H.S.Y., Wang, B.N.: Tessellations with the modular group from dynamics. Computers & Graphics 21, 523–534 (1997)
Barnsley, M.F.: Fractals Everywhere. Academic Press, San Diego (1988)
Ye, R.: Another choice for orbit traps to generate artistic fractal images. Computers & Graphics 26, 629–633 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ye, Rs., Zou, Yr., Lu, J. (2004). Fractal Tiling with the Extended Modular Group. In: Zhang, J., He, JH., Fu, Y. (eds) Computational and Information Science. CIS 2004. Lecture Notes in Computer Science, vol 3314. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30497-5_45
Download citation
DOI: https://doi.org/10.1007/978-3-540-30497-5_45
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24127-0
Online ISBN: 978-3-540-30497-5
eBook Packages: Computer ScienceComputer Science (R0)