Skip to main content

Cellular automata, matrix substitutions and fractals

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

It has been observed that the long time evolution of a cellular automata (CA) can generate fractal sets. In this paper, we define a broad class of CA, all of which have a limit set. Moreover, we present an algorithm which associates with a CA of the above defined class a substitution system which deciphers the self-similarity structure of the limit set.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. J.-P. Allouche, Finite automata in 1-D and 2-D physics, in:Number Theory of Physics, ed. J.-M. Lucu, P. Moussa and M. Waldschmidt.

  2. T. Bedford, Dynamics and dimension for fractal recurrent sets, J. London Math. Soc. 33(1986) 89–100.

    Google Scholar 

  3. F.M. Dekking, Recurrent sets, Adv. Math. 44(1982)78–104.

    Google Scholar 

  4. F.M. Dekking, Substitutions, branching processes and fractal sets, in:Fractal Geometry and Analysis, ed. J. Belair and S. Dubuc (Kluwer, 1991).

  5. F. v. Haeseler, H.-O, Peitgen and G. Skordev, Pascal's triangle, dynamical systems, and attractors, Ergodic Theory and Dynamical Systems 12(1992)479–486.

    Google Scholar 

  6. F. v. Haeseler, H.-O, Peitgen and G. Skordev, Linear cellular automata, substitutions, hierarchical iterated function systems and attractors, in:Fractal Geometry and Computer Graphics, ed. J.L. Encarnacao, H.-O. Peitgen, G. Sakas and G. Englert (Springer, Heidelberg, 1992).

    Google Scholar 

  7. F. v. Haeseler, H.-O. Peitgen and G. Skordev, On the fractal structure of limit sets of cellular automata and attractors of dynamical systems, submitted.

  8. G. Hedlund, Endomorphisms and automorphisms of the shift dynamical systems, Math. Syst. Th. 3(1969)320–375.

    Google Scholar 

  9. B. Mandelbrot,The Fractal Geometry of Nature (Freeman, New York, 1982).

    Google Scholar 

  10. B. Mandelbrot, Y. Gefen, A. Aharony and J. Peyriere, Fractals, their transfer matrices and their eigen-dimensional sequences, J. Phys. A: Math. Gen. 18(1985)335–354.

    Google Scholar 

  11. M. Queffelec,Substitutional Dynamical Systems, Lecture Notes in Math. 1294 (Springer, 1987).

  12. J. Shallit and J. Stolfi, Two methods for generating fractals, Comp. Graphics 13(1989)185–191.

    Google Scholar 

  13. S. Takahashi, Cellular automata and multifractals: dimension spectra of linear cellular automata, Physica D45(1990)36–48.

    Google Scholar 

  14. S. Takahashi, Self-similarity of linear cellular automata, J. Comp. Sci. 44(1992)114–140.

    Google Scholar 

  15. S. Willson, Cellular automata can generate fractals, Discr. Appl. Math. 8(1984)91–99.

    Google Scholar 

  16. S. Willson, The equality of fractional dimension for certain cellular automata, Physica D24(1987) 179–189.

    Google Scholar 

  17. S. Willson, Computing fractal dimensions for additive cellular automata, Physica D24(1987) 190–206.

    Google Scholar 

  18. S. Wolfram, Statistical mechanics and cellular automata, Rev. Mod. Phys. 55(1983)601–644.

    Google Scholar 

  19. S. Wolfram, Universality and complexity in cellular automata, Physica D10(1984)1–35.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Supported by DFG “Forschungsgruppe Dynamische Systems”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Haeseler, F.v., Peitgen, H.O. & Skordev, G. Cellular automata, matrix substitutions and fractals. Ann Math Artif Intell 8, 345–362 (1993). https://doi.org/10.1007/BF01530797

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01530797

Keywords