Characteristics of fractal cellular automata constructed from linear rules

  • Yoshihiko KayamaEmail author
Original Article


Cellular automata (CAs) have played a significant role in the study of complex systems. Recently, the recursive estimation of neighbors (REN) algorithm was proposed to extend a CA rule with a unit rule radius to rules with larger radii. This framework enables the construction of non-uniform CAs comprising cells that follow different CA rules. A non-uniform CA, referred to as fractal CA (F-CA), which comprises fractally arranged cells, inherits certain characteristics of basic CAs, including pattern replicability and time-reversibility of linear rules. In this paper, F-CAs based on linear rules, particularly the elementary CA #90 and #150, and life-like CA B1357S1357 and B1357S02468 are investigated. Cells in the F-CAs of #90 and B1357S1357 are separated into groups by their rule radius and each group has an independent lifetime. The explicitly constructed inverse rule of F-CA of #150 is more complex than that of F-CA. The complexity of the F-CA of B1357S02468 and its inverse CA is demonstrated by image scrambling. The F-CAs can be applied to encoding and decoding processes for encryption systems.


CA Non-uniform Life-like Reversible Inverse Encryption 



The author wishes to thank all the anonymous reviewers for valuable comments and suggestions. This research was supported in part by a grant from BAIKA Gakuen, Japan.


  1. 1.
    von Neumann J (1966) The theory of self-reproducing automata. In: Burks AW (ed) Essays on cellular automata. University of Illinois Press, ChampaignGoogle Scholar
  2. 2.
    Wolfram S (1986) Theory and applications of cellular automata. World Scientific, SingaporezbMATHGoogle Scholar
  3. 3.
    Hansen PB (1993) Parallel cellular automata: a model program for computational science. Concurr Pract Exp 5:425–448CrossRefGoogle Scholar
  4. 4.
    Ermentrout GB, Edelstein-Keshet L (1993) Cellular automata approaches to biological modelling. J Theor Biol 160:97–133CrossRefGoogle Scholar
  5. 5.
    Ganguly N, Sikdar BK, Deutsch A, Canright G, Chaudhuri PP (2003) A survey on cellular automataGoogle Scholar
  6. 6.
    Chopard B, Droz M (2005) Cellular automata modeling of physical systems. Cambridge University Press, CambridgezbMATHGoogle Scholar
  7. 7.
    Kayama Y (2016) Extension of cellular automata via the introduction of an algorithm for the recursive estimation of neighbors. Artif Life Robot 21(3):338–344CrossRefGoogle Scholar
  8. 8.
    Kayama Y (2016) Expansion of perception area in cellular automata using recursive algorithm. In: Proceeding of the fifteenth international conference on the simulation and synthesis of living systems, pp 92–99Google Scholar
  9. 9.
    Gerald E (2004) Classics on fractals. Westview Press, BoulderzbMATHGoogle Scholar
  10. 10.
    Trochet H (2009) A history of fractal geometry. University of St Andrews MacTutor History of Mathematics.
  11. 11.
    Mandelbrot BB, Pignoni R (1983) The fractal geometry of nature, vol 173. WH Freeman, New YorkGoogle Scholar
  12. 12.
    Briggs J (1992) Fractals: the patterns of chaos: a new aesthetic of art, science, and nature. Simon and Schuster, New YorkGoogle Scholar
  13. 13.
    Falconer K (2004) Fractal geometry: mathematical foundations and applications. Wiley, HobokenzbMATHGoogle Scholar
  14. 14.
    Song C, Havlin S, Makse HA (2005) Self-similarity of complex networks. Nature 433(7024):392CrossRefGoogle Scholar
  15. 15.
    Kayama Y (2018) Cellular automata in fractal arrangement. Artif Life Robot 23:395–401CrossRefGoogle Scholar
  16. 16.
    Wolfram S (1983) Statistical mechanics of cellular automata. Rev Mod Phys 55:601–644MathSciNetCrossRefGoogle Scholar
  17. 17.
    Wolfram S (2002) A new kind of science. Wolfram Media Inc, ChampaignzbMATHGoogle Scholar
  18. 18.
    Adamatzky A (ed) (2010) Game of life cellular automata. Springer, LondonzbMATHGoogle Scholar
  19. 19.
    Eppstein D (2010) Growth and decay in life-like cellular automata. In: Adamatzky A (ed) Game of life cellular automata. Springer, Berlin, pp 71–98CrossRefGoogle Scholar
  20. 20.
    Kayama Y (2011) Network representation of cellular automata. In: 2011 IEEE symposium on artificial life (ALIFE), pp 194–202Google Scholar
  21. 21.
    Fredkin E (1990) An informational process based on reversible universal cellular automata. Phys D Nonlinear Phenom 45(1–3):254–270MathSciNetCrossRefGoogle Scholar
  22. 22.
    Reynolds CW (1987) Flocks, herds and schools: a distributed behavioral model. ACM Siggraph Comput Graph 21(4):25–34CrossRefGoogle Scholar
  23. 23.
    Kayama Y, Koda Y, Yazawa I (2018) Fractal arrangement for 2D cellular automata and its implementation for outer-totalistic rules. In: Proceedings of thirteenth International Conference on Cellular Automata for Research and Industry, Springer, pp 328–339Google Scholar
  24. 24.
    Gardner M (1970) Mathematical games. Sci Am 223:102–123CrossRefGoogle Scholar
  25. 25.
    Berlekamp ER, Conway JH, Guy RK (1982) Winning ways for your mathematical plays, vol 2. AK Peters, NatickzbMATHGoogle Scholar

Copyright information

© International Society of Artificial Life and Robotics (ISAROB) 2019

Authors and Affiliations

  1. 1.Department of Media and InformationBAIKA Women’s UniversityIbarakiJapan

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