Skip to main content
SpringerLink
Log in

Two-dimensional cellular automata

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

A largely phenomenological study of two-dimensional cellular automata is reported. Qualitative classes of behavior similar to those in one-dimensional cellular automata are found. Growth from simple seeds in two-dimensiona! cellular automata can produce patterns with complicated boundaries, characterized by a variety of growth dimensions. Evolution from disordered states can give domains with boundaries that execute effectively continuous motions. Some global properties of cellular automata can be described by entropies and Lyapunov exponents. Others are undecidable.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Wolfram, Cellular automata as models for complexity,Nature 311:419 (1984).

    Google Scholar 

  2. T. Toffoli, Cellular automata mechanics, Ph.D. thesis and Technical Report 208, The Logic of Computers Group, University of Michigan (1977).

  3. G. Vichniac, Simulating physics with cellular automata,Physica 10D:96 (1984).

    Google Scholar 

  4. S. Wolfram, Statistical mechanics of cellular automata,Rev. Mod. Phys. 55:601 (1983).

    Google Scholar 

  5. O. Martin, A. Odlyzko, and S. Wolfram, Algebraic properties of cellular automata,Commun. Math. Phys. 93:219 (1984).

    Google Scholar 

  6. N. Packard, Cellular automaton models for dendritic crystal growth, Institute for Advanced Study, preprint (1985).

  7. S. Wolfram, Universality and complexity in cellular automata,Physica 10D:1 (1984).

    Google Scholar 

  8. E. R. Berlekamp, J. H. Conway, and R. K. Guy,Winning Ways for Your Mathematical Plays, Vol. 2 (Academic Press, New York, 1982), Chap. 25; M. Gardner,Wheels, Life and Other Mathematical Amusements (Freeman, San Francisco, 1983).

    Google Scholar 

  9. T. Toffoli, CAM: A high-performance cellular-automaton machine,Physica 10D:195 (1984).

    Google Scholar 

  10. J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983).

    Google Scholar 

  11. J. E. Hopcroft and J. D. Ullman,Introduction to Automata Theory, Languages, and Computation (Addison-Wesley, Reading, Massachusetts, 1979).

    Google Scholar 

  12. S. Wolfram, Computation theory of cellular automata,Commun. Math. Phys. 96:15 (1984).

    Google Scholar 

  13. L. Hurd, Formal language characterizations of cellular automaton limit sets, to be published.

  14. S. Willson, On convergence of configurations,Discrete Math. 23:279 (1978).

    Google Scholar 

  15. T. Yaku, The constructability of a configuration in a cellular automaton,J. Comput. Syst. Sci. 7:481 (1973); U. Golze, Differences between 1- and 2-dimensional cell spaces, inAutomata, Languages, Development, A. Lindenmayer and G. Rozenberg, eds. (North-Holland, Amsterdam, 1976).

    Google Scholar 

  16. J. Hardy, O. de Pazzis, and Y. Pomeau, Molecular dynamics of a classical lattice gas: transport properties and time correlation functions,Phys. Rev. A13:1949 (1976).

    Google Scholar 

  17. S. Wolfram, Cellular automata,Los Alamos Science (Fall 1983); Some recent results and questions about cellular automata, Institute for Advanced Study preprint (September 1983).

  18. S. Wolfram, Twenty problems in the theory of cellular automata,Phys. Scripta (to be published).

  19. S. Wolfram, Computer software in science and mathematics,Sci. Am. 251(3):188 (1984).

    Google Scholar 

  20. S. Willson, Comparing limit sets for certain increasing cellular automata, Mathematics Department, Iowa State University preprint (June 1984).

  21. S. Willson, Growth rates and fractional dimensions in cellular automata,Physica 10D:69 (1984).

    Google Scholar 

  22. N. Packard, Notes on a Go-playing program, unpublished (1984).

  23. Y. Sawada, M. Matsushita, M. Yamazaki, and H. Kondo, Morphological phase transition measured by “surface kinetic dimension” of growing random patterns,Phys. Scripta (to be published).

  24. J. M. Greenberg, B. D. Hassard, and S. P. Hastings, Pattern formation and periodic structures in systems modelled by reaction-diffusion equations,Bull. Am. Math. Soc. 84:1296 (1975); B. Madore and W. Freedman, Computer simulations of the Belousov-Zhabotinsky reaction,Science 222:615 (1983).

    Google Scholar 

  25. K. Prestonet al., Basics of cellular logic with some applications in medical image processing,Proc. IEEE 67:826 (1979).

    Google Scholar 

  26. J. W. Essam, Percolation theory,Rep. Prog. Phys. 43:833 (1980).

    Google Scholar 

  27. P. Grassberger, Chaos and diffusion in deterministic cellular automata,Physica 10D:52 (1984).

    Google Scholar 

  28. G. Vichniac, Cellular automaton dynamics for interface motion and ordering, M.I.T. report, to appear.

  29. R. Rosensweig, Fluid dynamics and science of magnetic liquids,Adv. Electronics Electron Phys. 48:103 (1979); Magnetic fluids,Sci. Am. 247(4):136 (1982).

    Google Scholar 

  30. G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system,Math. Syst. Theory 3:320 (1969); G. A. Hedlund, Transformations commuting with the shift, inTopological Dynamics, J. Auslander and W. H. Gottschalk, eds. (Benjamin, New York, 1968); S. Amoroso and Y. N. Patt, Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures,J. Comp. Syst. Sci. 6:448 (1972); M. Nasu, Local maps inducing surjective global maps of one-dimensional tessellation automata,Math. Syst. Theory 11:327 (1978).

    Google Scholar 

  31. H. Wang, Proving theorems by pattern recognition-II,Bell. Syst. Tech. J. 40:1 (1961).

    Google Scholar 

  32. R. Berger, The undecidability of the domino problem,Mem. Am. Math. Soc., No.66 (1966).

  33. R. Robinson, Undecidability and nonperiodicity for tilings of the plane,Inventiones Math. 12:177 (1971).

    Google Scholar 

  34. D. Ruelle,Thermodynamic Formalism (Addison-Wesley, Reading, Massachusetts, 1978), p. 68.

    Google Scholar 

  35. R. Penrose, Pentaplexity: a class of nonperiodic tilings of the plane,Math. Intelligencer 2:32 (1979); M. Gardner, Extraordinary nonperiodic tiling that enriches the theory of tiles,Sci. Am. 236(1):110 (1977); N. de Bruijn, Algebraic theory of Penrose's nonperiodic tilings of the plane,Nederl. Akad. Wetensch. Indag. Math. 43:39 (1981).

    Google Scholar 

  36. P. Kramer, Non-periodic central space filling with icosahedral symmetry using copies of seven elementary cells,Acta Crystallogr. A38:257 (1982).

    Google Scholar 

  37. M. Garey and D. Johnson,Computers and Intractability: A Guide to the Theory of NP-completeness (Freeman, San Francisco, 1979), p. 257.

    Google Scholar 

  38. J. Milnor, Entropy of cellular automaton-maps, Institute for Advanced Study preprint (May 1984).

  39. J. Milnor, Directional entropies in higher dimensions, rough notes (September 1984).

  40. N. Packard, Complexity of growing patterns in cellular automata, Institute for Advanced Study preprint (October 1983).

  41. J. Milnor, Notes on surjective cellular automaton-maps, Institute for Advanced Study preprint (June 1984).

  42. N. D. Mermin, The topological theory of defects in ordered media,Rev. Mod. Phys. 51:591 (1979).

    Google Scholar 

  43. A. Winfree and E. Winfree, Organizing centers in a cellular excitable medium, Purdue University preprint (July 1984) andPhysica D (to be published).

Download references

Author information

Authors and Affiliations

  1. The Institute for Advanced Study, 08540, Princeton, New Jersey

    Norman H. Packard & Stephen Wolfram

Authors
  1. Norman H. Packard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephen Wolfram
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported in part by the U.S. Office of Naval Research under Contract No. N00014-80-C-0657.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Packard, N.H., Wolfram, S. Two-dimensional cellular automata. J Stat Phys 38, 901–946 (1985). https://doi.org/10.1007/BF01010423

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01010423

Key words

Search

Navigation