Computational Complexity

2012 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Cellular Automata in Hyperbolic Spaces

  • Maurice Margenstern
Reference work entry

Article Outline


Definition of the Subject


The Locating Problem in Hyperbolic Tilings

Implementation of Cellular Automata in Hyperbolic Spaces

Complexity of Cellular Automata in Hyperbolic Spaces

On Specific Problems of Cellular Automata

Universality in Cellular Automata in Hyperbolic Spaces

The Connection with Tiling Problems

Future Directions




Span Tree Cellular Automaton Turing Machine Hyperbolic Space Splitting Method 
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.
This is a preview of subscription content, log in to check access.



The author is particularly in debt to Andrew Adamatzky for giving him the task to write this article.


Primary Literature

  1. 1.
    Chelghoum K, Margenstern M, Martin B, Pecci I (2004) Celluladr automata in the hyperbolic plane: proposal for a new environment. In: Proceedings of ACRI’2004, Amsterdam, 25–27 October 2004. Lecture Notes in Computer Sciences, vol 3305. Springer, Berlin, pp 678–687Google Scholar
  2. 2.
    Fraenkel AS (1985) Systems of numerations. Amer Math Mon 92:105–114MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Gromov M (1981) Groups of polynomial growth and expanding maps. Publ Math IHES 53:53–73MathSciNetMATHGoogle Scholar
  4. 4.
    Herrmann F, Margenstern M (2003) A universal cellular automaton in the hyperbolic plane.Theor Comput Sci 296:327–364MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Iwamoto C, Margenstern M (2003) A survey on the Complexity Classes in Hyperbolic Cellular Automata. In: Proceedings of SCI’2003, V, pp 31–35Google Scholar
  6. 6.
    Iwamoto C, Margenstern M, Morita K, Worsch T (2002) Polynomial‐time cellular automata in the hyperbolic plane accept accept exactly the PSPACE Languages. SCI’2002, Orlando, pp 411–416Google Scholar
  7. 7.
    Margenstern M (2000) New tools for cellular automata in the hyperbolic plane. J Univers Comput Sci 6(12):1226–1252MathSciNetMATHGoogle Scholar
  8. 8.
    Margenstern M (2002) A contribution of computer science to the combinatorial approach to hyperbolic geometry. In: SCI’2002, Orlando, USA, 14–19 July 2002Google Scholar
  9. 9.
    Margenstern M (2002) Revisiting Poincaré’s theorem with the splitting method. In: Bolyai’200, International Conference on Geometry and Topology, Cluj-Napoca, Romania, 1–3 October 2002Google Scholar
  10. 10.
    Margenstern M (2003) Implementing Cellular Automata on the Triangular Grids of the Hyperbolic Plane for New Simulation Tools. ASTC’2003, Orlando, 29 March–4 AprilGoogle Scholar
  11. 11.
    Margenstern M (2004) The Tiling of the Hyperbolic 4D Space by the 120-cell is Combinatoric. J Univers Comput Sci 10(9):1212–1238MathSciNetGoogle Scholar
  12. 12.
    Margenstern M (2006) A new way to implement cellular automata on the penta- and heptagrids. J Cell Autom 1(1):1–24MathSciNetMATHGoogle Scholar
  13. 13.
    Margenstern M (2007) A universal cellular automaton with five states in the 3D hyperbolic space. J Cell Autom 1(4):317–351MathSciNetGoogle Scholar
  14. 14.
    Margenstern M (2007) On the communication between cells of a cellular automaton on the penta- and heptagrids of the hyperbolic plane. J Cell Autom 1(3):213–232MathSciNetGoogle Scholar
  15. 15.
    Margenstern M (2007) Cellular Automata in Hyperbolic Spaces, vol 1: Theory. Old City Publishing, Philadelphia, p 422MATHGoogle Scholar
  16. 16.
    Margenstern M (2008) A Uniform and Intrinsic Proof that there are Universal Cellular Automata in Hyperbolic Spaces.J Cell Autom 3(2):157-180MathSciNetMATHGoogle Scholar
  17. 17.
    Margenstern M (2008) The domino problem of thehyperbolic plane is undecidable. Theor Comput Sci (to appear)Google Scholar
  18. 18.
    Margenstern M, Morita K (1999) A Polynomial Solution for 3-SAT in the Space of Cellular Automata in the Hyperbolic Plane. J Univers Comput Syst 5:563–573MathSciNetMATHGoogle Scholar
  19. 19.
    Margenstern M, Morita K (2001) NP problems are tractable in the space of cellular automata in the hyperbolic plane. Theor Comput Sci 259:99–128MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Margenstern M, Skordev G (2003) The tilings \( { \{p,q\} } \) of the hyperbolic plane are combinatoric. In: SCI'2003, V, pp 42–46Google Scholar
  21. 21.
    Margenstern M, Skordev G (2003) Tools for devising cellular automata in the hyperbolic 3D space. Fundamenta Informaticae 58(2):369–398MathSciNetMATHGoogle Scholar
  22. 22.
    Margenstern M, Song Y (2008) A new universal cellular automaton on the pentagrid. AUTOMATA’2008, Bristol, UK, 12–14 June 2008Google Scholar
  23. 23.
    Martin B (2005) VirHKey: a VIRtual Hyperbolic KEYboard with gesture interaction and visual feedback for mobile devices. In: MobileHCI’05, September, Salzburg, AustriaGoogle Scholar
  24. 24.
    Morgenstein D, Kreinovich V (1995) Which algorithms are feasible and which are not depends on the geometry of space-time. Geombinatorics 4(3):80–97MATHGoogle Scholar
  25. 25.
    Róka Z (1994) One-way cellular automata on Cayley Graphs. Theor Comput Sci 132:259–290Google Scholar
  26. 26.
    Stewart I (1994) A Subway Named Turing. Math Recreat Sci Am 90–92Google Scholar

Books and Reviews

  1. 27.
    Alekseevskij DV, Vinberg EB, Solodovnikov AS (1993) Geometry of spaces of constant curvature. In: Vinberg EB (ed) Geometry II, Encyclopedia of Mathematical Sciences, vol 29.Springer, BerlinGoogle Scholar
  2. 28.
    Berlekamp ER, Conway JH, Guy RK (1982) Winning Ways for Your Mathematical Plays.Academic PressGoogle Scholar
  3. 29.
    Bonola R (1912) Non-Euclidean Geometry. Open Court Publishing Company (also, (1955) Dover, New York)Google Scholar
  4. 30.
    Codd EF (1968) Cellular Automata. Academic Press, New YorkMATHGoogle Scholar
  5. 31.
    Coxeter HSM (1969) Introduction to Geometry. Wiley, New YorkMATHGoogle Scholar
  6. 32.
    Coxeter HSM (1974) Regular Complex Polytopes. Cambridge University Press, CambridgeMATHGoogle Scholar
  7. 33.
    Delorme M, Mazoyer J (eds) (1999) Cellular automata, a parallel model. Kluwer, p 460Google Scholar
  8. 34.
    Epstein DBA, Cannon JW, Holt DF, Levi SVF, Paterson MS, Thurston WP (1992) Word Processing in Groups. Jones and Barlett, BostonGoogle Scholar
  9. 35.
    Grünbaum B, Shephard GS (1987) Tilings and Patterns. Freeman, New YorkGoogle Scholar
  10. 36.
    Gruska J (1997) Foundations of computing. International Thomson Computer PressGoogle Scholar
  11. 37.
    Knuth DE (1998) The Art of Computer Programming, vol II: Seminumerical algorithms.Addison-WesleyGoogle Scholar
  12. 38.
    Meschkowski H (1964) Noneuclidean Geometry. Translated by Shenitzer A. Academic Press, New YorkMATHGoogle Scholar
  13. 39.
    Millman RS, Parker GD (1981) Geometry, a metric approach with models. SpringerGoogle Scholar
  14. 40.
    Minsky ML (1967) Computation: Finite and Infinite Machines. Prentice-Hall, Englewood CliffsMATHGoogle Scholar
  15. 41.
    Ramsay A, Richtmyer RD (1995) Introduction to Hyperbolic Geometry. SpringerGoogle Scholar
  16. 42.
    Toffoli T, Margolus N (1987) Cellular automata machines. MIT Press, CambridgeGoogle Scholar
  17. 43.
    von Neuman J (1966) Theory of self-reproducing automata. Edited and completed by A.W. Burks. The University of Illinois Press, UrbanaGoogle Scholar
  18. 44.
    Wolfram S (1994) Cellular Automata and Complexity. Addison-WesleyGoogle Scholar
  19. 45.
    Wolfram S (2002) A New Kind of Science. Wolfram MediaGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Maurice Margenstern
    • 1
  1. 1.Université Paul VerlaineMetzFrance