Inventiones mathematicae

, Volume 159, Issue 1, pp 119–132 | Cite as

A strongly aperiodic set of tiles in the hyperbolic plane

  • Chaim Goodman-Strauss


We construct the first known example of a strongly aperiodic set of tiles in the hyperbolic plane. Such a set of tiles does admit a tiling, but admits no tiling with an infinite cyclic symmetry. This can also be regarded as a “regular production system” [5] that does admit bi-infinite orbits, but admits no periodic orbits.


Production System Periodic Orbit Hyperbolic Plane Regular Production Cyclic Symmetry 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berger, R.: The undecidability of the Domino Problem. Mem. Am. Math. Soc. 66 (1966)Google Scholar
  2. 2.
    Block, J., Weinberger, S.: Aperiodic tilings, positive scalar curvature and amenability of spaces. J. Am. Math. Soc. 5, 907–918 (1992)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Epstein, D., et al.: Word processing in groups. Boston: Jones and Bartlett 1992Google Scholar
  4. 4.
    Goodman-Strauss, C.: Open questions in tilings. PreprintGoogle Scholar
  5. 5.
    Goodman-Strauss, C.: Regular production systems and triangle tilings. PreprintGoogle Scholar
  6. 6.
    Grünbaum, B., Shepherd, G.C.: Tilings and patterns. New York: W.H. Freeman and Co. 1987Google Scholar
  7. 7.
    Kari, J.: A small aperiodic set of Wang tiles. Discrete Math. 160, 259–264 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Margulis, G.A., Mozes, S.: Aperiodic tilings of the hyperbolic plane by convex polygons. Isr. J. Math. 107, 319–325 (1998)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Mozes, S.: Aperiodic tilings. Invent. Math. 128, 603–611 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Penrose, R.: Pentaplexity. Math. Intell. 2, 32–37 (1978)zbMATHGoogle Scholar
  11. 11.
    Robinson, R.M.: Undecidability and nonperiodicity of tilings in the plane. Invent. Math. 12, 177–209 (1971)zbMATHGoogle Scholar
  12. 12.
    Robinson, R.M.: Undecidable tiling problems in the hyperbolic plane. Invent. Math. 44, 259–264 (1978)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Wang, H.: Proving theorems by pattern recognition II. Bell System Tech. J. 40, 1–42 (1961)Google Scholar

Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ArkansasFayettevilleUSA

Personalised recommendations