Discrete & Computational Geometry

, Volume 46, Issue 2, pp 223–251 | Cite as

Rigidity of Quasicrystallic and Zγ-Circle Patterns



The uniqueness of the orthogonal Zγ-circle patterns as studied by Bobenko and Agafonov is shown, given the combinatorics and some boundary conditions. Furthermore we study (infinite) rhombic embeddings in the plane which are quasicrystallic, that is, they have only finitely many different edge directions. Bicoloring the vertices of the rhombi and adding circles with centers at vertices of one of the colors and radius equal to the edge length leads to isoradial quasicrystallic circle patterns. We prove for a large class of such circle patterns which cover the whole plane that they are uniquely determined up to affine transformations by the combinatorics and the intersection angles. Combining these two results, we obtain the rigidity of large classes of quasicrystallic Zγ-circle patterns.


Circle pattern Rigidity Quasicrystallic 


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  1. 1.
    Agafonov, S.I.: Imbedded circle patterns with the combinatorics of the square grid and discrete Painlevé equations. Discrete Comput. Geom. 29, 305–319 (2003) MATHMathSciNetGoogle Scholar
  2. 2.
    Agafonov, S.I.: Asymptotic behavior of discrete holomorphic maps z c, log (z). J. Nonlinear Math. Phys. 12, 1–14 (2005) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Agafonov, S.I.: Discrete Riccati equation, hypergeometric functions and circle patterns of Schramm type. Glasg. Math. J. A 47, 1–16 (2005) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Agafonov, S.I., Bobenko, A.I.: Discrete Z γ and Painlevé equations. Int. Math. Res. Not. 4, 165–193 (2000) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Aharonov, D.: The hexagonal packing lemma and discrete potential theory. Can. Math. Bull. 33, 247–252 (1990) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Aharonov, D.: The hexagonal packing lemma and the Rodin Sullivan conjecture. Trans. Am. Math. Soc. 343, 157–167 (1994) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bobenko, A.I.: Discrete conformal maps and surfaces. In: Clarkson, P.A., Nijhoff, F.W. (eds.) Symmetries and Integrability of Difference Equations. London Mathematical Society Lecture Notes Series, vol. 255, pp. 97–108. Cambridge University Press, Cambridge (1999) CrossRefGoogle Scholar
  8. 8.
    Bobenko, A.I., Hoffmann, T.: Hexagonal circle patterns and integrable systems: patterns with constant angles. Duke Math. J. 116, 525–566 (2003) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bobenko, A.I., Springborn, B.A.: Variational principles for circle patterns and Koebe’s theorem. Trans. Am. Math. Soc. 356, 659–689 (2004) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bobenko, A.I., Mercat, Ch., Suris, Yu.B.: Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function. J. Reine Angew. Math. 583, 117–161 (2005) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bobenko, A.I., Suris, Yu.B.: Discrete Differential Geometry. Integrable Structure. Graduate Studies in Mathematics, vol. 98. AMS, Providence (2008) MATHCrossRefGoogle Scholar
  12. 12.
    Bücking, U.: Approximation of conformal mappings by circle patterns and discrete minimal surfaces. Ph.D. thesis, Technische Universität Berlin (2007). Published online at http://opus.kobv.de/tuberlin/volltexte/2008/1764/
  13. 13.
    Bücking, U.: Approximation of conformal mapping by circle patterns. Geom. Dedic. 137, 163–197 (2008) MATHCrossRefGoogle Scholar
  14. 14.
    Doyle, P.G., Snell, J.L.: Random Walks and Electric Networks. Carus Mathematics Monographs, vol. 22. Math. Assoc. Am., Washington (1984) MATHGoogle Scholar
  15. 15.
    Dubejko, T.: Recurrent random walks, Liouville’s theorem, and circle packings. Math. Proc. Camb. Philos. Soc. 121, 531–546 (1997) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Duneau, M., Katz, A.: Quasiperiodic patterns. Phys. Rev. Lett. 54, 2688–2691 (1985) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Gähler, F., Rhyner, J.: Equivalence of the generalized grid and projection methods for the construction of quasiperiodic tilings. J. Phys. A 19, 267–277 (1986) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    He, Z.-X.: An estimate for hexagonal circle packings. J. Differ. Geom. 33, 395–412 (1991) MATHGoogle Scholar
  19. 19.
    He, Z.-X.: Rigidity of infinite disk patterns. Ann. Math. 149, 1–33 (1999) MATHCrossRefGoogle Scholar
  20. 20.
    He, Z.-X., Schramm, O.: Fixed points, Koebe uniformization and circle packings. J. Am. Math. Soc. 137, 369–406 (1993) MathSciNetGoogle Scholar
  21. 21.
    Kharevych, L., Springborn, B., Schröder, P.: Discrete conformal mappings via circle patterns. ACM Trans. Graph. 25, 412–438 (2006) CrossRefGoogle Scholar
  22. 22.
    Rodin, B., Sullivan, D.: The convergence of circle packings to the Riemann mapping. J. Differ. Geom. 26, 349–360 (1987) MATHMathSciNetGoogle Scholar
  23. 23.
    Schramm, O.: Rigidity of infinite (circle) packings. J. Am. Math. Soc. 4, 127–149 (1991) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Schramm, O.: Circle patterns with the combinatorics of the square grid. Duke Math. J. 86, 347–389 (1997) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Schreiber, V.: Quasicrystallic circle patterns, the discrete power function. Diplomarbeit (2008). Technische Universität Berlin Google Scholar
  26. 26.
    Senechal, M.: Quasicrystals and Geometry. Cambridge University Press, Cambridge (1995) MATHGoogle Scholar
  27. 27.
    Springborn, B.A.: Variational principles for circle patterns. Ph.D. thesis, Technische Universität Berlin (2003). Published online at http://opus.kobv.de/tuberlin/volltexte/2003/668/
  28. 28.
    Stephenson, K.: Introduction to Circle Packing: The Theory of Discrete Analytic Functions. Cambridge University Press, New York (2005) MATHGoogle Scholar
  29. 29.
    Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge University Press, Cambridge (2000) MATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

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