Rigidity of Infinite (Circle) Packings

  • Oded Schramm
Open Access
Part of the Selected Works in Probability and Statistics book series (SWPS)


The nerve of a packing is a graph that encodes its combinatorics. The vertices of the nerve correspond to the packed sets, and an edge occurs between two vertices in the nerve precisely when the corresponding sets of the packing intersect.

The nerve of a circle packing and other well-behaved packings, on the sphere or in the plane, is a planar graph. It was an observation of Thurston [Th1, Chapter 1; 13, Th2] that Andreev’s theorem [An1, An2] implies that given a finite planar graph, there exists a packing of (geometric) circles on the sphere whose nerve is the given graph. We refer to this fact as the circle packing theorem. The circle packing theorem also has a uniqueness part to it: if the graph is actually (the 1-skelaton of) a triangulation, then the circle packing is unique up to Möbius transformations.


Planar Graph Double Cover Circle Packing Rigidity Theorem Hexagonal Packing 
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.



I am deeply thankful to my teachers Bill Thurston and Peter Doyle, and to Richard Schwartz, Burt Rodin, and Zheng-Xu He for stimulating discussions relating to packings.


  1. [An1]
    E. M. Andreev, On convex polyhedra in Lobačevskiῐ spaces, Mat. Sb. (N.S.) 81 (1970), 445–478; English transl. in Math. USSR Sb. 10 (1970), 413–440.MathSciNetGoogle Scholar
  2. [An2]
    ——, On convex polyhedra of finite volume in Lobačevskiῐ space, Mat. Sb. (N.S.) 83 (1970), 256–260; English transl. in Math. USSR Sb. 12 (1970), 255–259.MathSciNetGoogle Scholar
  3. [BFP]
    I. Bárány, Z. Füredi, and J. Pach, Discrete convex functions and proof of the six circle conjecture of Fejes Tóth, Canad. J. Math. 36-3 (1984), 569–576.CrossRefGoogle Scholar
  4. [CR]
    I. Carter and B. Rodin, An inverse problem for circle packing and conformal mapping, preprint.Google Scholar
  5. [He1]
    Zheng-Xu He, An estimate for hexagonal circle packings, J. Differential Geom. (to appear).Google Scholar
  6. [He2]
    —, Solving Beltrami equations by circle packing, Trans. Amer. Math. Soc. (to appear).Google Scholar
  7. [Ro1]
    B. Rodin, Schwartz’s lemma for circle packings, Invent. Math. 89 (1987), 271–289.MathSciNetMATHCrossRefGoogle Scholar
  8. [Ro2]
    ——, Schwartz’s lemma for circle packings II, J. Differential Geom. 30 (1989), 539–554.MathSciNetMATHGoogle Scholar
  9. [RS]
    B. Rodin and D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987), 349–360.MathSciNetMATHGoogle Scholar
  10. [Sch1]
    O. Schramm, Packing two-dimensional bodies with prescribed combinatorics and applications to the construction of conformal and quasiconformal mappings, Ph.D. thesis, Princeton, 1990.Google Scholar
  11. [Sch2]
    —, Uniqueness and existence of packings with specified combinatorics, Israel J. Math. (to appear).Google Scholar
  12. [Ste]
    K. Stephenson, Circle packings in the approximation of conformal mappings, Bull. Amer. Math. Soc. 23 (1990), 407–415.MathSciNetMATHCrossRefGoogle Scholar
  13. [Th1]
    W. P. Thurston, The geometry and topology of 3-manifolds, Princeton Univ. Lecture Notes, Princeton, NJ.Google Scholar
  14. [Th2]
    —, The finite Riemann mapping theorem, invited talk at the International Symposium in Celebration of the Proof of the Bieberbach Conjecture, Purdue University, March 1985.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Mathematics DepartmentUniversity of California at San DiegoLa JollaUSA

Personalised recommendations