Selected Works of Oded Schramm pp 63-85 | Cite as

# Rigidity of Infinite (Circle) Packings

## Abstract

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.

## Keywords

Planar Graph Double Cover Circle Packing Rigidity Theorem Hexagonal Packing## Notes

### Acknowledgments

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.

## References

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*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 - [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 - [CR]I. Carter and B. Rodin,
*An inverse problem for circle packing and conformal mapping*, preprint.Google Scholar - [He1]Zheng-Xu He,
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*Solving Beltrami equations by circle packing*, Trans. Amer. Math. Soc. (to appear).Google Scholar - [Ro1]B. Rodin,
*Schwartz’s lemma for circle packings*, Invent. Math.**89**(1987), 271–289.MathSciNetMATHCrossRefGoogle Scholar - [Ro2]——,
*Schwartz’s lemma for circle packings II*, J. Differential Geom.**30**(1989), 539–554.MathSciNetMATHGoogle Scholar - [RS]B. Rodin and D. Sullivan,
*The convergence of circle packings to the Riemann mapping*, J. Differential Geom.**26**(1987), 349–360.MathSciNetMATHGoogle Scholar - [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 - [Sch2]—,
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*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