Rigidity of Infinite (Circle) Packings
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.
KeywordsHexagonal Bedding Doyle
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.
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