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A striking example:

Consider arrangements of disks in the plane so that no two disks overlap (except on their boundaries) and any two disks that touch share at least one common neighbor (so the incidence graph their complement is triangulated)a disjoint union of triangles (if we include a point at infinity). You can imagine trying to build a particular finite triangulated planar graph by placing different sized coins on the table.

Here's the theorem: Any such graph may be obtained. Further, the representation is unique up to Möbius (and anti-Möbius) transformations of the plane.

The proof of uniqueness is the striking bit. You think of the plane as the boundary of hyperbolic upper half space! Fill in each triangle of the original disks with a new disk tangent to them, and extend all the disks to half-balls. We view the surface of each ball as a plane in hyperbolic space, and consider the group of reflections across them. We the then apply the Mostow rigidity theorem to the quotient manifold, and obtain the result.

This observation is due to Thurston. See http://en.wikipedia.org/wiki/Circle_packing_theorem

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A striking example:

Consider arrangements of disks in the plane so that no two disks overlap (except on their boundaries) and any two disks that touch share at least one common neighbor (so the incidence graph is triangulated). You can imagine trying to build a particular finite triangulated planar graph by placing different sized coins on the table.

Here's the theorem: Any such graph may be obtained. Further, the representation is unique up to Möbius (and anti-Möbius) transformations of the plane.

The proof of uniqueness is the striking bit. You think of the plane as the boundary of hyperbolic upper half space! Fill in each triangle of the original disks with a new disk tangent to them, and extend all the disks to half-balls. We view the surface of each ball as a plane in hyperbolic space, and consider the group of reflections across them. We the apply the Mostow rigidity theorem to the quotient manifold, and obtain the result.

This observation is due to Thurston. See http://en.wikipedia.org/wiki/Circle_packing_theorem