The Riemann mapping theorem says that any simply connected domain $U$ that is not the whole $\mathbb{C}$ is biholomorphic to the unit disk. By uniformization theorem, $U$ carries a natural hyperbolic metric of constant curvature $-1$.

The Riemann mapping can be "approximated" using circle packings, as conjectured by Thurston and proved by Rodin and Sullivan. The picture is quite geometric, so I am wondering if the circle packing used here has some relation with the hyperbolic metric that can be given to $U$ (although in some sense the metric is "discretized").

Incidentally, while I was searching about this idea, I came across this MO question answered by Thurston himself, although the question is different but related. Thurston's answer isn't totally clear to me, but this concept of "using metric to understand some other notions" is quite appealing.

Some explanation, idea or reference will be of great help!


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