A circle packing proof of the Combinatorial Riemann Mapping Theorem

TL;DR

This paper proves the Combinatorial Riemann Mapping Theorem for bounded tilings using circle packing methods, extending classical techniques to a combinatorial setting.

Contribution

It introduces a circle packing proof for the Combinatorial Riemann Mapping Theorem applicable to bounded tilings, expanding the scope of circle packing techniques.

Findings

Established the theorem for bounded tilings

Extended circle packing methods to combinatorial settings

Provided a new proof approach for the theorem

Abstract

The traditional Riemann Mapping Theorem can be proved with circle packing techniques. We prove the Combinatorial Riemann Mapping Theorem for tilings of bounded size using circle packings.