Combinatorics encoding geometry: the legacy of Bill Thurston in the story of one theorem

TL;DR

This paper explores the historical and mathematical significance of Thurston's circle packing theorem, highlighting its impact on combinatorics, geometry, and topology within three-manifold theory.

Contribution

It provides a comprehensive overview of results related to Thurston's circle packing theorem and its influence on modern geometric and topological research.

Findings

Highlights the role of circle packings in understanding three-manifolds

Connects combinatorial encoding with geometric structures

Summarizes key developments stemming from Thurston's theorem

Abstract

This article presents a whirlwind tour of some results surrounding the Koebe-Andre'ev-Thurston Theorem, Bill Thurston's seminal circle packing theorem that appears in Chapter 13 of The Geometry and Topology of Three-Manifolds. It will appear as a chapter in the volume: In the tradition of Thurston: geometry and topology (ed. K. Ohshika and A. Papadopoulos), Springer, 2020.