An illustrated introduction to the arithmetic of Apollonian circle packings, continued fractions, and other thin orbits

TL;DR

This paper provides an accessible, illustrated overview of the interplay between Apollonian circle packings, thin groups, hyperbolic geometry, and number theory, emphasizing visual intuition and recent research developments.

Contribution

It offers a comprehensive, visual introduction to the arithmetic of Apollonian packings and their connections to thin groups and continued fractions, expanding on recent summer school material.

Findings

Illustrates the connection between circle packings and number theory.

Highlights the role of thin groups in geometric structures.

Provides visual intuition for complex arithmetic concepts.

Abstract

These notes cover and expand upon the material for two summer schools: The first, which was held at CIRM, Marseille, France, July 10-14, 2023, as part of "Renormalization and Visualization for packing, billiard and surfaces", was titled "Number theory as a door to geometry, dynamics and illustration". The second was held at NSU IMS in Singapore, June 3-7, 2024, as part of "Computational Aspects of Thin Groups", and was titled "Integral packings and number theory". Both courses were put together by a number theorist for students and researchers in other fields. They cover a web of ideas relating to Apollonian circle packings, integral orbits, thin groups, hyperbolic geometry, continued fractions, and Diophantine approximation. The connection of geometry and dynamics to number theory gives an opportunity to illustrate arithmetic by appealing to our visual intuition.