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

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.