Number Theory Through Geometry, Dynamics and Illustration - Lecture 1

Explore number theory through a dynamic and geometric lens in this lecture from the Centre International de Rencontres Mathématiques. Delve into continued fractions, Diophantine approximation, and Apollonian circle packings, focusing on both theoretical and experimental tools. Experience the role of visualization and illustration in mathematical research as the lecture emphasizes visual and dynamical approaches to background material. Cover topics such as continued fractions, quadratic forms, hyperbolic geometry, Minkowski space, and Kleinian groups. Investigate current research areas including the geometry of Diophantine approximation and continued fractions in the complex plane, algebraic starscapes, Schmidt arrangements, and the surprising relationships of Apollonian circle packings to these topics. Recorded during the thematic meeting "Renormalization and Visualization for packing, Billiard and Surfaces" on July 10, 2023, this 59-minute lecture is part of CIRM's Audiovisual Mathematics Library, offering features like chapter markers, keywords, abstracts, bibliographies, and multi-criteria search options for an enhanced learning experience.