Alex Kontorovich is a Professor of Mathematics at Rutgers University and served as the Distinguished Professor for the Public Dissemination of Mathematics at the National Museum of Mathematics in 2020–2021. Alex has received numerous awards for his illustrious mathematical career, including the Levi L. Conant Prize in 2013 for mathematical exposition, a Simons Foundation Fellowship, an NSF career award, and being elected Fellow of the American Mathematical Society in 2017. He currently serves on the Scientific Advisory Board of Quanta Magazine and as Editor-in-Chief of the Journal of Experimental Mathematics.

In this episode, Alex takes us from the ancient beginnings to the present day on the subject of circle packings. We start with the Problem of Apollonius on finding tangent circles using straight-edge and compass and continue forward in basic Euclidean geometry up until the time of Leibniz whereupon we encounter the first complete notion of a circle packing. From here, the plot thickens with observations on surprising number theoretic coincidences, which only received full appreciation through the craftsmanship of chemistry Nobel laureate Frederick Soddy. We continue on with more advanced mathematics arising from the confluence of geometry, group theory, and number theory, including fractals and their dimension, hyperbolic dynamics, Coxeter groups, and the local to global principle of advanced number theory. We conclude with a brief discussion on extensions to sphere packings.

Patreon: http://www.patreon.com/timothynguyen

I. Introduction

• 00:00: Biography

II. Setup

• 24:23: Triangles and tangent circles

III. Circle Packings

• 41:49: Iterating tangent circles: Apollonian circle packing

IV. Simple Geometry and Number Theory

• 1:04:51: Descartes’s Theorem

V. Group Theory, Hyperbolic Dynamics, and Advanced Number Theory

• 1:22:02: Generating circle packings through repeated inversions (through dual circles)

VI. Dimension Three: Sphere Packings

• 2:13:03: Setup + what Soddy built

VII. Conclusion

• 2:18:20: Wrap up

Image Credits: http://timothynguyen.org/image-credits/