Symmetries in Apollonian Circle Packings

Authors

• Clyde Kertzer, University of Colorado, BoulderFollow

Abstract

An Apollonian circle packing is generated from a Descartes quadruple (a set of four mutually tangent circles) by repeatedly filling the spaces between mutually tangent circles with further tangent circles. By studying the circles' curvatures $a,b,c,d$, two distinct types of symmetric packings appear: one where $a+b+c=d$ and one where $c=d$. We give complete parameterizations of these symmetric packings and count how many packings of each type are contained by a given enclosing circle.

Author Bio

Clyde Kertzer graduated from University of Colorado, Boulder, in 2025. He teaches mathematics at TARA high school in Boulder, CO. 

Faculty Sponsor

Katherine E. Stange

Recommended Citation

Kertzer, Clyde (2025) "Symmetries in Apollonian Circle Packings," Rose-Hulman Undergraduate Mathematics Journal: Vol. 26: Iss. 2, Article 1.
Available at: https://scholar.rose-hulman.edu/rhumj/vol26/iss2/1