Project IV, 2025-2026

Apollonian circle packings

Anna Felikson

Description:

An Apollonian packing is a tiling by mutually tangent circles. These attractive designs carry interesting geometric and number-theoretic properties (both classical and recently established). One can start from the classical Descartes circle Theorem, its numerous applications and modifications, then consider integrality properties, dual packings and related polyhedral packings. One can connect to continued fractions, golden ratio and Pythagorean triples, to reflection groups and arithmetics of imaginary quadratic fields...

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Prerequisites: Algebra II. Also, Geometry III would help (but is not required).

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Resources:

You can start by taking a look at the following:

Katherine Stange, The Farey structure of the Gaussian integers .
R. E. Schwartz, S. Tabachnikov, Descartes Circle Theorem, Steiner Porism, and Spherical Designs.

You can find many more references here.

email: Anna Felikson

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