Apollonian circle packings

Project IV, 2026-2027

Supervisor: 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,
Pythagorean triples,
reflection groups,
arithmetics of imaginary quadratic fields,

and many other objects and areas in mathematics.

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Mode of Operation and Evidence of Learning: The project will revolve around learning through reading with focus on the underlying theory and the development of conceptual understanding. Students will demonstrate their understanding by exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.

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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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