One can study an Apollonian circle packing (ACP) from many different angles. Such packings are certainly of interest in classical geometry. But ACP's encode fascinating information of an entirely different flavor: an infinite family of so-called integer ACP's encodes beautiful and mysterious number theoretic properties. The Apollonian gasket was first described by Gottfried Leibniz in the 17th century, and is a curved precursor of the 20th-century Sierpiński triangle. The Apollonian gasket also has deep connections to other fields of mathematics; for example, it is the limit set of Kleinian groups; and see also the Circle packing theorem.

(Rough idea): If all integers can be written as the sum of four squares then all integers should show up in some circle packings. If there is no “bias” in Apollonian circle packings, all packings should get roughly the same ratio of primes as all other packings and as the integers.

Geometry of the Apollonian Circle Packings Weiru Chen, Mo Jiao, Calvin Kessler, Amita Malik, Xin Zhang Introduction The ﬁne structures of Apollonian gaskets are encoded by local spatial statistics. In this article, we report our empirical results on some of such statistics, namely, pair correlation, nearest neighbor spacing and electrostatic ...

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