The Apollonian staircase

TL;DR

This paper investigates the distribution of minimal circle curvatures in primitive integral Apollonian packings, revealing a staircase pattern and probabilistic tendencies, using quadratic forms and number theory techniques.

Contribution

It introduces the Apollonian staircase as a new descriptive pattern for curvature distributions and connects geometric packings with number theoretic properties.

Findings

Distribution of c/n tends to the Apollonian staircase as n increases.

Probability of a circle being tangent to the outermost circle approaches 3/π.

Distribution smoothness varies with n being prime or composite, showing spikes related to prime divisors.

Abstract

A circle of curvature $n\in\mathbb{Z}^+$ is a part of finitely many primitive integral Apollonian circle packings. Each such packing has a circle of minimal curvature $-c\leq 0$, and we study the distribution of $c/n$ across all primitive integral packings containing a circle of curvature $n$. As $n\rightarrow\infty$, the distribution is shown to tend towards a picture we name the Apollonian staircase. A consequence of the staircase is that if we choose a random circle packing containing a circle $C$ of curvature $n$, then the probability that $C$ is tangent to the outermost circle tends towards $3/\pi$. These results are found by using positive semidefinite quadratic forms to make $\mathbb{P}^1(\mathbb{C})$ a parameter space for (not necessarily integral) circle packings. Finally, we examine an aspect of the integral theory known as spikes. When $n$ is prime, the distribution of $c/n$ is extremely smooth, whereas when $n$ is composite, there are certain spikes that correspond to prime divisors of $n$ that are at most $\sqrt{n}$.