论文标题
阿波罗尼亚楼梯
The Apollonian staircase
论文作者
论文摘要
\ Mathbb {z}^+$的曲率$ n \圆圈是有限的许多原始积分apollonian圆形包装的一部分。每个这样的包装都有一个最小曲率$ -C \ leq 0 $的圆圈，我们研究了所有原始整体包装中$ c/n $的分布，其中包含一个曲率$ n $的圆圈。作为$ n \ rightarrow \ infty $，该分布趋向于我们为阿波罗尼亚楼梯命名的图片。楼梯的结果是，如果我们选择一个随机圆包装，其中包含一个圆形曲率$ n $的圆圈$ c $，那么$ c $与最外面圆相切的概率趋向于$ 3/π$。通过使用积极的半芬酸二次形式来找到这些结果，使$ \ mathbb {p}^1（\ mathbb {c}）$一个参数空间（不一定是积分）圆形包装。最后，我们研究了称为尖峰的整体理论的一个方面。当$ n $是素数时，$ c/n $的分布非常平滑，而当$ n $是复合的时，某些尖峰对应于$ n $的prime除法，最多是$ \ sqrt {n} $。
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 $3/π$. 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 $\sqrt{n}$.
