On a Diophantine equation that generates all integral Apollonian Gaskets

Statistical Regularity of Apollonian Gaskets

Apollonian gaskets are formed by repeatedly filling the gaps between three mutually tangent circles with further tangent circles. In this paper we give explicit formulas for the the limiting pair correlation and the limiting nearest neighbor spacing of centers of circles from a fixed Apollonian gasket. These are corollaries of the convergence of moments that we prove. The input from ergodic theory is an extension of Mohammadi-Oh's Theorem on the equidisribution of expanding horospheres in infinite volume hyperbolic spaces.

Spatial Statistics of Apollonian Gaskets

Apollonian gaskets are formed by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We experimentally study the pair correlation, electrostatic energy, and nearest neighbor spacing of centers of circles from Apollonian gaskets. Even though the centers of these circles are not uniformly distributed in any `ambient' space, after proper normalization, all these statistics seem to exhibit some interesting limiting behaviors.

阿波罗尼奥斯垫片的豪斯多夫维数

Hausdorff dimension of the Apollonian gasket

本文针对阿波罗尼奥斯垫片豪斯多夫维数缺乏闭形式且难以严格计算的问题，提出一种高效估计抛物极限集维数的方法，首次将其计算至128位小数并给出严格误差界，且有望推广至其他抛物分形。

Sierpiński、Rauzy 和 Apollonian 垫片之间的显式插值

Explicit interpolations among the Sierpiński, Rauzy, and Apollonian gaskets

本文研究围绕仿射 Sierpiński 垫片的两个显式单参数族，分别实现到 Rauzy 垫片和等边 Apollonian 垫片的插值，并给出 Hausdorff 维数的变分公式及规范边界归一化模型。

The sensual Apollonian circle packing

The curvatures of the circles in integral Apollonian circle packings, named for Apollonius of Perga (262-190 BC), form an infinite collection of integers whose Diophantine properties have recently seen a surge in interest. Here, we give a new description of Apollonian circle packings built upon the study of the collection of bases of Z[i]^2, inspired by, and intimately related to, the `sensual quadratic form' of Conway.

Spinors, lattices, and classification of integral Apollonian disk packings

A parametrization of integral Descartes configurations (and effectively Apollonian disk packings) by pairs of two-dimensional integral vectors is presented. The vectors, called here tangency spinors defined for pairs of tangent disks, are spinors associated to the Clifford algebra for three-dimensional Minkowski space. A version with Pauli spinors is given. The construction provides a novel interpretation to the known Diophantine equation parametrizing integral Apollonian packings.