Kaleidoscopic Symmetries and Self-Similarity of Integral Apollonian Gaskets

TL;DR

This paper explores the kaleidoscopic symmetries and self-similar recursive structures of integral Apollonian gaskets, revealing their connection to modular group transformations and specific quadratic irrationals, with implications for Pythagorean triplet generation.

Contribution

It uncovers the modular group encoding of the gasket's self-similarity and links the asymptotic curvature scalings to quadratic irrationals with period-2 continued fractions.

Findings

Self-similar structure encoded in SL(2,Z) transformations

Asymptotic curvatures linked to quadratic irrationals with period-2 continued fractions

Nested kaleidoscopic patterns with three-fold symmetry for n=2

Abstract

We describe various kaleidoscopic and self-similar aspects of the integral Apollonian gaskets - fractals consisting of close packing of circles with integer curvatures. Self-similar recursive structure of the whole gasket is shown to be encoded in transformations that forms the modular group $SL(2,Z)$. The asymptotic scalings of curvatures of the circles are given by a special set of quadratic irrationals with continued fraction $[n+1: \overline{1,n}]$ - that is a set of irrationals with period-2 continued fraction consisting of $1$ and another integer $n$. Belonging to the class $n=2$, there exists a nested set of self-similar kaleidoscopic patterns that exhibit three-fold symmetry. Furthermore, the even $n$ hierarchy is found to mimic the recursive structure of the tree that generates all Pythagorean triplets