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arXiv · 2104.13198

Kaleidoscopic Symmetries and Self-Similarity of Integral Apollonian Gaskets

Indubala I Satija
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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

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Indubala I Satija. 2021-04-21. Kaleidoscopic Symmetries and Self-Similarity of Integral Apollonian Gaskets. https://arxiv.org/abs/2104.13198

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