2021/04/21 by Indubala I. Satija, Satija, Indubala I · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Fractal and DNA sequence analysis #General Mathematics (math.GM) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2104.13198
openalex publication_date 2021/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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: 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