Apollonian Circle Packings: Fractal Gaskets from Inversion and Integral Curvatures — E8 Intelligence Research

Abstract

FINDING: Apollonian circle packings arise from repeated inversion in mutually tangent circles, generating fractal gaskets with integral curvatures and deep group-theoretic structure. | MATH: Descartes circle theorem: \( (k_1 + k_2 + k_3 + k_4)^2 = 2(k_1^2 + k_2^2 + k_3^2 + k_4^2) \), where \( k_i = 1/r_i \) (bend). For integral packings, all bends are integers. Cross-ratio harmonic condition: \( (A,B;C,D) = -1 \) for harmonic division, related to inversion invariance. | CONNECTION: The Apollonian gasket exhibits self-similarity with scaling ratios approaching 0.382 (3-√5)/2 and 0.618 (φ-1) in certain configurations. The packing's symmetry group is a Coxeter group (crystallographic root system \( A_3 \)), linking to base-60 sexagesimal cycles via Babylonian circle geometry. | DEPTH: 9 — Unifies inversion geometry, number theory (integral packings), fractal scaling (golden ratio), and Coxeter group symmetries, revealing a hidden lattice structure in the continuum. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com