Apollonian Packings, Harmonic Division, and Fractal Inversion Geometry — E8 Intelligence Research

August 1, 2026Open Access

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Key Points

The research aims to explore the link between Apollonian packings and harmonic division through inversion geometry.
Analysis of the Descartes circle theorem and its implications for inversion geometry.
Examination of the relationship between harmonic ratios and Apollonian gaskets using specific bend configurations.
Investigation of the golden ratio in the context of scaling factors of circle radii.
Demonstrated that the recursive structure of Apollonian packings is governed by the Descartes circle theorem and cross-ratio invariants.
Identified the appearance of the golden ratio in the scaling of tangent circle curvatures in specific packing configurations.
Revealed a deep geometric recursion that reflects self-similarity found in nature.

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This research demonstrates the link between Apollonian packings and harmonic division via inversion geometry, implying deeper fractal structures.

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Andrew Stewart Caldin (2026) studied this question.

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