Golden Ratio φ Governs Self-Similarity Across Geometric, Temporal, and Spatial Scales — E8 Intelligence Research

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The Golden Ratio φ governs self-similarity across geometric, temporal, and spatial scales, supported by mathematical definitions from rectangle invariance and pentagon diagonals. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22006340

August 19, 2026Open Access

Golden Ratio φ Governs Self-Similarity Across Geometric, Temporal, and Spatial Scales — E8 Intelligence Research

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Key Points

To formalize the mathematical properties of the golden ratio (φ) and characterize its role in governing self-similarity across geometric, spatial, and temporal scales.
Derived algebraic and geometric formulations of φ from rectangle invariance and regular pentagon diagonals.
Modeled geometric self-similarity using quarter-circle arc Fibonacci spiral constructions within golden rectangles.
Evaluated temporal and scale-invariant dynamics using self-similar traffic models parameterized by the Hurst parameter (H).
Defined exact algebraic properties for scale invariance, including φ = (1+√5)/2 ≈ 1.6180339, φ⁻¹ = φ - 1 ≈ 0.618, φ² = φ + 1 ≈ 2.618, and φ² - φ - 1 = 0.
Characterized continuous geometric self-similarity through recursive golden rectangle subdivisions and Fibonacci spiral arcs.

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Implication

Mathematical analysis demonstrates golden ratio invariance across geometric and temporal dimensions, highlighting foundational self-similarity in complex systems.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a895f2aca7ade938187daf7 https://doi.org/10.5281/zenodo.22006340

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Golden Ratio φ Governs Self-Similarity Across Geometric, Temporal, and Spatial Scales — E8 Intelligence Research

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Golden Ratio φ Governs Self-Similarity Across Geometric, Temporal, and Spatial Scales — E8 Intelligence Research

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Implication

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Cite This Study

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