Golden Ratio and Continued Fractions: Optimal Packing via Slowest Convergence — E8 Intelligence Research

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Continued fraction convergents of square roots and Fibonacci-type ratios underpin optimal packing and phyllotaxis, with the golden ratio representing the slowest-converging and most irrational case. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22470010

September 6, 2026Open Access

Golden Ratio and Continued Fractions: Optimal Packing via Slowest Convergence — E8 Intelligence Research

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Key Points

To characterize how continued fraction convergents of irrational numbers govern optimal space packing and divergence angles in phyllotaxis.
Analyzed recurrence relations for simple continued fraction convergents and periodic continued fractions of square roots.
Evaluated the approximation properties and convergence rate of the golden ratio using Hurwitz's theorem.
Demonstrated that the golden ratio achieves the slowest possible convergence because all its partial quotients equal 1, making it the most irrational number under Hurwitz's bound.
Showed that Fibonacci convergents directly define the characteristic divergence angle of approximately 137.507 degrees required for space-filling spirals.

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Implication

Theoretical analysis demonstrates optimal packing via slowest continued fraction convergence in phyllotactic systems, highlighting the mathematical mechanism behind spiral patterns.

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Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9fd78a58e84d0ff5b46455 https://doi.org/10.5281/zenodo.22470010

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Golden Ratio and Continued Fractions: Optimal Packing via Slowest Convergence — E8 Intelligence Research

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Discussion

Cite This Study

Golden Ratio and Continued Fractions: Optimal Packing via Slowest Convergence — E8 Intelligence Research

Key Points

No abstract yet

Implication

Discussion

Cite This Study

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