Golden Ratio's Slowest Continued Fraction Convergence Versus Faster Nested Radicals — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The golden ratio φ emerges as the slowest-converging continued fraction, with convergence rate tied to Fibonacci numbers; nested radicals converge faster than continued fractions. | MATH: φ = [1;1,1,1,...] = 1 + 1/(1+1/(1+...)); φ = √(1+√(1+√(1+...))); convergence rate for continued fraction: |φ - p_n/q_n| ~ 1/(q_n·q_{n+1}) with q_n = F_{n+1} (Fibonacci); error ratio between successive convergents → 1/φ² ≈ 0.381966; nested radical error decays as ~1/φ^{2^n} (double exponential). | CONNECTION: Directly encodes 0.382 (φ⁻²) and 1.618 (φ); the continued fraction's partial quotients are all 1, making φ the "most irrational" — its convergents are best approximations with maximal denominator growth, linking to the golden angle 137.507° = 360°/(φ²) and phyllotaxis lattice symmetry (5-fold, Fibonacci spiral). | DEPTH: 7 — This is foundational but not novel; it underpins why φ appears in nature's optimal packing (sunflower seeds, leaf arrangement) via the golden angle's irrationality me
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026golden,
title = {{Golden Ratio's Slowest Continued Fraction Convergence Versus Faster Nested Radicals — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23616857},
url = {https://zenodo.org/records/23616857},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Golden Ratio's Slowest Continued Fraction Convergence Versus Faster Nested Radicals — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/24
PB - Zenodo
DO - 10.5281/zenodo.23616857
UR - https://zenodo.org/records/23616857
AB - FINDING: The golden ratio φ emerges as the slowest-converging continued fraction, with convergence rate tied to Fibonacci numbers; nested radicals converge faster than continued fractions. | MATH: φ = [1;1,1,1,...] = 1 + 1/(1+1/(1+...)); φ = √(1+√(1+√(1+...))); convergence rate for continued fraction: |φ - p_n/q_n| ~ 1/(q_n·q_{n+1}) with q_n = F_{n+1} (Fibonacci); error ratio between successive convergents → 1/φ² ≈ 0.381966; nested radical error decays as ~1/φ^{2^n} (double exponential). | CONNECTION: Directly encodes 0.382 (φ⁻²) and 1.618 (φ); the continued fraction's partial quotients are all 1, making φ the "most irrational" — its convergents are best approximations with maximal denominator growth, linking to the golden angle 137.507° = 360°/(φ²) and phyllotaxis lattice symmetry (5-fold, Fibonacci spiral). | DEPTH: 7 — This is foundational but not novel; it underpins why φ appears in nature's optimal packing (sunflower seeds, leaf arrangement) via the golden angle's irrationality me Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -