Golden Ratio's Continued Fractions Bridge Fibonacci Convergence and Quintic Solvability — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The golden angle's continued-fraction structure is not directly addressed in these papers, but the convergence rate of Fibonacci ratios to φ (1.618…) is implicitly tied to the optimality of continued fraction approximants, and the Rogers-Ramanujan continued fraction provides an algebraic bridge to quintic solvability. | MATH: Golden ratio φ = (1+√5)/2 ≈ 1.6180339887; its conjugate ψ = (1−√5)/2 ≈ −0.6180339887. Fibonacci convergence rate: |F_{n+1}/F_n − φ| ~ 1/(φ^{2n}) (since F_n ~ φ^n/√5). Continued fraction of φ = [1;1,1,1,…] — the slowest-converging simple continued fraction, giving optimal rational approximants with error ~ 1/(q_n^2) where q_n = F_n. Rogers-Ramanujan continued fraction: R(q) = q^{1/5}/(1 + q/(1 + q²/(1 + q³/(…)))) — for q = e^{−2π}, R(e^{−2π}) = √(5−√5)/2 − (√5−1)/2, an algebraic number involving √5. | CONNECTION: The golden angle = 2π(1−1/φ) ≈ 2.399963… rad ≈ 137.507764° — its continued fraction [2;1,1,1,…] yields convergents 2, 3, 5/2, 8/3, 13/5, 21/8,… (
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 Continued Fractions Bridge Fibonacci Convergence and Quintic Solvability — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23490667},
url = {https://zenodo.org/records/23490667},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Golden Ratio's Continued Fractions Bridge Fibonacci Convergence and Quintic Solvability — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/20
PB - Zenodo
DO - 10.5281/zenodo.23490667
UR - https://zenodo.org/records/23490667
AB - FINDING: The golden angle's continued-fraction structure is not directly addressed in these papers, but the convergence rate of Fibonacci ratios to φ (1.618…) is implicitly tied to the optimality of continued fraction approximants, and the Rogers-Ramanujan continued fraction provides an algebraic bridge to quintic solvability. | MATH: Golden ratio φ = (1+√5)/2 ≈ 1.6180339887; its conjugate ψ = (1−√5)/2 ≈ −0.6180339887. Fibonacci convergence rate: |F_{n+1}/F_n − φ| ~ 1/(φ^{2n}) (since F_n ~ φ^n/√5). Continued fraction of φ = [1;1,1,1,…] — the slowest-converging simple continued fraction, giving optimal rational approximants with error ~ 1/(q_n^2) where q_n = F_n. Rogers-Ramanujan continued fraction: R(q) = q^{1/5}/(1 + q/(1 + q²/(1 + q³/(…)))) — for q = e^{−2π}, R(e^{−2π}) = √(5−√5)/2 − (√5−1)/2, an algebraic number involving √5. | CONNECTION: The golden angle = 2π(1−1/φ) ≈ 2.399963… rad ≈ 137.507764° — its continued fraction [2;1,1,1,…] yields convergents 2, 3, 5/2, 8/3, 13/5, 21/8,… ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -