Convergence Rates of Golden Ratio Representations and the Golden Angle — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Golden angle derives from golden ratio; Fibonacci convergence to φ is linear with rate 1/φ ≈ 0.618; nested radicals and continued fractions converge at different rates, with continued fractions exhibiting optimal (slowest algebraic) convergence. | MATH: Golden angle = 2π(1 − 1/φ) = 2π/φ² ≈ 137.5077° (or equivalently 2π − 2π/φ). Fibonacci ratio convergence: Fₙ₊₁/Fₙ → φ, error term ~ (−1/φ)ⁿ, giving asymptotic error ratio |eₙ₊₁/eₙ| = 1/φ ≈ 0.618. Continued fraction for φ: φ = 1 + 1/(1 + 1/(1 + …)) — all partial quotients = 1, making it the *slowest* converging continued fraction (worst-case for rational approximation). Nested radical: φ = √(1 + √(1 + √(1 + …))) converges faster — error decays doubly exponentially (rate ~ 1/φ² per nesting level squared). | CONNECTION: The convergence rate 1/φ = 0.618 is the *complement* of φ (φ − 1 = 1/φ). This is the same ratio appearing in phyllotaxis (golden angle = 2π/φ² ≈ 137.5°), which optimizes seed packing via the continued fraction of φ
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- Golden-Threshold Homological Convergence — E8 Intelligence Research
- E8-Golden Threshold Convergence — E8 Intelligence Research
- Golden Ratio's Slowest Continued Fraction Convergence Versus Faster Nested Radicals — E8 Intelligence Research
- Golden Ratio Absent in D3's Irreducible Representations — E8 Intelligence Research
- Golden Ratio's Continued Fractions Bridge Fibonacci Convergence and Quintic Solvability — E8 Intelligence Research
BibTeX
@misc{caldin2026convergence,
title = {{Convergence Rates of Golden Ratio Representations and the Golden Angle — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23553555},
url = {https://zenodo.org/records/23553555},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Convergence Rates of Golden Ratio Representations and the Golden Angle — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23553555
UR - https://zenodo.org/records/23553555
AB - FINDING: Golden angle derives from golden ratio; Fibonacci convergence to φ is linear with rate 1/φ ≈ 0.618; nested radicals and continued fractions converge at different rates, with continued fractions exhibiting optimal (slowest algebraic) convergence. | MATH: Golden angle = 2π(1 − 1/φ) = 2π/φ² ≈ 137.5077° (or equivalently 2π − 2π/φ). Fibonacci ratio convergence: Fₙ₊₁/Fₙ → φ, error term ~ (−1/φ)ⁿ, giving asymptotic error ratio |eₙ₊₁/eₙ| = 1/φ ≈ 0.618. Continued fraction for φ: φ = 1 + 1/(1 + 1/(1 + …)) — all partial quotients = 1, making it the *slowest* converging continued fraction (worst-case for rational approximation). Nested radical: φ = √(1 + √(1 + √(1 + …))) converges faster — error decays doubly exponentially (rate ~ 1/φ² per nesting level squared). | CONNECTION: The convergence rate 1/φ = 0.618 is the *complement* of φ (φ − 1 = 1/φ). This is the same ratio appearing in phyllotaxis (golden angle = 2π/φ² ≈ 137.5°), which optimizes seed packing via the continued fraction of φ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -