The Golden Ratio's Irrationality: Continued Fractions, Stern-Brocot Trees, and Ergodic Theory — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The golden ratio is the "most irrational" number because its continued fraction expansion is all 1s, making it the slowest-converging rational approximant; the Stern-Brocot tree and Farey sequence provide the canonical lattice of rationals whose uniform distribution properties are governed by infinite ergodic theory. | MATH: Golden ratio φ = (1+√5)/2 ≈ 1.6180339887; its continued fraction φ = [1;1,1,1,...] = 1 + 1/(1+1/(1+...)); conjugate φ' = (1−√5)/2 ≈ −0.6180339887. Stern-Brocot tree: mediant of a/b and c/d is (a+c)/(b+d); every positive rational appears exactly once. Farey sequence F_n: all reduced fractions between 0 and 1 with denominator ≤ n. Irrational rotation number α: the map x → x+α mod 1; for α = φ−1 ≈ 0.618, the orbit is the most uniformly spread (Kronecker's theorem, three-gap theorem). The dichotomy result (arXiv:1009.1823) shows even Stern-Brocot and Farey sequences are uniformly distributed mod 1 under canonical weightings, with asymptotic for Lebesgue measur
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 = {{The Golden Ratio's Irrationality: Continued Fractions, Stern-Brocot Trees, and Ergodic Theory — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635180},
url = {https://zenodo.org/records/23635180},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Golden Ratio's Irrationality: Continued Fractions, Stern-Brocot Trees, and Ergodic Theory — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635180
UR - https://zenodo.org/records/23635180
AB - FINDING: The golden ratio is the "most irrational" number because its continued fraction expansion is all 1s, making it the slowest-converging rational approximant; the Stern-Brocot tree and Farey sequence provide the canonical lattice of rationals whose uniform distribution properties are governed by infinite ergodic theory. | MATH: Golden ratio φ = (1+√5)/2 ≈ 1.6180339887; its continued fraction φ = [1;1,1,1,...] = 1 + 1/(1+1/(1+...)); conjugate φ' = (1−√5)/2 ≈ −0.6180339887. Stern-Brocot tree: mediant of a/b and c/d is (a+c)/(b+d); every positive rational appears exactly once. Farey sequence F_n: all reduced fractions between 0 and 1 with denominator ≤ n. Irrational rotation number α: the map x → x+α mod 1; for α = φ−1 ≈ 0.618, the orbit is the most uniformly spread (Kronecker's theorem, three-gap theorem). The dichotomy result (arXiv:1009.1823) shows even Stern-Brocot and Farey sequences are uniformly distributed mod 1 under canonical weightings, with asymptotic for Lebesgue measur Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -