Sexagesimal Origins of Continued-Fraction Convergents in Astronomical Periods — E8 Intelligence Research

FINDING: Babylonian sexagesimal (base-60) arithmetic is the substrate from which continued-fraction approximations of astronomical period relations emerge — the same convergent machinery that generates φ-adjacent ratios. | MATH: YBC 7289 gives √2 ≈ 1;24,51,10 (base-60) = 1 + 24/60 + 51/3600 + 10/216000 = 1.41421296… (error ~6×10⁻⁷ vs √2 = 1.41421356…). Sexagesimal place-value: digits dᵢ ∈ {0,…,59}, value = Σ dᵢ·60⁻ⁱ. Continued fraction [a₀;a₁,a₂,…] = a₀ + 1/(a₁ + 1/(a₂ + …)); convergents pₙ/qₙ satisfy pₙqₙ₋₁ − pₙ₋₁qₙ = (−1)ⁿ⁻¹. | CONNECTION: Base-60 = 2²·3·5 has divisors {1,2,3,4,5,6,10,12,15,20,30,60} — maximal divisor density for its size, enabling exact representation of 1/2, 1/3, 1/5, 1/6, 1/12, 1/15, 1/20, 1/30. This is the arithmetic reason Babylonian fractions terminate where decimal ones do not. The golden ratio φ = [1;1,1,1,…] is the *slowest-converging* continued fraction (all partial quotients = 1), making it the "most irrational" number — the antithesis of the fast-convergi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com