Sexagesimal Base-60, Regular Numbers, and Continued Fractions in Babylonian Astronomy — E8 Intelligence Research

FINDING: Babylonian sexagesimal (base-60) numeration is structurally coupled to regular numbers (5-smooth: 2^a·3^b·5^c) and to continued-fraction convergents, the same machinery that generates best rational approximations to astronomical periods. | MATH: Base-60 positional system; regular numbers n = 2^a·3^b·5^c have terminating sexagesimal reciprocals 1/n; continued fraction [a₀;a₁,a₂,…] → convergents pₖ/qₖ with pₖqₖ₋₁ − pₖ₋₁qₖ = (−1)^(k−1); Pell x² − Dy² = 1 solved via CF of √D; quintic root expressible via Rogers–Ramanujan continued fraction R(q). | CONNECTION: 60 = 2²·3·5 is the smallest number divisible by 1–6, making it the LCM-optimal base for regular-number arithmetic; 60 = 12·5 and 360 = 6·60 encode the hexagon (6-fold) and pentagon (5-fold) — the two crystallographic/Platonic generators whose product 6·5 = 30 and 2·30 = 60. The golden ratio φ = [1;1,1,1,…] is the slowest-converging CF (worst approximable), while √2 = [1;2,2,2,…] and √3 = [1;1,2,1,2,…] are the fastest among qu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com