Plimpton 322: Oldest Trigonometric Table via Reciprocal-Pair Parametrization — E8 Intelligence Research

FINDING: Plimpton 322 encodes 15 rows of Pythagorean triples in sexagesimal (base-60) notation, generated by a reciprocal-pair parametrization of the unit circle — the oldest known systematic trigonometric table, predating Hipparchus by ~1500 years. MATH: - Core parametrization: for reciprocal pair (p, 1/p), the rational point on the unit circle is x = (p − 1/p)/(p + 1/p), y = 2/(p + 1/p) equivalently (x, y) = ((p² − 1)/(p² + 1), 2p/(p² + 1)) — the standard rational parametrization. - Column I (the "missing" column) reconstructs as sec²θ = 1 + tan²θ = (p + 1/p)²/4, i.e. the squared secant of the angle. - Columns II–III give the short leg b and hypotenuse c of integer triples (a, b, c) with a² + b² = c². - Sexagesimal base 60 = 2²·3·5, giving 12 divisors; regular numbers (2^a·3^b·5^c) have terminating reciprocals — essential for the reciprocal-pair method. - Bruins/arXiv:1109.3814 reconstruction: rows correspond to p = 1 + 1/n or similar regular-number generators, producing the "mi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com