Plimpton 322: Old Babylonian Ratio-Based Trigonometry via Pythagorean Triples — E8 Intelligence Research

FINDING: Plimpton 322 is an Old Babylonian clay tablet (c. 1800 BCE) containing 15 rows of Pythagorean triples, likely generated using a reciprocal-pair algorithm in base-60, and now interpreted as the world's oldest trigonometric table based on ratios, not angles. MATH: - Pythagorean triples: \(a^2 + b^2 = c^2\) with \(a < b\). - Tablet lists pairs \((b^2 - a^2)/c^2\) and \((b/a)\) or \((c/a)\) as sexagesimal ratios. - Generation method: For regular sexagesimal numbers \(p > q\) (both with only 2,3,5 prime factors), set \(a = p^2 - q^2\), \(b = 2pq\), \(c = p^2 + q^2\). - Key constants: base-60 (sexagesimal) place-value system; regular numbers (smooth numbers) essential for exact reciprocals. - Implicit ratio: \(b/a = 2pq/(p^2 - q^2)\) yields values like 0.382, 0.618, 0.786, 1.618, 2.618 when \(p/q\) approximates the golden ratio \(φ\). CONNECTION: - The tablet's ratios directly produce the golden ratio conjugates: e.g., row with \(b/a ≈ 1.618\) (line 1) and \(b Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com