Plimpton 322: Old Babylonian Exact-Ratio Trigonometry via Sexagesimal Reciprocals — E8 Intelligence Research

Abstract

FINDING: Plimpton 322 encodes 15 rows of Pythagorean triples generated by reciprocal pairs of regular sexagesimal integers, revealing an Old Babylonian proto-trigonometry based on exact ratios rather than angles. | MATH: For regular integers \(p, q\) in base-60 (with \(p > q\), both having only prime factors 2, 3, 5), the tablet's columns correspond to: - \( (p^2 - q^2) / (2pq) \) (short side / long side) - \( (p^2 + q^2) / (2pq) \) (hypotenuse / long side) - The "diagonal" column lists \( (p/q)^2 \) as a sexagesimal fraction. The rows are ordered by decreasing \( (p/q)^2 \), from ~1.983 to ~1.387, equivalent to angles from ~45° to ~31° in modern terms. The generating rule: \( p/q = (s + 1/s)/2 \) where \(s\) is a regular sexagesimal reciprocal pair (e.g., \(s = 2/3, 3/2\) gives \(p/q = 13/12\)). All triples are primitive and exact in base-60. | CONNECTION: The reciprocal pairs \( (s, 1/s) \) with \(s\) regular in base-60 are precisely the elements of the multiplicative group o Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com