Plimpton 322: A Systematic Base-60 Table of Pythagorean Triples via Reciprocal Pairs — E8 Intelligence Research

FINDING: Plimpton 322 encodes a systematic table of Pythagorean triples generated by reciprocal pairs in base-60, with cotangent values forming a regular decreasing sequence. MATH: - Triples satisfy \(a^2 + b^2 = c^2\) with \(a < b\). - Tablet lists columns: (b, a, c) in base-60 sexagesimal. - Key ratio: \((θ) = b/a\) decreases from ~0.79 to ~0.01 across 15 rows. - Generation method: For regular sexagesimal reciprocals \((p, q)\) with \(p > q\), set \(a = p^2 - q^2\), \(b = 2pq\), \(c = p^2 + q^2\). - Base-60 constants: 1, 2, 3, 5, 60, 3600 appear as factors; all \(p, q\) are regular numbers (only prime factors 2,3,5). CONNECTION: - Cotangent values (e.g., row 1: ~0.786) approximate \(1/φ ≈ 0.618\) and \(φ^2 ≈ 2.618\) in some rows, but not exact. - Base-60 system aligns with 5-fold and 6-fold symmetries (factors 2,3,5) — crystallographic root systems \(A_2\), \(B_2\), \(G_2\) have angles 30°, 45°, 60° whose cotangents are regular in base-60. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com