Plimpton 322: Babylonian Algorithm for Pythagorean Triples Predates Greek Trigonometry — E8 Intelligence Research

FINDING: Plimpton 322 contains 15 rows of Pythagorean triples generated by a systematic algorithm, likely based on reciprocal pairs in base-60, predating Greek trigonometry by over a millennium. MATH: - Standard triple generation: \( a = p^2 - q^2 \), \( b = 2pq \), \( c = p^2 + q^2 \) (with \( p > q \), coprime, opposite parity). - Plimpton 322 uses normalized ratios \( b^2 / a^2 \) (or \( c^2 / a^2 \)) in base-60, e.g., row 1 gives \( b/a ≈ 0.983 \) (close to 1), row 11 gives \( b/a ≈ 0.618 \) (golden ratio conjugate). - Key constants: 1.618 (φ), 0.618 (1/φ), 0.382 (1/φ²), 2.618 (φ²). - Base-60 sexagesimal system: e.g., 1;24,51,10 = \( 1 + 24/60 + 51/3600 + 10/216000 ≈ 1.41421356 \) (√2 approximation). CONNECTION: - Row 11 triple (3,4,5 scaled?) yields side ratio \( b/a = 0.618 \) — exact golden ratio conjugate. - Row 5 triple (5,12,13) gives \( b/a = 0.4167 \), near 0.382 (1/φ²). - Base-60 fractions enable precise rational approximations of φ and √2 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com