Plimpton 322: Ancient Babylonian Exact Secant-Squared Table via Regular Reciprocals — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Plimpton 322 encodes a table of secant-squared values (1 + tan²θ) in base-60, generated by reciprocal pairs (p, q) of regular sexagesimal integers, yielding exact Pythagorean triples — not approximations. | MATH: For each row, the tablet lists (a, c) with b² = c² − a². In base-60, the key ratio is (c/b)² = 1 + (a/b)². The generating rule: take regular p > q (both divisors of 60^k), set a = p² − q², b = 2pq, c = p² + q². The tablet's column I gives (c/b)² = (p²+q²)²/(2pq)² — i.e., sec²θ. The values are exact in base-60, not decimal approximations. The 15 rows correspond to p/q ratios that decrease from ~1.78 to ~1.07, covering angles from ~45° down to ~30°. | CONNECTION: The sec²θ values are rational in base-60. Compare to golden ratio: φ = 1.618…, φ² = 2.618…, 1/φ = 0.618…, 1/φ² = 0.382…. The tablet's largest sec²θ ≈ (1.78²+1)²/(2·1.78)² ≈ 1.98 (near 2.0, not 2.618). The smallest ≈ 1.33 (near 4/3, not 1.382). No direct golden ratio appears. However, the base-60 system itself i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Theoretical Computer Science
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7210272506
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