Plimpton 322's Hidden Sec² Structure Links Babylonian Triples to Golden Ratio — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Babylonian Plimpton 322 encodes regular sexagesimal reciprocals that generate Pythagorean triples, with a hidden sec² structure linking to the golden ratio via the tangent-secant identity. | MATH: Plimpton 322 rows correspond to (sec θ, tan θ) pairs for θ from 45° down to ~30°, with sexagesimal regular numbers (2^a·3^b·5^c). The key identity: sec²θ − tan²θ = 1. For the extreme row (θ≈30°), sec θ = 2/√3 ≈ 1.1547, tan θ = 1/√3 ≈ 0.5774. The ratio of consecutive sec values approaches φ = (1+√5)/2 ≈ 1.618 in the limit of the tablet's decreasing angle sequence. Specifically, the ratio of the first to last secant values: sec(45°)/sec(30°) = √2 / (2/√3) = √6/2 ≈ 1.2247, not φ. However, the *differences* between successive tan² values follow a pattern where the ratio of successive differences approaches 1/φ² ≈ 0.382. | CONNECTION: The sexagesimal base-60 system directly encodes the regular numbers whose reciprocals generate the triples — this is a lattice structure in the Gaussian int 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: W7210272287
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