Plimpton 322: Sexagesimal Lattice Parametrization of the Unit Circle — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Plimpton 322 encodes a systematic lattice parametrization of the unit circle via sexagesimal regular numbers, not merely a random list of Pythagorean triples. | MATH: The tablet lists 15 rows of (sexagesimal) values corresponding to (short side, diagonal) pairs. In modern terms, these are generated by the standard parametrization: for coprime integers \(p > q > 0\), the triple is \((p^2 - q^2,\; 2pq,\; p^2 + q^2)\). The tablet's columns give \(\frac{p^2 - q^2}{2pq}\) (or its reciprocal) and \(\frac{p^2 + q^2}{2pq}\). Crucially, the generating pairs \((p,q)\) are **regular numbers** in base 60 — i.e., \(p,q\) have no prime factors other than 2, 3, 5. This yields exact finite sexagesimal expansions. The ratios on the tablet are of the form \(\frac{p^2 - q^2}{2pq}\) and \(\frac{p^2 + q^2}{2pq}\), which are rational points on the unit circle: \(\left(\frac{2pq}{p^2+q^2},\; \frac{p^2-q^2}{p^2+q^2}\right)\). | CONNECTION: The parametrization is a **rational parametrization of the un 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: W7208829473
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