Plimpton 322's Sexagesimal Algorithm as Rational Circle Parametrization — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Plimpton 322 encodes Pythagorean triples via a sexagesimal algorithm equivalent to rational parametrization of the unit circle, with a proposed new generative rule b = M · QM. MATH: - Standard reconstruction: For coprime p > q > 0, triples (a, b, d) satisfy a = p2 − q2, b = 2pq, d = p2 + q2 (or swapped). - Rational parametrization: x = 2t/(1+t2), y = (1−t2)/(1+t2) for t = q/p (or reciprocal), mapping t ∈ ℚ to points on x2 + y2 = 1. - New algorithm (arXiv 2001.11141): b = M · QM, with M, QM ∈ ℤ+, generating the tablet's rows — data‑driven, not assuming the standard p,q form. - Sexagesimal base‑60 appears in column headers (squares of sides, diagonals) — e.g., values like 1;59, 2;49, etc., implying use of reciprocal pairs and regular numbers (smooth in base 60). CONNECTION: - The parametrization t = tan(θ/2) yields ratios: for t = 1/√3 →
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- Plimpton 322: Babylonian Sexagesimal Parametrization of the Unit Circle — E8 Intelligence Research
- MERLIN SCIENCE — Plimpton 322 as a Sexagesimal Lattice of Rational Points on the Unit C — E8 Intelligence Research
- Sexagesimal Rational Parametrization of the Unit Circle — E8 Intelligence Research
- Plimpton 322's Sexagesimal Ratios: A Reconstruction Algorithm for Missing Pairs — E8 Intelligence Research
- Plimpton 322 Refined: Reconstructing Missing Pairs in Babylonian Sexagesimal Triples — E8 Intelligence Research
BibTeX
@misc{caldin2026plimpton,
title = {{Plimpton 322's Sexagesimal Algorithm as Rational Circle Parametrization — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23553644},
url = {https://zenodo.org/records/23553644},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Plimpton 322's Sexagesimal Algorithm as Rational Circle Parametrization — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23553644
UR - https://zenodo.org/records/23553644
AB - FINDING: Plimpton 322 encodes Pythagorean triples via a sexagesimal algorithm equivalent to rational parametrization of the unit circle, with a proposed new generative rule b = M · Q_M. MATH: - Standard reconstruction: For coprime p > q > 0, triples (a, b, d) satisfy a = p^2 − q^2, b = 2pq, d = p^2 + q^2 (or swapped). - Rational parametrization: x = 2t/(1+t^2), y = (1−t^2)/(1+t^2) for t = q/p (or reciprocal), mapping t ∈ ℚ to points on x^2 + y^2 = 1. - New algorithm (arXiv 2001.11141): b = M · Q_M, with M, Q_M ∈ ℤ^+, generating the tablet's rows — data‑driven, not assuming the standard p,q form. - Sexagesimal base‑60 appears in column headers (squares of sides, diagonals) — e.g., values like 1;59, 2;49, etc., implying use of reciprocal pairs and regular numbers (smooth in base 60). CONNECTION: - The parametrization t = tan(θ/2) yields ratios: for t = 1/√3 → Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -