Plimpton 322 as Reciprocal Pairs, Not Trigonometry — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Plimpton 322 encodes Pythagorean triples generated by a rational parametrization of the unit circle, likely via reciprocal pairs (x, 1/x) in base‑60, not a modern trigonometric table. MATH: - Standard parametrization: x = 2t/(1+t2), y = (1−t2)/(1+t2) for t ∈ ℚ. - Plimpton 322 columns correspond to b2/d2 = 1 − (a/d)2 with a2 + b2 = d2. - Proposed algorithm (arXiv:2001.11141): b = M · QM, with M, QM ∈ ℤ+, generating all rows via factor pairs — equivalent to choosing t = p/q with p, q from sexagesimal regular numbers (2,3,5‑smooth). - Base‑60 constants: reciprocals of regular numbers (e.g., 1/60, 1/30, 1/20, 1/15, 1/12, 1/10) yield exact finite sexagesimal expansions — the key constraint.
CONNECTION: - The rational parametrization t ↦ (2t/(1+t2), (1−t2)/(1+t2)) is the stereographic projection from
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- Plimpton 322 Decoded: A Sexagesimal Table of Regular Reciprocal Pairs Generating Pythagorean Triples — E8 Intelligence Research
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- Babylonian Tablet Plimpton 322: Ancient Sexagesimal Trigonometry Predates Greeks — E8 Intelligence Research
- Plimpton 322 Refined: Reconstructing Missing Pairs in Babylonian Sexagesimal Triples — E8 Intelligence Research
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BibTeX
@misc{caldin2026plimpton,
title = {{Plimpton 322 as Reciprocal Pairs, Not Trigonometry — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635289},
url = {https://zenodo.org/records/23635289},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Plimpton 322 as Reciprocal Pairs, Not Trigonometry — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635289
UR - https://zenodo.org/records/23635289
AB - FINDING: Plimpton 322 encodes Pythagorean triples generated by a rational parametrization of the unit circle, likely via reciprocal pairs (x, 1/x) in base‑60, not a modern trigonometric table. MATH: - Standard parametrization: x = 2t/(1+t^2), y = (1−t^2)/(1+t^2) for t ∈ ℚ. - Plimpton 322 columns correspond to b^2/d^2 = 1 − (a/d)^2 with a^2 + b^2 = d^2. - Proposed algorithm (arXiv:2001.11141): b = M · Q_M, with M, Q_M ∈ ℤ^+, generating all rows via factor pairs — equivalent to choosing t = p/q with p, q from sexagesimal regular numbers (2,3,5‑smooth). - Base‑60 constants: reciprocals of regular numbers (e.g., 1/60, 1/30, 1/20, 1/15, 1/12, 1/10) yield exact finite sexagesimal expansions — the key constraint. CONNECTION: - The rational parametrization t ↦ (2t/(1+t^2), (1−t^2)/(1+t^2)) is the stereographic projection from Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -