Babylonian Tablet Plimpton 322: Ancient Sexagesimal Trigonometry Predates Greeks — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Plimpton 322 encodes exact sexagesimal reciprocal pairs that generate Pythagorean triples via a tangent half-angle parametrization of the unit circle, predating Greek trigonometry by a millennium. | MATH: The tablet lists 15 rows of (a, c) values with b² = c² − a². The underlying parametrization is equivalent to: x = (1 − t²)/(1 + t²), y = 2t/(1 + t²), where t = p/q in sexagesimal. The reciprocal pairs (p, q) satisfy p·q = 60^k (regular numbers), yielding exact rational points on the unit circle. Key ratios: c/a values range from 1.983 to 1.387 (sexagesimal: 1;59,00,15 to 1;23,13,46). The diagonal-to-short-side ratio (c/b) corresponds to tan(θ/2) = q/p, with θ the acute angle. | CONNECTION: The reciprocal pairs are regular sexagesimal numbers — their prime factors are only 2, 3, 5 — linking directly to base-60 arithmetic. The tangent half-angle substitution is the rational parametrization of the unit circle, whose integer solutions (Pythagorean triples) are lattice poi
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- Plimpton 322 Refined: Reconstructing Missing Pairs in Babylonian Sexagesimal Triples — E8 Intelligence Research
- Babylonian Tablet Plimpton 322: Exact Sexagesimal Pythagorean Triples from 45° to 30° — E8 Intelligence Research
- Plimpton 322: Babylonian Sexagesimal Parametrization of the Unit Circle — E8 Intelligence Research
- Sumerian Base-60 Trigonometry: Plimpton 322's Exact Rational Triples Outperform Modern Sine Methods — E8 Intelligence Research
- Plimpton 322 as Reciprocal Pairs, Not Trigonometry — E8 Intelligence Research
BibTeX
@misc{caldin2026babylonian,
title = {{Babylonian Tablet Plimpton 322: Ancient Sexagesimal Trigonometry Predates Greeks — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23447128},
url = {https://zenodo.org/records/23447128},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Babylonian Tablet Plimpton 322: Ancient Sexagesimal Trigonometry Predates Greeks — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23447128
UR - https://zenodo.org/records/23447128
AB - FINDING: Plimpton 322 encodes exact sexagesimal reciprocal pairs that generate Pythagorean triples via a tangent half-angle parametrization of the unit circle, predating Greek trigonometry by a millennium. | MATH: The tablet lists 15 rows of (a, c) values with b² = c² − a². The underlying parametrization is equivalent to: x = (1 − t²)/(1 + t²), y = 2t/(1 + t²), where t = p/q in sexagesimal. The reciprocal pairs (p, q) satisfy p·q = 60^k (regular numbers), yielding exact rational points on the unit circle. Key ratios: c/a values range from 1.983 to 1.387 (sexagesimal: 1;59,00,15 to 1;23,13,46). The diagonal-to-short-side ratio (c/b) corresponds to tan(θ/2) = q/p, with θ the acute angle. | CONNECTION: The reciprocal pairs are regular sexagesimal numbers — their prime factors are only 2, 3, 5 — linking directly to base-60 arithmetic. The tangent half-angle substitution is the rational parametrization of the unit circle, whose integer solutions (Pythagorean triples) are lattice poi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -