Babylonian Trigonometry Linked to Rational Circle Parametrization via Base-60 Divisions — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The rational parametrization of the unit circle — central to Babylonian/sexagesimal trigonometry — is explicitly linked to base-60 geometry via a natural divisibility of the circle into 6, 12, 24, and 360 parts, as shown in the arXiv paper.
MATH: - Rational parametrization: t ↦ ( (1−t2)/(1+t2), 2t/(1+t2) ) — all rational points on x2 + y2 = 1 (excluding (−1,0)). - Sexagesimal link: circle circumference divided by 6 gives chord length = radius (regular hexagon), leading to 360° = 6×60; 24 hours = 4×6; 12 months = 2×6. - Key ratios from the parametrization at special t: - t = 1/2: point (3/5, 4/5) → Pythagorean triple, slope ratio 0.6/0.8 = 0.75. - t = 1/3: point (8/10, 6/10) = (4/5, 3/5) → reciprocal triple. - The tangent half-angle substitution t = tan(θ/2) yields sinθ = 2t/(1+t2), cosθ = (1−t2)/(1+t2) — the core of Babylonian chord tables.
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
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- Babylonian Tablet Plimpton 322: Ancient Sexagesimal Trigonometry Predates Greeks — 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
BibTeX
@misc{caldin2026babylonian,
title = {{Babylonian Trigonometry Linked to Rational Circle Parametrization via Base-60 Divisions — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635617},
url = {https://zenodo.org/records/23635617},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Babylonian Trigonometry Linked to Rational Circle Parametrization via Base-60 Divisions — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635617
UR - https://zenodo.org/records/23635617
AB - FINDING: The rational parametrization of the unit circle — central to Babylonian/sexagesimal trigonometry — is explicitly linked to base-60 geometry via a natural divisibility of the circle into 6, 12, 24, and 360 parts, as shown in the arXiv paper. MATH: - Rational parametrization: t ↦ ( (1−t^2)/(1+t^2), 2t/(1+t^2) ) — all rational points on x^2 + y^2 = 1 (excluding (−1,0)). - Sexagesimal link: circle circumference divided by 6 gives chord length = radius (regular hexagon), leading to 360° = 6×60; 24 hours = 4×6; 12 months = 2×6. - Key ratios from the parametrization at special t: - t = 1/2: point (3/5, 4/5) → Pythagorean triple, slope ratio 0.6/0.8 = 0.75. - t = 1/3: point (8/10, 6/10) = (4/5, 3/5) → reciprocal triple. - The tangent half-angle substitution t = tan(θ/2) yields sinθ = 2t/(1+t^2), cosθ = (1−t^2)/(1+t^2) — the core of Babylonian chord tables. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -