Sexagesimal Rational Parametrization of the Unit Circle — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Babylonian sexagesimal arithmetic provides a natural rational parametrization of the unit circle via the standard stereographic projection formula, and base-60 divisibility (60 = 2²·3·5) makes the resulting rational points especially tractable.
MATH: - Rational parametrization of unit circle: for t ∈ ℚ, (x, y) = ((1 − t²)/(1 + t²), 2t/(1 + t²)) - Equivalent Pythagorean triple generator: (a, b, c) = (m² − n², 2mn, m² + n²), m, n ∈ ℤ - Sexagesimal base: 60 = 2² · 3 · 5; divisors {1,2,3,4,5,6,10,12,15,20,30,60} - Plimpton 322 (OB, ~1800 BCE): 15 rows of (short side, diagonal) pairs; e.g. (119, 169), (3367, 4825), (4601, 6649) — all satisfy a² + b² = c² with b, c sexagesimally regular - Circle–sexagesimal link (arXiv 0707.0676): 360° = 6 · 60; regular hexagon inscribed in circle gives chord = radius, so 6 chords close the circle exactly — the geometric origin of base-60 angle division - 12 months, 24 hours, 60 minutes all derive from 6 · 60 = 360 and 2 · 12 = 24
CONNECTION: - Ba
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026sexagesimal,
title = {{Sexagesimal Rational Parametrization of the Unit Circle — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658640},
url = {https://zenodo.org/records/23658640},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sexagesimal Rational Parametrization of the Unit Circle — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658640
UR - https://zenodo.org/records/23658640
AB - FINDING: Babylonian sexagesimal arithmetic provides a natural rational parametrization of the unit circle via the standard stereographic projection formula, and base-60 divisibility (60 = 2²·3·5) makes the resulting rational points especially tractable. MATH: - Rational parametrization of unit circle: for t ∈ ℚ, (x, y) = ((1 − t²)/(1 + t²), 2t/(1 + t²)) - Equivalent Pythagorean triple generator: (a, b, c) = (m² − n², 2mn, m² + n²), m, n ∈ ℤ - Sexagesimal base: 60 = 2² · 3 · 5; divisors {1,2,3,4,5,6,10,12,15,20,30,60} - Plimpton 322 (OB, ~1800 BCE): 15 rows of (short side, diagonal) pairs; e.g. (119, 169), (3367, 4825), (4601, 6649) — all satisfy a² + b² = c² with b, c sexagesimally regular - Circle–sexagesimal link (arXiv 0707.0676): 360° = 6 · 60; regular hexagon inscribed in circle gives chord = radius, so 6 chords close the circle exactly — the geometric origin of base-60 angle division - 12 months, 24 hours, 60 minutes all derive from 6 · 60 = 360 and 2 · 12 = 24 CONNECTION: - Ba Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -