Babylonian Base-60 Math: Exact Fractions, Reciprocal Pairs, and Plimpton 322's Golden Ratio Link — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Babylonian base-60 (sexagesimal) positional notation encodes exact fractions (1/60, 1/3600) and yields the ratio 0.618 (1/φ) via reciprocal pairs, with Plimpton 322 showing Pythagorean triples in base-60. | MATH: Base-60 place value: \( a \cdot 60^n + b \cdot 60^{n-1} + \dots \); reciprocal pairs (e.g., 1/60 = 0.01666…, 60/1 = 60); Plimpton 322 contains 15 rows of integer Pythagorean triples \((a,b,c)\) satisfying \(a^2 + b^2 = c^2\), with ratios \(b/a\) approximating \(\tan(\theta)\) for \(\theta\) from ~45° to ~30°. Key constants: 60, 3600, 216000 (60³); the sexagesimal expansion of 1/φ ≈ 0.6180339… = 0;36,36,36… (base-60: 36/60 + 36/3600 + 36/216000…). | CONNECTION: Direct link to golden ratio: 1/φ = 0.6180339… = 0;36,36,36… in base-60 — a repeating sexagesimal pattern. Also, 60° is the interior angle of an equilateral triangle (crystallographic 3-fold symmetry), and 360° (6×60) is the full circle — base-60 naturally encodes hexagonal/trigonal symmetry (root system A₂, hexa Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Theoretical Computer Science
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204111776
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