Babylonian Roots to Weyl Reflections: A Unified Symmetry — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Babylonian sexagesimal quadratic methods prefigure Vieta jumping's root-pair symmetry, which itself mirrors Weyl group reflections in root systems. | MATH: Babylonian normal form x2 − bx + c = 0 with base-60 arithmetic; solution via x = b/2 ± √((b/2)2 − c) (equivalent to modern quadratic formula). Vieta jumping: for integer solutions (a,b) to a2 + b2 = k(ab+1), the transformation (a,b) ↦ (a, ka − b) preserves the equation — this is a reflection in the root lattice of type A2 (or A1 × A1 for the IMO 1988 case). The invariant k is fixed; the jump is a Weyl group action on the solution set. | CONNECTION: The ratio k in Vieta jumping is constrained to perfect squares (IMO 1988: k = q2). For q=1, the golden ratio emerges: a/b → ϕ = 1.618... as solutions grow. The base-60 system yields exact fractions with denominators 2a 3b 5c — the same primes generating the crystallographic root s
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026babylonian,
title = {{Babylonian Roots to Weyl Reflections: A Unified Symmetry — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23410063},
url = {https://zenodo.org/records/23410063},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Babylonian Roots to Weyl Reflections: A Unified Symmetry — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23410063
UR - https://zenodo.org/records/23410063
AB - FINDING: Babylonian sexagesimal quadratic methods prefigure Vieta jumping's root-pair symmetry, which itself mirrors Weyl group reflections in root systems. | MATH: Babylonian normal form x^2 − bx + c = 0 with base-60 arithmetic; solution via x = b/2 ± √((b/2)^2 − c) (equivalent to modern quadratic formula). Vieta jumping: for integer solutions (a,b) to a^2 + b^2 = k(ab+1), the transformation (a,b) ↦ (a, ka − b) preserves the equation — this is a reflection in the root lattice of type A_2 (or A_1 × A_1 for the IMO 1988 case). The invariant k is fixed; the jump is a Weyl group action on the solution set. | CONNECTION: The ratio k in Vieta jumping is constrained to perfect squares (IMO 1988: k = q^2). For q=1, the golden ratio emerges: a/b → ϕ = 1.618... as solutions grow. The base-60 system yields exact fractions with denominators 2^a 3^b 5^c — the same primes generating the crystallographic root s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -