Babylonian Roots to Weyl Reflections: A Unified Symmetry — E8 Intelligence Research

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 \(x^2 - bx + c = 0\) with base-60 arithmetic; solution via \(x = \frac{b}{2} \pm \sqrt{(\frac{b}{2})^2 - c}\) (equivalent to modern quadratic formula). Vieta jumping: for integer solutions \((a,b)\) to \(a^2 + b^2 = k(ab+1)\), the transformation \((a,b) \mapsto (a, ka - b)\) preserves the equation — this is a reflection in the root lattice of type \(A_2\) (or \(A_1 \times A_1\) 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: \(a/b \to \phi = 1.618...\) 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