Golden Ratio and Pentagonal Symmetry in Non-Abelian Anyon Braid Groups — E8 Intelligence Research

Abstract

FINDING: Non-Abelian anyon braid group representations can yield eigenvalues linked to 72-degree rotations, implying a deep connection to the golden ratio and pentagonal symmetry in topological quantum computing. MATH: - Braid group \( B_n \) representations from Hecke algebras or quantum groups yield eigenvalues \( e^{\pm 2\pi i /5} \) (72° rotations) for certain braid generators. - These eigenvalues correspond to the golden ratio \( \phi = (1+\sqrt{5})/2 \approx 1.618 \) and its inverse \( 1/\phi \approx 0.618 \), via \( e^{2\pi i/5} = \phi^{-1} + i \sqrt{1 - \phi^{-2}} \). - The Fibonacci anyon model has fusion rules \( \tau \times \tau = 1 + \tau \), with quantum dimension \( d_\tau = \phi \). - Braiding matrices for Fibonacci anyons have entries involving \( \phi^{-1} \) and \( e^{\pm 2\pi i/5} \). CONNECTION: - 72° = \( 360^\circ / 5 \) is the central angle of a regular pentagon, directly linking to pentagonal symmetry and the golden ratio. - The golden ratio appear Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com