The Birth of $E_8$ out of the Spinors of the Icosahedron

TL;DR

This paper demonstrates that the exceptional $E_8$ root system can be constructed purely from three-dimensional icosahedral geometry using Clifford algebra, providing a new geometric perspective on these complex symmetries.

Contribution

It introduces a novel spinorial approach that constructs $E_8$ and other exceptional root systems from 3D geometry, challenging traditional higher-dimensional views.

Findings

$E_8$ roots derived from 3D icosahedral group elements

Spinorial construction links 3D and 4D root systems

Provides geometric explanation for automorphism groups

Abstract

$E_8$ is prominent in mathematics and theoretical physics, and is generally viewed as an exceptional symmetry in an eight-dimensional space very different from the space we inhabit; for instance the Lie group $E_8$ features heavily in ten-dimensional superstring theory. Contrary to that point of view, here we show that the $E_8$ root system can in fact be constructed from the icosahedron alone and can thus be viewed purely in terms of three-dimensional geometry. The $240$ roots of $E_8$ arise in the 8D Clifford algebra of 3D space as a double cover of the $120$ elements of the icosahedral group, generated by the root system $H_3$. As a by-product, by restricting to even products of root vectors (spinors) in the 4D even subalgebra of the Clifford algebra, one can show that each 3D root system induces a root system in 4D, which turn out to also be exactly the exceptional 4D root systems. The spinorial point of view explains their existence as well as their unusual automorphism groups. This spinorial approach thus in fact allows one to construct all exceptional root systems within the geometry of three dimensions, which opens up a novel interpretation of these phenomena in terms of spinorial geometry.