G2 and the rolling ball

Abstract

Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2: Its Lie algebra g2 acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of radii is 1:3. Using the split octonions, we devise a similar, but more global, picture of G2: it acts as the symmetries of a `spinorial ball rolling on a projective plane', again when the ratio of radii is 1:3. We describe the incidence geometry of both systems, and use it to explain the mysterious 1:3 ratio in simple, geometric terms.