G2, split octonions, and the rolling ball
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 infinitesimally as the symmetries of a ball rolling on another ball, but only when the ratio of radii is 1:3 or 3:1. Using the split octonions and the `divisors-of-zero distribution' of Agrachev, we devise a similar, but more global, picture of G2: it acts as the symmetries of a `fermionic ball rolling on a projective plane', again only when the ratio of radii is 1:3 or 3:1. We describe the incidence geometry of this system, and show how it sheds light on the role of this mysterious ratio, 1:3. This is joint work with Jim Dolan.