# G2 and the Rolling Ball

*Algebra/Geometry/Topology Seminar*

*by John Huerta*

*Institution:*Australian National University

*Date: Fri 12th October 2012*

*Time: 3:15 PM*

*Location: 213 Richard Berry*

*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, \(G_2\). Its Lie algebra \(\mathfrak{g}_2\) 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 \(G_2\): 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.