  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 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 explain this
ratio in simple terms, use the dot product and cross product of split octonions
to describe the G2 incidence geometry, and show how a form of geometric
quantization applied to this geometry lets us recover the imaginary split
octonions and these operations.
