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.
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