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Understanding the exceptional Lie groups as the symmetry groups of simplerobjects is a long-standing program in mathematics. Here, we explore one famousrealization of the smallest exceptional Lie group, G2. Its Lie algebra actslocally as the symmetries of a ball rolling on a larger ball, but only when theratio of radii is 1:3. Using the split octonions, we devise a similar, but moreglobal, picture of G2: it acts as the symmetries of a 'spinorial ball rollingon a projective plane', again when the ratio of radii is 1:3. We explain thisratio in simple terms, use the dot product and cross product of split octonionsto describe the G2 incidence geometry, and show how a form of geometricquantization applied to this geometry lets us recover the imaginary splitoctonions and these operations.

G2 and the Rolling Ball