G2and the rolling ball

John C. Baez, John Huerta

    Research output: Contribution to journalArticlepeer-review

    19 Citations (Scopus)


    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 g2acts 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 G2incidence geometry, and show how a form of geometric quantization applied to this geometry lets us recover the imaginary split octonions and these operations.

    Original languageEnglish
    Pages (from-to)5257-5293
    Number of pages37
    JournalTransactions of the American Mathematical Society
    Issue number10
    Publication statusPublished - 2014


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