ABSTRACT
This chapter briefly describes some “external” applications of the octonions. The compact semisimple Lie (that is, continuous) groups over the real and complex numbers have been famously classified. In the 1950s, Freudenthal found that four particular “geometries” could all be defined over ℝ, ℂ, ℍ, O: the geometries of the elliptic and projective planes, and 5-dimensional symplectic geometry, which were already known, and his new “metasymplectic” geometry. The automorphisms of the four geometries over the four rings form his celebrated “Magic Square”. The chapter discusses the Octonion projective plane, and defines the octonion projective plane to have points and lines specified by 3 × 3 Hermitian idempotents of trace 1 over the octonions, with point e lying on the line f just when e ◦ f = 0.
