  We elucidate the geometry of matrix models based on simple formally real
Jordan algebras. Such Jordan algebras give rise to a nonassociative geometry
that is a generalization of Lorentzian geometry. We emphasize constructions for
the exceptional Jordan algebra and the exceptional Jordan C*-algebra and
describe the projective spaces related to the exceptional cubic matrix model
and the E_6 matrix model. The resulting projective spaces are shown to be
exceptional versions of projective twistor space, thus revealing the existence
of exceptional twistor string theories that are dual to octonionic matrix
models.
