Geometry Of The Octonions, The

Describes rotations in three dimensions and in various symmetry groups. This title offers an introduction to the properties of the octonions, with emphasis on their geometric structure. It includes applications such as: the exceptional Lie groups, octonionic projective spaces, and applications to particle physics.
There are precisely two further generalizations of the real and complex numbers, namely, the quaternions and the octonions. The quaternions naturally describe rotations in three dimensions. In fact, all (continuous) symmetry groups are based on one of these four number systems. This book provides an elementary introduction to the properties of the octonions, with emphasis on their geometric structure. Elementary applications covered include the rotation groups and their spacetime generalization, the Lorentz group, as well as the eigenvalue problem for Hermitian matrices. In addition, more sophisticated applications include the exceptional Lie groups, octonionic projective spaces, and applications to particle physics including the remarkable fact that classical supersymmetry only exists in particular spacetime dimensions.