This book investigates the geometry of the quaternion and octonion
algebras. Following a comprehensive historical introduction, the
special properties of 3- and 4-dimensional Euclidean spaces are
illuminated using quaternions, leading to enumerations of the
corresponding finite groups of symmetries.
The second half of the book discusses the less familiar octonion
algebra, concentrating on its remarkable "triality
symmetry" after an appropriate study of Moufang loops. The
arithmetics of the quaternions and octonions are also described,
and the book concludes with a new theory of octonion
factorization.
Topics covered include:
- history
- the geometry of complex numbers
- quaternions and 3-dimensional groups
- quaternions and 4-dimensional groups
- the Hurwitz integral quaternions
- the composition algebras
- Moufang loops
- octonions and 8-dimensional geometry
- integral octonions
- the octonion projective plane
