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
General
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