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On Quaternions and Octonions

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
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" "A resonant spike above background noise in one parameter as another parameter is varied is a frequent indicator!" -Geoffrey Dixon, Mathematical Intelligencer , May 2004 "Those readers who are fascinated by the links between geometry and groups will find that this book gives them new insights. " -Hugh Williams, The Mathematical Gazette , July 2004"

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