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

ISBN 1568811349   Pages 160 ISBN13 9781568811345 (What's this?)   Volumes 001 Publisher Taylor & Francis Inc   Weight (grammes) 362 Imprint A K Peters   Published in Natick Format Hardback   Height (mm) 230 Publication date 01 Jul 2001   Width (mm) 157 Library of Congress QA196.C66   Spine width (mm) 15 DEWEY 516.35   Academic level Undergraduate, Postgraduate, Professional / Scholarly DEWEY edition DC21

		Preface
I		The Complex Numbers		1
1		Introduction		3
2		Complex Numbers and 2-Dimensional Geometry		11
II		The Quaternions		21
3		Quaternions and 3-Dimensional Groups		23
4		Quaternions and 4-Dimensional Groups		41
5		The Hurwitz Integral Quaternions		55
III		The Octonions		65
6		The Composition Algebras		67
7		Moufang Loops		83
8		Octonions and 8-Dimensional Geometry		89
9		The Octavian Integers O		99
10		Automorphisms and Subrings of O		119
11		Reading O Mod 2		133
12		The Octonion Projective Plane OP[superscript 2]		143
		Bibliography		149
		Index		153

" "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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