On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry - Hardcover

Conway, John H.; Smith, Derek A.

Hardcover

ISBN 10: 1568811349 ISBN 13: 9781568811345

Publisher: AK Peters/CRC, 2003

Synopsis

This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces 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 authors also describe the arithmetics of the quaternions and octonions. The book concludes with a new theory of octonion factorization. Topics covered include the geometry of complex numbers, quaternions and 3-dimensional groups, quaternions and 4-dimensional groups, Hurwitz integral quaternions, composition algebras, Moufang loops, octonions and 8-dimensional geometry, integral octonions, and the octonion projective plane.

About this title

Review:

"Conway and Smith’s book is a wonderful introduction to the normed division algebras ... They develop these number systems from scratch, explore their connections to geometry, and even study number theory in quaternionic and octonionic versions of the integers. ... a lucid and elegant introduction. ... remarkably self-contained. It assumes no knowledge of number theory, string theory, Lie theory, or lower-case Gothic letters."
―John C. Baez, Bulletin of the American Mathematical Society, January 2005

"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

"This is a beautiful and fascinating book on the geometry and arithmetic of the quaternion algebra and the octonion algebra. ... most intriguing to read: it is an excellent exposition of very attractive topics, and it contains several new and significant results."
―Theo Grundhöfer, Mathematical Reviews, 2003

Synopsis:

An investigation of the geometry of quaternion and octonion algebras, this work is intended for mathematicians, physicists and crystallographers at any level - from undergraduate to professional - who are interested in symmetries of low-dimensional space. The text can also be used as a text for graduate courses in many mathematical fields, including geometry, group theory, algebra and number theory. Topics covered include: the geometry of complex numbers; quaternions and 3-dimensional groups; quaternions and 4-dimensional groups; the Hurwitz integral quaternions; Moufang loops; octonions and 8-dimensional geometry; integral octonions; and the octonion projective plane.

About this edition

• Publisher: AK Peters/CRC
• Publication date: 2003
• ISBN 10: 1568811349
• ISBN 13: 9781568811345
• Binding: Hardcover
• Edition number: 1
• Number of pages: 172