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On Quaternions and Octonions [Hardcover]

John H. Conway , Derek A. Smith
4.3 out of 5 stars  See all reviews (3 customer reviews)
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Book Description

Jan. 23 2003 1568811349 978-1568811345 1

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.

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"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

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2 of 2 people found the following review helpful
3.0 out of 5 stars Good, but something is better July 25 2003
Conway is an excellent mathematician and an extremely lucid author. No criticism should be given to any of his writings. In the case of quaternions (and octonians), a much better, more complete, and more powerful view is achieved by seeing them in the larger setting of geometric algebra. The geometric algebra gives direct access to all the results and all the geometry of these algebras, and does so in an intuitive and useful way. I suggest that the new book by Chris Doran and Anthony Lasenby called "Geometric Algebra for Physicists" is a better place, generally, to get acquainted with these issues deeply. It isn't a criticism of Conway. It's just an advantage of seeing things in the right context.
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1 of 1 people found the following review helpful
5.0 out of 5 stars A model of exposition March 25 2003
By Sally
John Conway's books are always well written, and this could serve as a model for other mathematics authors. I don't need to know that much about quaternions and octonions, but I found myself working through most of the book and the beautiful mathematics it covers. The only thing that disappoints is the dreadful cover and the difficulty getting hold of a copy in a bookstore. But then I guess Amazon.com exists to help people get their hands on stuff they might never see in a bookstore.
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5.0 out of 5 stars Examining 3- and 4-dimensional Euclidean spaces April 18 2003
On Quaternions And Octonions: Their Geometry, Arithmetic, And Symmetry is a collaboratively presented treatise by John H. Conway and Derek A. Smith on the geometry of the quaternion and octonion algebras. Examining 3- and 4-dimensional Euclidean spaces, enumerating the corresponding finite group of symmetries, analyzing the arithmetics of quaternions and octonions and much more, this impressive presentation sheds new light on the geometry of complex numbers and is a scholarly addition to Advanced Mathematics reference collections and reading lists.
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