An investigation of the geometry of quaternion and octonion algebras, this text is intended for mathematicians who are interested in the symmetries of low-dimensional space. Topics covered include Moufang loops, integral octonions and the octonion projective plane.

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.

"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