On Quaternions and Octonions (英語) ハードカバー – 2003/1/1
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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.
"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
Amazon.com で最も参考になったカスタマーレビュー (beta) （「Early Reviewer Program」のレビューが含まれている場合があります）
The style is simple and lucid, assuming you are a mathematician.
therir work is an appreciable advance of the field.
in this hastily compiled piece of trash. No one who
does not already understand the material on octonians
will be able to penetrate this unannotated formulary.
The same goes for the entire second half of the book.
[The first half is just a rehash of material so familiar
that there is no need to see it in print for the N+1 st time. ]
The only worthwhile entry here is the reference guiding
readers to John Baez's article on octonians which:
a) is available free online and
b) actually explains in a reader friendly way the
history, math, and applications involved.
The authors (not to mention editors) should be ashamed
at such a sloppy treatment of this rich and historically
interesting episode in mathematics.
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