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On Quaternions and Octonions [Hardcover]

by John Horton Conway, Derek Smith
3.9 out of 5 stars  See all reviews (7 customer reviews)

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Book Description

January 23, 2003 1568811349 978-1568811345 1st

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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On Quaternions and Octonions + Rotations, Quaternions, and Double Groups (Dover Books on Mathematics) + A Primer of Quaternions (Classic Reprint)
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Editorial Reviews


"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

Product Details

  • Hardcover: 159 pages
  • Publisher: A K Peters/CRC Press; 1st edition (January 23, 2003)
  • Language: English
  • ISBN-10: 1568811349
  • ISBN-13: 978-1568811345
  • Product Dimensions: 9.3 x 6.2 x 0.6 inches
  • Shipping Weight: 12.6 ounces (View shipping rates and policies)
  • Average Customer Review: 3.9 out of 5 stars  See all reviews (7 customer reviews)
  • Amazon Best Sellers Rank: #193,401 in Books (See Top 100 in Books)

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Customer Reviews

3.9 out of 5 stars
3.9 out of 5 stars
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Most Helpful Customer Reviews
62 of 71 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 2 people found the following review helpful
5.0 out of 5 stars For afficionados December 28, 2013
Format:Hardcover|Verified Purchase
I love it. There are many fascinating, deep and unique tidbits from John Conway himself.
The style is simple and lucid, assuming you are a mathematician.
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1 of 2 people found the following review helpful
5.0 out of 5 stars quaternions and octonions revisited October 19, 2013
Format:Kindle Edition|Verified Purchase
Candid review by one of the masters of the subject. The text is accessible to undergraduate students, very concise and clearly written.
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2 of 4 people found the following review helpful
3.0 out of 5 stars People either love Conway or hate him? September 27, 2009
This books gives a window into the newer notation in group theory.
Sometimes things that are "obvious" to Conway and his co-arthor,
just aren't to the rest of us.
But in contrast to that he gives concrete examples
of new approaches that are beyond classical Coexter
and Cartan type approaches.
If you are looking for physics applications to quantum mechanics
for modern group theory,
you might want to try another book,
but if you want an idea of what a Moufang loop is or why
octonions are not associative, then you might like this book.
Some time in this century we may even get
a chance to understand Freudenthal's metasymplectic geometry?
This book for me is sort of a study guide to
what i should try to learn for the future?
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