This book investigates the geometry of the quaternion and octonion algebras. Following a comprehensive historical introduction, the special properties of 3- and 4-dimensional Euclidean spaces are illuminated 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 arithmetics of the quaternions and octonions are also described, and the book concludes with a new theory of octonion factorization. Topics covered include: - history - the geometry of complex numbers - quaternions and 3-dimensional groups - quaternions and 4-dimensional groups - the Hurwitz integral quaternions - the composition algebras - Moufang loops - octonions and 8-dimensional geometry - integral octonions - the octonion projective plane
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" ""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"
""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"
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First Sentence
We suppose you understand the real numbers! Read the first page
Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
We suppose you understand the real numbers! Read the first page
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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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Most Helpful Customer Reviews
1 of 1 people found the following review helpful
By Albear
Format:Hardcover
I am surprised to find myself the first reviewer of this excellent book. I am not a mathematician, although I did many maths courses as part of my degree many many years ago. The authors, Conway and Smith, lead naturally from Real and Complex numbers, through Quarternions and Octonians, addressing interesting variations on the way. I had many 'Oh now I see!' moments from their weaving together of ideas from geometry, arithmetic and symmetry. They classify the groups related to the symmetries of the objects using notions and notations from The Symmetry of Things (Conway, Burgiel and Goodman-Strauss), and relating them to Coxeter's classification (Regualr Polytopes). As they bring material together they address previously unsolved problems making the classification nearer to completion than previous works (as I understand it). I would think this book will become a valuable reference work.
This is not a book for the casual reader. Although the explanation is excellent, there is a lot of notation, and I must admit that I am still struggling to get a grip on the final chapter on 'Reading O Mod 2'.
This is not a book for the casual reader. Although the explanation is excellent, there is a lot of notation, and I must admit that I am still struggling to get a grip on the final chapter on 'Reading O Mod 2'.
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
3.4 out of 5 stars
5 reviews
62 of 69 people found the following review helpful
3.0 out of 5 stars
Good, but something is better
25 July 2003
By Jim Curry
- Published on Amazon.com
Format:Hardcover
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.
16 of 27 people found the following review helpful
5.0 out of 5 stars
A model of exposition
25 Mar 2003
By Sally
- Published on Amazon.com
Format:Hardcover
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.
1 of 3 people found the following review helpful
3.0 out of 5 stars
People either love Conway or hate him?
27 Sep 2009
By R. Bagula
- Published on Amazon.com
Format:Hardcover
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?
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?
›
Go to Amazon.com to see all 5 reviews
3.4 out of 5 stars
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