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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Editorial Reviews
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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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Most Helpful Customer Reviews
62 of 70 people found the following review helpful
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
By Sally
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
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?
13 of 25 people found the following review helpful
Format:Hardcover
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.
13 of 30 people found the following review helpful
By reader
Format:Hardcover|Amazon Verified Purchase
If you want a good laugh take a look at chapter six
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.
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.
Citations (learn more)
This book cites 7
books:
See all 7 books this book cites
1
book
cites this book:
- Rotations, Quaternions, and Double Groups (Dover Books on Mathematics) by Simon L. Altmann on page 10, and Back Matter
- Studies in Modern Algebra (Mathematical Association of America, Studies in Mathematics, Vol 2) by A. A. Albert in Back Matter
- Topological Geometry by Ian R. Porteous in Back Matter
- Sir William Rowan Hamilton by Thomas L. Hankins in Back Matter
- Life of Sir William Rowzo Hamilton (History , Philosophy & Sociology of Science) by R. Graves in Back Matter
See all 7 books this book cites
- Mathematics of Physics and Engineering by Edward K. Blum on page 433, and Back Matter
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