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On Quaternions and Octonions

A K Peters
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Given Conway's idiosyncratic taste in mathematical topics (witness his Surreal Numbers for example), as well as his famous style and wit, one expects this book to be, well, different. And so it is. The reader is taken on a grand tour of some pretty deep algebraic arcana, with a good deal of history thrown in along the way, as well as plenty of motivation and examples. And it's all done in an immensely entertaining way. By and large, the material in these pages is geared to evoke pleasure and surprise by means of a light and readable writing style, a profusion of examples, and a huge number of (very cool) calculations. The authors entertain and teach us while obviously indulging themselves in sheer fun (it already starts with their dedication on page v : this is not going to be another dry tome). But in the second part of the book Conway and Smith go on to introduce and describe a subtle but pervasive phenomenon, "triality symmetry," which attests to their ideological goal of placing these algebras in a proper, deeper context. So the book is both fun and deep.

Conway and Smith are interested in the full "geometry and arithmetic of the quaternion and octonion algebras" (p. xi). While the objects taking center stage are unapologetically algebraic (and why, pray tell, should any one need to apologize for algebra?!), there is a great deal of geometry in it, as well as connections with number theory. But, to be sure, the book's algebraic sweep is truly remarkable as is evidenced by the fact that the index lists 67 adjectives to "group." (See also our samplings, below.)

So what exactly are octonions? Most of us run across quaternions from time to time of course, but octonions? They constitute an 8-dimensional composition algebra (page 5) which is in fact the largest player in Hurwitz' classification of such algebras which (zowie!) we get only in dimensions 1, 2, 4, and 8; yes, of course, the dimension 4 case is the quaternions. But only on p. 65, underscoring the authors' belief in motivation, do we find the explicit definition of the octonions as an algebra over the real numbers. We see there that they are generated by seven non-trivial units (and 1, of course, making for an 8-dimensional linear basis) satisfying a set of relations of the same flavor as but more complex than that for Hamilton's quaternions. By the way, this definition on page 65 is the only other thing on a page announcing the start of Part III, "The Octonions and Their Applications to 7- and 8-Dimensional Geometry." This part of the book contains the critical discussion of triality symmetry (page 90 ff.) as a unifying notion abstracted from the behavior of the main players.

As mentioned already above, there is a lot of history in the book. It's not just Hurwitz and Hamilton either: Cartan, Cayley, Coxeter, Dickson, Freudenthal, Tits, and (even) John von Neumann (to name a few) make appearances throughout the book. And the explicit examples, calculations, and excursions are not to be missed. But maybe the most distinctive feature along these lines is the profusion of group theory; see e.g. page 35: "The Projective Group Tells Us All." Here we find (again, to name but a few) the octahedral rotation group (as a chiral group), the diplo-octahedral group (as a diploid group), and the tetra-octahedral group. Turn the page to find pyritohedral group, the holo-octahedral group, and the n-gonal holo-antiprismatic group. And this is really only the beginning

We close our review with a sample from page 49: "The subject of chirality is more subtle than it seems... [a] distinction is made by some chemists — that between 'shoe chirality' and 'potato chirality.' We expect neither shoes nor potatoes to be achiral, but yet we discriminate left shoes from right shoes but not 'left potatoes' from 'right potatoes!' Why is this?" (The answer is easy and elegant.)

Michael Berg ( is professor of mathematics at Loyola Marymount University in Los Angeles, CA. His research interests are algebraic number theory and non-archimedian Fourier analysis.

Date Received: 
Friday, June 6, 2003
Include In BLL Rating: 
John Horton Conway and Derek A. Smith
Publication Date: 
Michael Berg
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Modify Date: 
Thursday, January 3, 2008