The Geometry of the Octonions

By way of motivation for what is to come, consider, first, something nice and familiar: for a fixed positive integer \(n\), let \(X\) denote the set \(\mathbb{R}^n\) of \(n\)-tuples of real numbers, denoted here by boldface letters. If \(\mathbf u\) is an element of \(X\), denote by \(|\mathbf{u}|\) the usual Euclidean norm of \(\mathbf u\). We now ask: for which positive integers \(n\) can we define a bilinear operation \(\mathbf{u}*\mathbf{v}\) on \(X\) such that \(|\mathbf{u}*\mathbf{v}| = |\mathbf{u}| |\mathbf{v}|\) for all \(\mathbf{u}\) and \(\mathbf{v}\)?

The answer to this question was given by Hurwitz in 1898: this is possible only when \(n= 1, 2, 4\) or \(8\). For \(n= 1\) and \(n=2\), the existence of such an operation is obvious, since in these two cases \(X\) can be identified with the set \(\mathbb{R}\) of real or \(\mathbb{C}\) of numbers, on each of which set there is a well-known norm-preserving operation of multiplication. Likewise, when \(n= 4\), we can identify \(X\) with the set \(\mathbb{H}\) of quaternions \( a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k}\) (with \(a, b, c, d \in\mathbb{R}\)) and use the well-known definition of multiplication (which is associative, but not commutative) of these elements. (The letter \(\mathbb{H}\) honors Hamilton, who discovered these numbers while walking, and was so excited about his discovery that he carved the defining conditions into the Brougham bridge in Dublin.) The quaternions are not as well-known as the complex numbers, of course, but they are mentioned in lots of undergraduate abstract algebra textbooks, and so most undergraduate mathematics majors probably at least see the definition of them at some point in their education.

But what about \(n = 8\)? It turns out that there is a number system, complete with norm-preserving multiplication, that works here too: namely, the set of octonions, denoted \(\mathbb{O}\). These are much less well-known than the quaternions, and not as well behaved: they are neither commutative nor associative. In fact, the physicist John Baez, in a 2001 paper, gave the following delightful and often-quoted assessment of them:

The real numbers are the dependable breadwinner of the family, the complete ordered field we all depend on. The complex numbers are a slightly flashier but still respectable younger brother… The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative. (Emphasis in the original.)

Because octonions are so unusual, my guess is that many majors graduate without ever even seeing them defined. This is a pity, because (as we’ll soon see) this algebraic system, despite its algebraic deficiencies, is very useful indeed, both in other branches of mathematics and in physics. The authors of the book under review, in fact, believe that the octonions “will ultimately be seen as the key to a unified field theory in physics.”

Discussions of the octonions are not particularly easy to find in the standard undergraduate textbook literature. On Quaternions and Octonions by Conway and Smith, reviewed in this column but unseen by me, puts octonions front and center, but not many books do. Out of curiosity, I spent fifteen minutes or so browsing through the indices of some of the books on my shelf, and found only two that defined them (Stillwell’s Mathematics and its History and (very succinctly) Jacobson’s Basic Algebra I), and only a few others that even mentioned them at all: Naïve Lie Theory, also by Stillwell; Matrix Groups for Undergraduates by Tapp; and A Guide to Groups, Rings and Fields by Gouvea. There is also an article discussing (but not defining) them: The Strangest Numbers in String Theory by Baez and Huerta, that appears in The Best Writing in Mathematics 2012, but of course this doesn’t constitute “textbook literature”. Doubtless there are other textbooks that mention them that either are not in my possession, or which I missed in my hurried and incomplete search, but the fact still remains that a student looking to find a definition in the pages of a mathematics textbook will have to search harder than he or she would, for example, to find out what a group is. Therefore this book, as that famously inaccurate sentence goes, “fills a much needed gap in the literature”.

The book is divided into three parts (“Number Systems”, “Symmetry Groups” and “Applications”, respectively) each of which contains several chapters. The first (and by far the shortest) part reviews the history, algebra and geometry of the complex numbers and the quaternions, and then introduces the octonions in a way that emphasizes the relationships between them and these two algebraic systems. Just as a typical quaternion \( a + b\mathbf{i}+c\mathbf{j}+d\mathbf{k}\) can be rewritten as \((a+b\mathbf{i})+(c+d\mathbf{i})\mathbf{j}\) and can therefore be thought of as a “complex number with complex coefficients”, an octonion can be thought of as a “complex number with quaternion coefficients”: namely, we take a new “imaginary unit” \(\mathbf{I}\) with \(\mathbf{I}^2=-1\), independent of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\), and consider expressions of the form \(q_1 +q_2\mathbf{I}\), where the \(q_i\) are quaternions. Rather than invent new names for \(\mathbf{iI}, \mathbf{jI},\mathbf{kI}\), the author, “[s]ince we are running out of letters”, simply retains this notation for the products. An octonion, therefore, is simply a real linear combination of \(1\) and the seven independent imaginary units \(\mathbf{i},\mathbf{j}, \mathbf{k}, \mathbf{I}, \mathbf{iI}, \mathbf{jI}, \mathbf{kI}\). Just as, in the ring of quaternions, multiplication of the imaginary units can be remembered by the simple mnemonic device of writing them (in that order) around a circle, we have for the octonions another such device for remembering the rules for taking the product of these seven imaginary units; instead of a circle, however, we use the familiar circle/triangle diagram for the Fano 7-point plane in projective geometry. So already we begin to see a hint of interesting things lurking in the background.

Since a complex number can be identified with an ordered pair of real numbers, a quaternion can be identified with an ordered pair of complex numbers, and an octonion can be identified with an ordered pair of quaternions, a natural question to ask is: why not repeat this process with ordered pairs of octonions, thus getting a 16-dimensional number system? This process, called the Cayley-Dickinson process, is also discussed in Part I of this book, and it leads to a new number system called the sedenions, but this is not a division algebra; in fact, there are zero divisors. This is all related to another theorem of Hurwitz, which generalizes the result with which I opened this review: if \(A\) is a composition algebra (i.e., an algebra (not necessarily associative, but having an identity) together with a non-degenerate quadratic form \(Q\) such that \(Q(xy) = Q(x)Q(y)\) for all \(x,y\in A\), then, if \(Q\) is positive or negative definite, \(A\) must be one of \(\mathbb{R},\mathbb{C}, \mathbb{H}, \mathbb{O}\). Dray and Manogue round out Part I of the text by mentioning this result and discussing three new algebras, the split complex numbers, split quaternions and split octonions; these are composition algebras where the quadratic form is not positive definite.

This first part of this book is a splendid introduction to the octonions — it requires minimal prerequisites (some familiarity with the complex numbers is basically all that is required), and is written in a clear and chatty style that will appeal to readers. In a nice homey touch, the first-named author even provides a photograph of himself at the aforementioned Brougham bridge in Dublin.

Part II of the book is devoted to symmetry groups and their connections to the systems \(\mathbb{R},\mathbb{C}, \mathbb{H}, \mathbb{O}\). The authors’ approach throughout much of this part of the book is to focus on specific “small” symmetry groups and try to develop some insight, generally geometric, into what these groups “look like”. The chapters in part II are pitched at a somewhat higher level than the ones in part I, but even here the prerequisites are kept to a minimum: a good background in linear and abstract algebra (groups, anyway) should get the reader fairly far. Some background in physics would also be a plus, since references to that subject are sprinkled throughout the chapters: Minkowski spacetime, Lorentz transformations, Pauli spin matrices, boosts, etc.

The authors start with orthogonal groups and consider as specific examples \(\mathrm{SO}(n)\) for \(n = 2, 3, 4\), and also two groups related to Minkowski space in relativity theory, namely \(\mathrm{SO}(3,1)\) and \(\mathrm{SO}(4,2)\). They then proceed to unitary groups and look specifically at \(\mathrm{U}(1), \mathrm{SU}(2), \mathrm{SU}(3)\), and another relativity-related group, \(\mathrm{SU}(2,2)\). A couple of real symplectic groups (preserving an anti-symmetric product rather than the usual Euclidean symmetric one) are discussed next.

This sort of thing has interested me ever since I first learned, from Michael Artin’s Algebra, that \(\mathrm{SU}(2)\), the special unitary group of 2 x 2 complex matrices, is a sphere in four-dimensional real space, and is also a double cover of \(\mathrm{SO}(3)\); it is therefore nice to have a wide assortment of specific matrix groups discussed in one place.

Things don’t end with these chapters, however. Quaternions and octonions enter the picture in chapter 9. It turns out, for example, that there are analogues of the special unitary groups and symplectic groups over \(\mathbb H\) and \(\mathbb O\); some special cases are worked out, via some fairly intricate calculations, and shown to be double coverings of previously seen matrix groups.

All these matrix groups naturally make one think about Lie groups and Lie algebras, and these are the topics of the next two chapters. Chapter 10 provides a quick but quite informative overview, without proofs of course (that would take a book of its own) of Lie groups and Lie algebras and the classification theory of simple Lie algebras. Purists may complain about the fairly informal, manifold-free, definition of a Lie group as a “group whose elements depend smoothly on some number of parameters”, but the authors are not writing a treatise on Lie theory, so this didn’t really bother me.

As all good Lie theorists know, there are four infinite families of simple complex Lie algebras, and, with only five specific exceptions, any simple complex Lie algebra is in one of these. The five exceptions are called “exceptional”, and it turns out that all five of these exceptional Lie algebras are related to the octonions, a subject that is (for four of them, anyway; one of them is beyond the scope of the text) addressed in the next chapter, the final one in Part II of the book.

The final (and most sophisticated, I thought) part of the book consists of four chapters, all independent of one another, each one addressing an application of the octonions to mathematics or physics. So, for example, we have a chapter on eigenvalue problems with octonions, and another which touches on connections to topology (Hopf mappings), geometry (the octonionic projective line and projective plane) and algebra (quaternionic and octonionic integers).

The chapter on physics applications will (if my own experience is typical) prove very heavy going for people whose background in this subject is lacking; I knew I was stepping into the deep end of the pool when I read the chapter’s first sentence (“We begin by considering angular momentum in quantum mechanics”), and the subsequent sentence “Here’s a crash course in quantum mechanics” three pages later didn’t exactly function as a life jacket. Undoubtedly, however, there will be lots of better-prepared people who will get far more out of this chapter than I am capable of doing.

Given the subject matter of this book, it is not likely to get used as an undergraduate text, and in fact the book really doesn’t have the feel of such a text; there are no exercises, and although there are some separately marked theorems and proofs, mostly the discussion is discursive and computational. However, anybody interested in the topics covered here should find this book to be a valuable reference.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.