On Quaternions and Octonions

“On Quaternions and Octonions”, J.H. Conway & D.A. Smith, 2003: 1568811349 (Blackwell’s, amazon.com, amazon.co.uk).

This is a book about mathematical beauty. Not the facile surface beauty of Lissajous figures or fractals, but a beauty that is visible only to the inner eye. We are in the world of concepts that are almost too simple to understand, whose visible manifestations (polyhedra, wallpaper patterns) are only consequences of the underlying reality and not pictures of the reality itself. It is a lot like theology: for God cannot be seen, is too simple for human comprehension, and his visible manifestations are nothing more than shadows and signposts to a reality that they cannot circumscribe or define.

Most people know about complex numbers: they can be written as x+iy, where x and y are real numbers and i has no meaning but obeys the rule that i²=-1, and you can do almost everything with them that you can with real numbers. In fact, a lot of mathematics is much easier with complex numbers than it is without them.

A bored intelligent schoolchild can imagine that there will be something further, with three numbers rather than two. It took Sir William Hamilton eight years to discover that there is no such thing; but that if you use four numbers, there is: x+iy+jz+kw, with i²=-1, j²=-1, k²=-1, and ijk=-1, then there is. Quaternions turn out to be a good way of representing transformations in 3-dimensional and 4-dimensional space. The writers of 3-D computer games use them when computing the effect of multiple rotations.

You can sort of guess what octonions might be.

It begins to seem that we’re getting more and more Victorian. Victorian mathematics is a bit like the Albert Memorial: it delights in intricacy and abundance of detail. A Victorian likes nothing better than manipulating equations with many variables and dozens of terms, rearranging them for page after page: overtime for typographers.

Twentieth and twenty-first century mathematics is spare and bony. So bony that (like the comic-book characters who are so strong that “even their muscles have muscles”) even its bones have bones. No sooner has one mathematician abstracted everything numeric from numbers – and left a beautiful skeleton behind, that fits into all manner of hitherto unrelated bodies and gives them shape and motion – than the next mathematician abstracts most of the content from his predecessor’s abstraction until its symbols become like those shadows that fill your sight when you have stared at a neon sign too long: shadows that float in the vision and dodge away when you try to look at them. If the Victorian ideal was decoration at every scale, the late-twentieth-century ideal is a theorem that can be expressed in only three symbols and proved in six lines (which it will take you a year to understand fully). It can be confidently predicted that the twenty-first-century ideal will be a theorem that is expressed by a blank piece of paper and cannot be understood even after a lifetime of study.

“On Quaternions and Octonions” is a book about bones. It categorizes real numbers (R), complex numbers (C), quaternions (Q) and octonions (O) as “algebras” (a term that has only a passing relation to what one means by “algebra” at school). It looks at them geometrically – showing, for instance, that whether you can have the equivalent of unique factorisation of integers in these algebras depends on what you define “integers” to be and (with stunning simplicity) on how much space there is between them; thus abolishing pages and pages of number-theory texts with a simple picture.

A rotation in 3 dimensions can be represented by a single quaternion; rotations in 4 dimensions can be represented by a pair of them. And so Conway & Smith look at 3-dimensional rotations geometrically. It shows (this is classic stuff but I have never seen it so well presented) that the regular polyhedra (cube, tetrahedron, icosahedron) can be characterised by the way you can rotate them and get back to what you started. For instance, you can rotate a cube a quarter-turn round the centre of a face and you’ll get back to the same cube. You can rotate it a half-turn round the mid-point of one of its sides: same result. You can rotate it a third of a revolution round one of its diagonals : same result (I always need to pick up a real cube to see this one). Those numbers, 423, define the symmetry of the cube (and its dual, the octahedron).

But then, by relating these rotations to reflections (think of how the two angled mirrors of a kaleidoscope generate a whole group of rotations), the question “what regular polyhedra are there?” turns into a question about what spherical triangles you can have whose angles obey certain rules. So by enumerating a few possible combinations of numbers you can find all the regular polyhedra that exist and prove that you can’t have missed any out.

In the spirit of “the only ones possible”, Conway and Smith also give Hurwitz’s Theorem, which shows that each of R, C, Q and O comes from “doubling” the one before it in the series, that the series has to stop after O, and that no other algebras of this kind can exist. They go on to explore 4-dimensional geometry, the 7- and 8-dimensional geometry of O, and to prove some new results on factorisation in O.

“On Quaternions and Octonions” is not a textbook. It assumes that you already know what groups, rings and fields are. When an idea is presented, it is presented once: not two or three times with additional exercises to ram the point home. Some peripheral concepts, such as the orbifold notation, are used without elaborate proofs: you can spend some enjoyable time working out for yourself how they work and why they are the way they are.

The result is not only a bony book but a chewy one: you can come back to the same page day after day and understand a little more of it each time. Anyone who has at least a first-year undergraduate grounding in algebra will find this book rewarding and enjoyable: something to come back to at intervals and get a little more out of each time.