The missing link...

New Scientist vol 176 issue 2368 - 09 November 2002, page 30

1, 2, 4, 8... What comes next? Nothing. If we're talking numbers, the obvious next term is 16. But if we're talking a particular kind of algebra, there is no next term. And it turns out that this is highly significant. The ultimate number - the humble 8 - lies at the heart of a mathematical system known as the octonions, and this system appears to be the key that will allow physicists to fit quantum theory and gravity together. Strange as it may seem, the number 8 may provide us with a "theory of everything".

The tale of the octonions begins in the mid-16th century. Until that time, mathematicians had thought that numbers were God-given, a done deal. No one could contemplate inventing a new number. But around 1550 the Italian algebraists Girolamo Cardano and Raphael Bombelli did just that, by writing down the square root of -1. It took about 400 years to sort out what the thing meant, but only 300 to convince mathematicians that it was too useful to be ignored.

By the 1800s, Cardano and Bombelli's concoction had crystallised into a new kind of number, i, whose square is -1. The square of a "real number" - the usual kind that we all know - is always positive. So whatever i may be, it's not a real number, and mathematicians call it an "imaginary" number to make this clear. A combination of real and imaginary numbers, like 4 + 5i, is said to be "complex".

We live in a curious Universe in which, as physicist Eugene Wigner memorably announced, mathematics is "unreasonably effective". Complex numbers may seem weird, but they turn out to be a marvellous tool for understanding physics. Problems of heat, light, sound, vibration, elasticity, gravity, magnetism, electricity and fluid flow all succumbed to this complex weaponry - but only for physics in two dimensions.

Our own Universe, however, has three dimensions of space - if not more. So, since the two-dimensional system of complex numbers was so effective for two-dimensional physics, might there be an analogous three-dimensional number system that could be used for physics in the real world?

The answer is a resounding no. The Irish mathematician William Rowan Hamilton spent years trying to find a three-dimensional number system - but with no success. Then, on 16 October 1843 he had a flash of insight: don't look in three dimensions, look in four. And it worked. Hamilton named his new numbers "quaternions".

Two months later, having heard about quaternions from Hamilton, John Graves - a British mathematician and an old college friend of Hamilton's - announced he had found an eight-dimensional number system. He called it the "octaves". But before Graves could publish, the British lawyer-mathematican Arthur Cayley made the same discovery, and published it as an addendum to an otherwise awful paper on elliptic functions. He called the system "octonions".

The discovery of the octonions was ever after credited to the wrong person (they are often known as Cayley numbers, even today), but it didn't really matter because nobody took any notice of them anyway. The octonions appeared to be nothing more than Victorian mathematical whimsy.

Graves was not to be put off though, and spent a long while convinced that his method of going from 4 to 8 could be repeated, leading to algebras with dimensions of 16, 32, 64 and so on for any power of 2. He called his 16-dimensional algebra the sedenions, but he couldn't find a way to make it - or any of the others - work, and began to doubt whether it could exist.

His doubt was well-founded. We now know that those four algebras, of dimensions 1, 2, 4 and 8, are the only ones that behave remotely like ordinary real numbers. The reason is that, with increasing numbers of dimensions, these systems obey fewer and fewer algebraic laws - the amount of algebraic structure keeps decreasing. Put rather too simply, by the time we reach Graves's sedenions, there's pretty much no algebraic structure left.

Real, complex, quaternion, octonion; 1, 2, 4, and 8 dimensions: even by mathematical standards this is an odd set of tools. These four number systems have several features in common, the most striking being that they are "division algebras". There are many number systems in which notions of addition, subtraction and multiplication hold good: when these notions are applied to the integers (... -2, -1, 0, 1, 2, 3, ...), for example, they transform two integers into another integer. But the same can't be said for division: divide some integers by others, for example, and the result is often not an integer. But in these four number systems, you can always divide and yet remain within the same system.

And that's not the only mathematical operation that sets them apart. Numbers in these systems are the only ones to have a "norm", effectively the number's distance from the origin (see Graphic). With the complex numbers, the norm of x + iy is x² + y². Because of the existence of a norm, and their divisibility, these number systems are known as "normed division algebras".

This is all very pretty, to mathematicians at least. But surely the only really important cases are the real and complex numbers. Well, not quite: the quaternions have shown up in some useful if esoteric researches - fields such as abstract algebra and topology.

But it's certainly true that the octonions remained in the shadows for a long time. In 1925 Wigner, working with the mathematician John von Neumann, tried to make the octonions the basis of quantum mechanics. But he failed, and the octonions slipped back into obscurity. Until now, that is.

Rather surprisingly, the octonions have revealed themselves as the most important system of all. That's because they are crucial to string theory, the best candidate for a physical theory of everything. After 150 years, physics is finally telling us the purpose of the octonions: they are essential to space and time.

String theory is an attempt to marry the large-scale geometry of Einstein's general relativity to the small-scale uncertainties inherent in quantum theory. Both these theories are brilliantly successful in their own domain. But they can't be fitted together: put into the same framework, they effectively contradict each other. So the search has been on for a unified theory that modifies them well enough to fit them together consistently but doesn't destroy their existing successes.

The current front runner in this search is known as string theory. Very roughly, the traditional idea that a fundamental particle is a featureless point is rejected, and particles are modelled instead as tiny loops of energy - the aforementioned strings. The loops can vibrate in ways that give them integer quantum numbers, such as spin, charge and charm.

But all this only works if the loop is a many-dimensional surface that protrudes beyond the familiar four-dimensional space-time, and one of the burning questions is just how many dimensions there are. At the moment, finding the answer seems to depend on finding the number of dimensions where the theories work most elegantly. And though physicists have not pinned it down precisely, they have noticed that something rather pleasing occurs when they work with 3, 4, 6 and 10 dimensions. Interestingly, each of these numbers is 2 greater than that of a normed division algebra: subtract 2 from 3, 4, 6 and 10, and you get 1, 2, 4 and 8. And that's no coincidence: these algebras are a vital part of the theory.

Consider, for example, the relationship between two mathematical objects: vectors and spinors. A vector is essentially a way to describe the size and orientation of something. Velocity, for example, is a vector because it describes a body's speed and the direction in which it is moving. The spinor is a more esoteric mathematical gadget invented by Paul Dirac to describe electron spin. It turns out that the relationship between vectors and spinors holds precisely (and only) in space-times of 3, 4, 6, and 10 dimensions. This happens because, in 3, 4, 6, and 10-dimensional string theory, every spinor can be represented using two numbers in the associated normed division algebra. This doesn't happen for any other number of dimensions, and it has lots of nice consequences for physics.

So we have four candidate string theories here: real, complex, quaternionic, and octonionic. The one that is thought to have the best chance of corresponding to reality is the 10-dimensional one, because it neatly avoids a mass of mathematical obstacles while allowing the physics to work properly. And, in this system, the relationships between the properties of matter are specified by the octonions: if this particular theory really does correspond to reality, then our Universe is built from pairs of octonions.

If 10 dimensions turns out to be not quite enough, however, it seems that the octonions will still be found to play a vital role in the theory of everything. The other very fashionable candidate string theory, "M-theory", involves 11-dimensional space-time. Although that means the vector-spinor relationship won't hold, something almost as good does. In M-theory, the extra dimensions don't need to be curled up tightly, so the restriction to six extra dimensions can be relaxed to allow a seventh, but again it doesn't work without the octonions.

In order to reduce the perceptible part of space-time from 11 dimensions to the familiar four (three space and one time), we have to hide seven of them. We do that by rolling them up so tightly that they can't be detected. And how do you do that? You make use of the octonions' symmetry.

The idea of symmetry - a property that allows you to move something in a certain way and leave it looking the same - has turned out to be central to physics, especially the quantum world. All our theories of fundamental particles, and their strange properties such as spin, charge and charm, which come in whole-number chunks, boil down to symmetries. And the use of octonionic symmetry in M-theory even gives a purpose to a mathematical peculiarity, discovered around the same time as the octonions, whose existence has always mystified mathematicians (see "The eightfold way"). So the efficacy of the octonions here is doubly pleasing.

While the octonions started out as mathematical curiosities, and were almost entirely ignored for 150 years, their time has come. They are no longer quaint Victoriana, but a hefty clue to a possible theory of everything. Daunting though their mathematics is, physicists are beginning to take up this new set of tools and work with it. A paper published this year by John Baez of the University of Califonia, Riverside, has prompted much Web-based discussion between string theorists. It all boils down to one extraordinary realisation: the humble 8 is no longer just a number. It's our key to the Universe.

The eightfold way

The best candidate we have for a theory of everything is string theory. This idea suggests that the fundamental particles are loops of energy that exist in many more dimensions than the four we experience. Multidimensional loops can in principle take on lots of shapes. The big task facing physicists is to find the right one.

In string theory, as in old-fashioned quantum theory, a key principle to pin down correct theories is symmetry. Physicists often settle on a particular shape as the correct description of something because it has the right kinds of symmetry: the Universe seems to like symmetric characteristics. And in string theory, it turns out, the symmetry of the octonions is crucial.

A symmetry of something is a way to transform it so that when you've finished it looks the same as it did at the start. If you take a featureless circular disc, for example, and rotate it through any angle, it looks just the same. This is an example of a continuous symmetry. But with a square, only 90° rotations will do that.

And it's not only objects such as circles and squares that can have symmetry operations applied to them. Hard as it may be to envisage, algebras have symmetries too. The collection of all symmetries of a given shape or algebra or whatever, is called its symmetry group. In the 19th century, the Norwegian mathematician Sophus Lie captured such symmetries using an algebraic structure that is known nowadays as a Lie group. An example is the set of rotations of an object in three-dimensional space. One symmetry in that set would be a rotation that turns this magazine through 180° until it's upside down.

Lie groups - the fundamental kinds of symmetry - can be divided into four main families. For instance, one family consists of the rotation group in the plane, the rotation group in space, the rotation group in four-dimensional space, and so on. Each dimension of space corresponds to one member of the family. Another family describes all the possible ways to distort space of n dimensions while keeping straight lines straight. Again, there is one such group for each dimension n. These are "linear mappings" and they do things like (in the case n = 2) stretch the plane in a north-south direction while leaving east-west unchanged; or tilt the north-south axis by 45° while leaving the east-west one unchanged. It's as if someone has leaned against the vertical axis and pushed it over. A similar thing happens with larger n.

But there are five curious symmetry groups that don't fit into any family. The very existence of these "exceptional" Lie groups, which rejoice in the names G2, F4, E6, E7 and E8, is a puzzle and a pain to mathematicians, who like everything to fit into some pattern or other. One exasperated mathematician even declared them a "brutal act of Providence".

For decades, no one could find any use for the exceptional groups, or any reason for their existence, and it was tempting simply to ignore them. However, it has now been realised that all five of them can be explained in terms of the octonions. In effect, they form a small family of their own, one with only five members. And it looks as though the octonions actually hold together what may prove to be the theory of everything.

For the 11 dimensions of M-theory to be reduced to the four that we experience, physicists need to carry out some particular mathematical transformations on space-time. The only way to do that is with the exceptional Lie group G2. And what else is this heroic group that rescues a vital theory of physics? It's the symmetry group of the octonion algebra.

The octonions have eight units: an ordinary number and seven others called e₁, e₂ and so on up to e₇. The square of any of these is -1. The multiplication rule for the units is determined by the "Fano plane", shown on the left. Suppose you want to multiply e₃ by e₇, say. Look at the diagram for the corresponding points, find the line that joins them, and you will see that there is a third point on the line, e₁. Following the arrows, you go from e₃ to e₇ to e₁, so e₃ × e₇ = e₁. If the ordering is the other way round, throw in a minus sign: e₇ × e₃ = -e₁. What's more, all the lines are considered to loop back to the start, so e₁ × e₃ = e₇, and e₃ × e₁ = -e₇. Do this for all the possible pairs of units and you know how to multiply octonions.