by Ian Stewart
The tale of the octonions lives in the heady realms of abstract algebra, and it is the topic of a beautiful mathematical survey published in 2001 by the American mathematician John Baez. I have drawn heavily on Baez’s insights here. I’ll do my best to convey the bizarre yet elegant wonders that inhabit this curious interface between mathematics and physics. As with the ghost of Hamlet’s father, a disembodied voice beneath the stage, much of the mathematical action must happen out of sight of the audience. Bear with me, and don’t worry too much about the odd piece of unexplained jargon. Sometimes we just need a convenient word to keep track of the main players.
A few reminders may help to set the scene. The step-by-step expansion of the number system has woven in and out of our tale of the quest for symmetry. The first step was the discovery (or invention) in the mid-sixteenth century of complex numbers, in which –1 has a square root. Until that time, mathematicians had thought that numbers were God-given, unique, and done. No one could contemplate inventing a new number. But around 1550, Cardano and Bombelli did just that, by writing down the square root of a negative number. 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 baroque concoction had crystallized into a new kind of number, with a new symbol: i. Complex numbers may seem weird, but they turn out to be a marvelous tool for understanding mathematical physics. Problems of heat, light, sound, vibration, elasticity, gravity, magnetism, electricity, and fluid flow all succumb to the complex weaponry—but only for physics in two dimensions.
Our own universe, however, has three dimensions of space—or so we thought until recently. Since the two-dimensional system of complex numbers is so effective for two-dimensional physics, might there be an analogous three-dimensional number system that can be used for genuine physics? Hamilton spent years trying to find one, with absolutely no success. Then, on 16 October 1843, he had a flash of insight: don’t look in three dimensions, look in four, and carved his equations for quaternions into the stonework of Brougham Bridge.
Hamilton had an old friend from college, John Graves, who was an algebra buff. It was probably Graves who got Hamilton excited about extensions of the number system in the first place. Hamilton wrote his buddy a long letter about quaternions the day after he had vandalized the bridge. Graves was initially perplexed, and wondered how legitimate it was to invent multiplication rules off the top of one’s head. “I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties,” he wrote back. But he also saw the potential of the new idea and wondered how far it might be pushed: “If with your alchemy you can make three pounds of gold, why stop there?”
It was a good question, and Graves set about answering it. Within two months he wrote back to say that he had found an eight-dimensional number system. He called it the “octaves.” Associated with it was a remarkable formula about sums of eight squares, to which I will shortly return. He tried to define a 16-dimensional number system, but met what he called an “unexpected hitch.” Hamilton said that he would help to bring his friend’s discovery to public attention, but he was too busy exploring quaternions to do so. Then he noticed a potential embarrassment: multiplication of octaves does not obey the associative law. That is, the two ways to form products of three octaves, (ab)c and a(bc), are usually different. After much soul-searching, Hamilton had been willing to dispense with the commutative law, but throwing away the associative law might be going too far.
Now Graves had some serious bad luck. Before he could publish, Cayley independently made the same discovery, and in 1845 he published it as an addendum to an otherwise awful paper on elliptic functions—so riddled with errors that it was removed from his collected works. Cayley called his system “octonions.”
Graves was unhappy at being beaten to publication, and it so happened that a paper of his own was shortly due to be published in the journal where Cayley had announced his discovery. So Graves added a note to his paper, pointing out that he had come across the same idea two years before, and Hamilton backed him up by publishing a brief note confirming that his friend should be granted priority. Despite the record being set straight, the octonions quickly acquired the name “Cayley numbers,” which is still widely used. Many mathematicians now use Cayley’s terminology, calling the system “octonions,” but give credit to Graves. It’s a better name than “octaves” anyway, because it resembles “quaternions.”
The algebra of octonions can be described in terms of a remarkable diagram known as the Fano plane. This is a finite geometry composed of seven points joined in threes by seven lines, and it looks like this:
The Fano plane, a geometry with seven points and seven lines.
One line has to be bent into a circle to draw it in the plane, but that doesn’t matter. In this geometry, any two points are joined by a line, and any two lines meet at a point. There are no parallel lines. The Fano plane was invented for a totally different purpose, but it turned out to encapsulate the rules for multiplying octonions.
The octonions have eight units: the ordinary number 1, and seven others called e1, e2, e3, e4, e5, e6, and e7. The square of any of these is –1. The diagram determines the multiplication rule for the units. Suppose you want to multiply e3 by e7. Look in the diagram for points 3 and 7 and find the line that joins them. On it, there is a third point, which in this case is 1. Following the arrows, you go from 3 to 7 to 1, so e3e7 = e1. If the ordering is back to front, throw in a minus sign: e7e3 = –e1. Do this for all possible pairs of units, and you know how to do arithmetic with octonions. (Addition and subtraction are always easy, and division is determined by multiplication.)
Graves and Cayley didn’t know about this connection with finite geometry, so they had to write out a multiplication table for octonions. The Fano plane pattern was discovered later.
For many years, the octonions were merely a minor curiosity. Unlike quaternions, they had no geometrical interpretation and no application in science. Even within pure mathematics, nothing seemed to follow from them; no wonder they fell into obscurity. But all this would change with the realization that the octonions are the source of the most bizarre algebraic structures known to mathematics. They explain where Killing’s five exceptional Lie groups—G2, F4, E6, E7, and E8—really come from. And the group E8, the largest of the exceptional Lie groups, shows up twice in the symmetry group that forms the basis of 10-dimensional string theory, which has unusually pleasant properties and is thought by many physicists to be the best candidate yet for a Theory of Everything.
If we agree with Dirac that the universe is rooted in mathematics, then we could say that a plausible Theory of Everything exists because E8 exists, and E8 exists because the octonions exist. Which opens up an intriguing philosophical possibility: the underlying structure of our universe, which we know to be very special, is singled out by its relationship to a unique mathematical object: the octonions.
Beauty is truth, truth beauty. The Pythagoreans and Platonists would have loved this evidence of the pivotal role of mathematical patterns in the structure of our world. The octonions have a haunting, surreal mathematical beauty, which Dirac would have seized upon as a reason why 10-dimensional string theory has to be true. Or, if unhappily proved false, is nevertheless more interesting than whatever is true. But we have learned that beautiful theories need not be true, and until the verdict on super-strings is in, this possibility must remain pure conjecture.
Whatever its importance in physics, the circle of ideas surrounding the octonions is pure gold for mathematics.
The connection between the octonions and the exceptional Lie groups is just one of many strange relationships between various generalizations of the quaternions and the frontiers of today’s physics. I want to explore some of these connections in enough depth for you to appreciate how remarkable they are. And I’m go
ing to start with some of the oldest exceptional structures in mathematics, formulas about sums of squares.
One such formula derives naturally from the complex numbers. Every complex number has a “norm,” the square of its distance from the origin. The Pythagorean theorem implies that the norm of x + iy is x2 + y2. The rules for multiplying complex numbers, as laid down by Wessel, Argand, Gauss, and Hamilton, tell us that the norm has a very pretty property. If you multiply two complex numbers together, then the norms multiply too. In symbols, (x2 + y2)(u2 + v2) = (xv + yu)2 + (xu – yv)2. A sum of two squares times a sum of two squares is always a sum of two squares. This fact was known to the Indian mathematician Brahmagupta around 650, and to Fibonacci in 1200.
The early number theorists were fascinated by sums of two squares, because they distinguished two different types of prime number. It is easy to prove that if an odd number is the sum of two squares, then it must be of the form 4k + 1 for some integer k. The remaining odd numbers, which are of the form 4k + 3, cannot be represented as the sum of two squares. However, it is not true that every number of the form 4k + 1 is a sum of two squares, even if we allow one of the squares to be zero. The first exception is the number 21.
Fermat made a very beautiful discovery: these exceptions can never be prime. He proved that on the contrary, every prime number of the form 4k + 1 is a sum of two squares. By applying the above formula for multiplying sums of two squares together, it then follows that an odd number is a sum of two squares if and only if every prime factor of the form 4k + 3 occurs to an even power. For instance, 45 = 32 + 62 is a sum of two squares. Its prime factorization is 3 × 3 × 5, and the prime factor 3, which has the form 4k + 3 with k = 0, occurs to the power two—an even number. The other factor, 5, occurs to an odd power, but that’s a prime of the form 4 k + 1 (with k = 1), so it doesn’t cause any trouble.
On the other hand, the exception 21 is equal to 3 × 7, which are both primes of the form 4 k + 3, and here each occurs to the power 1, which is odd—so that’s why 21 doesn’t work. Infinitely many other numbers don’t work for the same reason.
Later, Lagrange used similar methods to prove that every positive whole number is a sum of four squares (zero permitted). His proof used a clever formula discovered by Euler in 1750. It is similar to the one above, but for sums of four squares. A sum of four squares times a sum of four squares is itself a sum of four squares. There can be no such formula for sums of three squares, because there exist pairs of numbers that are both sums of three squares but whose product is not. However, in 1818 Degen found a product formula for sums of eight squares. It is the same formula that Graves discovered using octonions. Poor Graves—his discovery of octonions, which was original, was credited to someone else; his other discovery, the eight-squares formula, turned out not to be original.
There is also a trivial product formula for sums of one square—that is, squares. It is x2 y2 = (xy)2. This formula does for real numbers what the two-squares formula does for complex numbers: it proves that the norm is “multiplicative”—the norm of a product is the product of the norms. Again, the norm is the square of the distance from the origin. The negative of any number has the same norm as its positive.
What of the four-squares formula? It does the same thing for quaternions. The four-dimensional analogue of the Pythagorean theorem (yes, there is such a thing) tells us that a general quaternion x + iy + jz + kw has norm x2 + y2 + z2 + w2, a sum of four squares. The quaternionic norm is also multiplicative, and this explains Lagrange’s four-squares formula.
You will probably be ahead of me by now. Degen’s eight-squares formula has a similar interpretation for octonions. The octonion norm is multiplicative.
Something very curious is going on here. We have four types of evermore-elaborate number system: the reals, complexes, quaternions, and octonions. These have dimensions 1, 2, 4, and 8. We have formulas that say that a sum of squares times a sum of squares is a sum of squares: these apply to 1, 2, 4, or 8 squares. The formulas are closely related to the number systems. More intriguing still is the pattern of the numbers.
1, 2, 4, 8—what comes next?
If the pattern continued, we would confidently expect to find an interesting 16-dimensional number system. Indeed, such a system can be constructed in a natural way, called the Cayley–Dickson process. If you apply that process to the reals, you get the complexes. Apply it to the complexes, you get the quaternions. Apply it to the quaternions, you get the octonions. And if you plow ahead and apply it to the octonions, you get the sedenions, a 16-dimensional number system, followed by algebras of dimension 32, 64, and so on, doubling at every step.
So there is a 16-squares formula, then?
No. The sedenion norm is not multiplicative. Product formulas for sums of squares exist only when the number of squares involved is 1, 2, 4, or 8. The law of small numbers strikes again: the apparent pattern of powers of two grinds to a halt.
Why? Basically, the Cayley–Dickson process slowly destroys laws of algebra. Every time you apply it, the resulting system is not quite as well behaved as the previous one. Step by step, law by law, the elegant real number system descends into anarchy. Let me explain in more detail.
The four number systems have other features in common aside from their norms. Their most striking feature, which qualifies them as generalizations of the real numbers, is that they are “division algebras.” There are many algebraic systems in which notions of addition, subtraction, and multiplication are valid. But in these four systems, you can also divide. The existence of a multiplicative norm makes them “normed division algebras.” For a while, Graves thought his method of going from 4 to 8 could be repeated, leading to normed division algebras with 16, 32, 64 dimensions, any power of two. But he hit a snag with the sedenions and began to doubt whether a 16-dimensional normed division algebra could exist. He was right: we now know that there exist only four normed division algebras, of dimensions 1, 2, 4, and 8. And there is no 16-squares formula like Graves’s eight-squares formula or Euler’s four-squares formula.
Why is this? At every step along the chain of powers of two, the new number systems lose a certain amount of structure. The complex numbers are not ordered along a line. The quaternions fail to obey the algebraic rule ab = ba, the “commutative law.” The octonions fail to obey the associative law (ab)c = a(bc), though they do obey the “alternative law” (ab)a = a(ba). The sedenions fail to form a division algebra and have no multiplicative norm either.
This is far more fundamental than just a failure of the Cayley–Dickson process. In 1898, Hurwitz proved that the only normed division algebras are our four old friends. In 1930, Max Zorn proved that these same four algebras are the only alternative division algebras. They truly are exceptional.
This is the sort of thing pure mathematicians, with their Platonist instincts, love. But the only really important cases for the rest of humanity seemed to be the real and complex numbers, which were of massive practical importance. The quaternions did show up in some useful if esoteric applications, but the octonions shunned the limelight of applied science. They seemed to be a pure-mathematical dead end, the sort of pretentious intellectual nonsense you would expect from people with their heads in the clouds.
The history of mathematics shows repeatedly that it is dangerous to dismiss some clever or beautiful idea merely because it has no obvious utility. Unfortunately, this does not stop people from dismissing such ideas, often because they are beautiful or clever. The more “practical” people consider themselves to be, the more they tend to heap scorn on mathematical concepts that arise from abstract questions, invented “for their own sake” instead of addressing some real-world issue. The prettier the concept is, the greater the scorn, as if prettiness itself were a reason to be ashamed.
Such declarations of uselessness are hostages to fortune. It takes only one new application, one new scientific advance, and the despised concept may suddenly plonk itself down on center stage—no longer useless, but es
sential.
The examples are endless. Cayley himself said his matrices were completely useless, but today no branch of science could function without them. Cardano declared complex numbers to be “as subtle as they are useless,” but no engineer or physicist could function in a world that lacked complex numbers. Godfrey Harold Hardy, England’s leading mathematician in the 1930s, was immensely pleased that number theory had no practical application, and in particular that it could not be used in warfare. Today number theory is used to encrypt messages into code, a technique that is vital to secure Internet commerce, and even more vital to the military.
So it is turning out with the octonions. They may yet become a compulsory topic in mathematics courses, and even more so in physics. It is now emerging that the octonions are of central importance in the theory of Lie groups—especially those of interest in physics, especially the five exceptional Lie groups G2, F4, E6, E7, and E8, with their weird dimensions 14, 52, 78, 133, and 248. Their very existence is a puzzle. One exasperated mathematician declared them a brutal act of Providence.
Lovers of nature enjoy revisiting well-known beauty spots and finding a new vantage point . . . halfway down a waterfall, just along a ledge leading off to the side of the well-worn footpath, on a promontory overlooking a blue ocean vista. In the same way, mathematicians like to revisit old topics and look at them from new points of view. As our perspective on mathematics changes, we can often reinterpret old concepts in new, insightful ways. This is not just a matter of mathematical tourism, gazing open-mouthed at the ineffable from a different angle. It provides new, powerful ways to tackle old and new problems. Nowhere has this tendency been more apparent, or more informative, than in the theory of Lie groups.
Recall that Killing organized almost all simple Lie groups into four infinite families, two of which are really parts of one larger family, the special orthogonal groups SO(n) in even and odd dimensions. The other two are the special unitary groups SU(n) and the symplectic groups Sp(2n).