Why Beauty is Truth

by Ian Stewart

  We now know that these families are all variations on the same theme. They consist of all n × n matrices satisfying a particular algebraic condition—they are “skew-Hermitian.” The only difference is that you have to use matrices of real numbers to get the orthogonal Lie algebras, matrices of complex numbers to get the unitary Lie algebras, and matrices of quaternions to get the symplectic Lie algebras. These algebras come in infinite families because matrices come in infinitely many sizes. It is wonderful to see that the Lie algebras corresponding to the natural transformations of Hamilton’s version of mechanics, his first great discovery, have a natural description in terms of quaternions, his last.

  It does make you wonder what happens when you use octonions as entries in the matrix. Unfortunately, because of the lack of associativity, you don’t get a new infinite family of simple Lie algebras. Actually, that should be “fortunately,” since we know that no such family exists. But when you play the right games with octonions, and have the law of small numbers on your side, you can get Lie algebras with a vengeance.

  The first hint that this might be the case came in 1914, when Élie Cartan answered an obvious question and got a surprising answer. A guiding principle in mathematics and physics is that if you have some interesting object, the first thing to ask is what its symmetry group looks like. The symmetry group of the real number system is trivial, consisting of only the identity transformation “do nothing.” The symmetry group of the complex number system contains the identity and one mirror symmetry, which transforms i into –i. The symmetry group of the quaternions is SU(2), which is very nearly the rotation group SO(3) in real three-dimensional space.

  Cartan asked, What is the symmetry group of the octonions?

  If you are a Cartan, you can also answer this question. The symmetry group of the octonions is the smallest exceptional simple Lie group, the one known as G2. The 8-dimensional system of octonions has a 14-dimensional symmetry group. The exceptional normed division algebra is directly related to the first of the exceptional Lie groups.

  To proceed further, we need to take on board one more idea, which goes back to the Renaissance—but to the artists, not the mathematicians.

  In those days, mathematics and art were rather close; not just in architecture but in painting. The Renaissance painters discovered how to apply geometry to perspective. They found geometric rules for drawing images on paper that really looked like three-dimensional objects and scenes. In so doing, they invented a new and extremely beautiful kind of geometry.

  The work of earlier artists does not look quite realistic to our eyes. Even a painter like Giotto (Ambrogio Bondone) managed to produce works with an almost photographic quality, but on close analysis the perspective is not completely systematic. It was Filippo Brunelleschi who, in 1425, formulated a systematic mathematical method for obtaining accurate perspective, which he then taught to other artists. By 1435 we find the first book on the topic, Leone Alberti’s Della Pittura.

  The method was brought to perfection in the paintings of Piero della Francesca, who was also a consummate mathematician. Piero wrote three books on the mathematics of perspective. And it would be impossible not to mention Leonardo da Vinci, whose Trattato della Pittura begins by stating, “Let no one who is not a mathematician read my works,” an echo of the slogan “Let no one ignorant of geometry enter” which legendarily was placed over the door of Plato’s academy in ancient Greece.

  The essence of perspective is the notion of “projection,” by which a three-dimensional scene is rendered on a flat sheet of paper by (conceptually) drawing each point of the scene to the viewer’s eye, and seeing where that line meets the paper. A key idea is that projections distort shapes in ways not permitted by Euclid. In particular, projection can turn parallel lines into lines that meet.

  How projection makes parallels meet at the horizon.

  We see this effect every day. When we stand on a bridge and see a long, straight railroad track or highway disappearing into the distance, the lines converge and seem to meet at the horizon. The real lines remain the same distance apart, but perspective causes the perceived distance to shrink as the lines get farther away from us. In a mathematical idealization, infinitely long parallel lines in a plane also meet if they are suitably projected. But the place where they meet is not the image of anything in the plane—it can’t be; they don’t meet in the plane. It is the apparent “horizon” toward which the lines, and the plane, extend. On the plane itself, the horizon is infinitely distant, but its projection is a perfectly sensible line across the middle of the picture.

  This line is known as the “line at infinity.” Like the square root of minus one, it is a fiction, but an extremely useful one. The kind of geometry that emerges is called projective geometry, and in the spirit of Klein’s Erlangen program, it is the geometry of those features of a scene that are not changed by projections. Every artist who makes perspective drawings with a horizon line and “vanishing points” to organize his or her images to look like real objects is using projective geometry.

  In a projective plane, geometry is very elegant. Any two points can be joined by a unique line, just as in Euclid’s geometry. But any two distinct lines meet, too, at exactly one point. Parallels, which so exercised Euclid, do not exist.

  If this reminds you of the Fano plane, you’re right. The Fano plane is a finite projective geometry.

  From Renaissance perspective to the exceptional Lie groups is now but a short step. The projective plane that was implicit in Alberti’s methods was made explicit as a new kind of geometry. In 1636, Girard Desargues, an army officer who later became an architect and engineer, published Proposed Draft of an Attempt to Treat the Results of a Cone Intersecting a Plane. It sounds like a book on conics, and it was, but instead of using the traditional Greek geometry, Desargues used projective methods. Just as Euclidean geometry could be turned into algebra by using Descartes’s coordinates (x, y), a pair of real numbers, so projective geometry could be turned into algebra by letting x or y become infinite (in a cleverly controlled manner, involving ratios of three coordinates and setting 1 ÷ 0 = infinity).

  What you can do with real numbers, you can also do with complex numbers, so now you get the complex projective plane. And if those work, why not try the quaternions or the octonions?

  There are problems—the obvious methods don’t work, because of the lack of commutativity. But in 1949, the mathematical physicist Pascual Jordan found a meaningful way to construct an octonionic projective plane with 16 real dimensions. In 1950, the group theorist Armand Borel proved that the second exceptional Lie group F4 is the symmetry group of the octonionic projective plane—much like the complex plane, but formed from two 8-dimensional “rulers” labeled with octonions, not real numbers.

  So now there was an octonionic explanation of two of the five exceptional Lie groups. What about the other three—E6, E7, and E8?

  The view of the exceptional Lie groups as brutal acts of a malicious deity was fairly widespread until 1959, when Hans Freudenthal and Jacques Tits independently invented the “magic square,” and explained E6, E7, and E8.

  The rows and columns of the magic square correspond to the four normed division algebras. Given any two normed division algebras, you look in the corresponding row and column, and what the magic square gives you—following a technical mathematical recipe—is a Lie group. Some of these groups are straightforward; for example, the Lie group corresponding to the real row and the real column is the group SO(3) of rotations in three-dimensional space. If both row and column correspond to the quaternions, you get the group SO(12) of rotations in twelve-dimensional space, which to mathematicians is just as familiar. But if you look in the octonion row or column, the entries are the exceptional Lie groups F4, E6, E7, and E8. The missing exceptional group G2 is also intimately associated with the octonions—as we’ve already seen, it is their symmetry group.

  So now the general opinion is that the exceptional Lie groups exist bec
ause of the wisdom of the deity in permitting the octonions to exist. We should have known. As Einstein remarked, the Lord is subtle but not malicious. All five exceptional Lie groups are the symmetries of various octonionic geometries.

  Around 1956, the Russian geometer Boris Rosenfeld, perhaps thinking about the magic square, conjectured that the three remaining exceptional groups E6, E7, and E8 are also the symmetry groups of projective planes. In place of the octonions, however, you have to use the following structures:

  • For E6: the bioctonions, built from complex numbers and octonions.

  • For E7: the quateroctonions, built from quaternions and octonions.

  • For E8: the octooctonions, built from octonions and octonions.

  The only slight snag was that no one knew how to define sensible projective planes over such combinations of number systems. But there was some evidence that the idea made sense. As matters currently stand, we can now prove Rosenfeld’s conjecture, but only by making use of the groups to construct the projective planes. This is not very satisfactory, because the idea was to go the other way, from the projective planes to the groups. Still, it’s a start. In fact, for E6 and E7 there now exist independent ways to construct the projective planes. Only E8 is still holding out.

  Were it not for the octonions, the Lie group story would be more straightforward, as Killing originally hoped, but nowhere near as interesting. Not that we mortals get to choose: the octonions, and all the associated paraphernalia, are there. And in some obscure way, the existence of the universe may depend on them.

  The connection between the octonions and life, the universe, and everything emerges from string theory. The key feature is the need for extra dimensions to hold the strings. Those extra dimensions can in principle have lots of shapes, and the big question is to find the right shape. In old-fashioned quantum theory, a key principle is symmetry, and that’s the case in string theory too. So of course Lie groups get in on the act. Everything hinges on those Lie groups of symmetries, and again the exceptional groups stick out—not as sore thumbs, but as opportunities for unusual coincidences that could help make the physics work.

  Which gets us back to the octonions.

  Here’s an example of their influence. In the 1980s, physicists noticed that a rather nice relationship occurs in space-times of 3, 4, 6, and 10 dimensions. Vectors (directed lengths) and spinors (algebraic gadgets originally created by Paul Dirac in his theory of electron spin) are very neatly related in these dimensions, and only these. Why? It turns out that the vector–spinor relationship holds precisely when the dimension of space-time is 2 greater than that of a normed division algebra. Subtract 2 from 3, 4, 6, and 10, and what you get is 1, 2, 4, and 8.

  The mathematical point is that 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 a number of nice consequences for physics. So we have four candidate string theories here: real, complex, quaternionic, and octonionic. And it so happens that among these possible string theories, the one that is currently thought to have the best chance of corresponding to reality is the 10-dimensional one, specified by the octonions. If this 10-dimensional theory really does correspond to reality, then our universe is built from octonions.

  And that’s not the only place where these strange “numbers,” barely clinging to that name because they just satisfy enough of the rules of algebra, are influential. That fashionable new candidate string theory, M-theory, involves 11-dimensional space-time. In order to reduce the perceptible part of space-time from 11 dimensions to the familiar 4, we have to throw away 7 by rolling them up so tightly that they can’t be detected. And how do you do that for 11-dimensional supergravity? You make use of the exceptional Lie group G2, the symmetry group of the octonions.

  There they are again: no longer quaint Victoriana but a hefty clue to a possible Theory of Everything. It’s an octonionic world.

  16

  SEEKERS AFTER TRUTH AND BEAUTY

  Was Keats right? Is beauty truth, and truth beauty?

  The two are intimately connected, possibly because our minds react similarly to both. But what works in mathematics need not work in physics, and vice versa. The relationship between mathematics and physics is deep, subtle, and puzzling. It is a philosophical conundrum of the highest order—how science has uncovered apparent “laws” in nature, and why nature seems to speak in the language of mathematics.

  Is the universe genuinely mathematical? Are its apparent mathematical features mere human inventions? Or does it seem mathematical to us because mathematics is the deepest aspect of its infinitely complex nature that we are able to understand?

  Mathematics is not some disembodied version of ultimate truth, as many used to think. If anything emerges from our tale, it is that mathematics is created by people. We can readily identify with their triumphs and their tribulations. Who could fail to be moved by the appalling deaths of Abel and Galois, both at the age of 21? One was deeply loved but never earned enough money to marry; the other, brilliant and unstable, fell in love but was rejected, and perhaps died because of that love. Today’s medical advances would have saved Abel, and might even have helped Hamilton stay sober.

  Because mathematicians are human and live ordinary human lives, the creation of new mathematics is partly a social process. But neither mathematics nor science is wholly the result of social processes, as social relativists often claim. Both must respect external constraints: logic, in the case of mathematics, and experiment, in the case of science. However desperately mathematicians might want to trisect an angle by Euclidean methods, the plain fact is that it is impossible. However strongly physicists might want Newton’s law of gravity to be the ultimate description of the universe, the motion of the perihelion of Mercury proves that it’s not.

  This is why mathematicians are so stubbornly logical, and obsessed by concerns that most people could not care less about. Does it really matter whether you can solve a quintic by radicals?

  History’s verdict on this question is unequivocal. It does matter. It may not matter directly for everyday life, but it surely matters to humanity as a whole—not because anything important rests on being able to solve quintic equations, but because understanding why we can’t opens a secret doorway to a new mathematical world. If Galois and his predecessors had not been obsessed with understanding the conditions under which an equation can be solved by radicals, humanity’s discovery of group theory would have been greatly delayed, and perhaps might not have happened.

  You may not encounter groups in your kitchen or on your drive to work, but without them today’s science would be severely curtailed, and our lives would be far different. Not so much in gadgetry like jumbo jets or GPS navigation or cell phones—though those are part of the story too—but in insight into nature. No one could have predicted that a pedantic question about equations could reveal the deep structure of the physical world, but that is what happened.

  The clear message of history is a simple one. Research on deep mathematical issues should not be rejected or denigrated merely because those issues seem to have no direct practical use. Good mathematics is more valuable than gold, and where it comes from is mostly irrelevant. What counts is where it leads.

  The astonishing thing is that the best mathematics usually leads somewhere unexpected, and a lot of it turns out to be vital for science and technology, even though it was originally invented for some totally different purpose. The ellipse, studied by the Greeks as a section of a cone, was the clue that led, via Kepler, from Tycho Brahe’s observations of the motion of Mars to Newton’s theory of gravity. Matrix theory, whose inventor Cayley apologized for its uselessness, became an essential tool in statistics, economics, and virtually every branch of science. The octonions may be the inspiration for a Theory of Everything. Of course, the theory of superstrings may turn out to be just a pretty piece of mathematics
with no relevance to physics. If so, the existing uses of symmetry in quantum theory still demonstrate that group theory provides deep insights into nature, even though it was developed to answer a question in pure mathematics.

  Why is mathematics so useful for purposes that its inventors never intended?

  The Greek philosopher Plato said that “God ever geometrizes.” Galileo said much the same thing: “Nature’s great book is written in mathematical language.” Johannes Kepler set out to find mathematical patterns in planetary orbits. Some of them led Newton to his law of gravitation; others were mystical nonsense.

  Many modern physicists have commented on the astonishing power of mathematical thought. Wigner alluded to the “unreasonable effectiveness” of mathematics as a way to understand nature; the phrase appears in the title of an article he wrote in 1960. In the body of the article he said he would tackle two main points:

  The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories.

  And:

  The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

  Paul Dirac believed that in addition to being mathematical, nature’s laws also had to be beautiful. In his mind, beauty and truth were two sides of the same coin, and mathematical beauty gave a strong clue to physical truth. He even went so far as to say he would prefer a beautiful theory to a correct one, and that he valued beauty above simplicity: “The research worker, in his efforts to express the fundamental laws of nature in mathematical form, should strive mainly for mathematical beauty. He should still take simplicity into consideration in a subordinate way to beauty . . . where they clash the latter must take precedence.”