Why Beauty is Truth (39 page)

He was nearly right. In 1894, the French geometer Élie Cartan noticed that Killing's two 56-dimensional algebras are really the same algebra viewed in two different ways. That means that there are only five exceptional simple Lie algebras, corresponding to five exceptional simple Lie groups: Killing's old friend G
₂
, and four others now called F
₄
, E
₆
, E
₇
, and E
₈
.

This is an exceedingly curious answer. The infinite families are reasonable enough; they are all related to various natural types of geometry in any number of dimensions. But the five exceptional Lie groups seem unrelated to anything geometric, and their dimensions are bizarre. Why are spaces of dimensions 14, 56, 78, 133, and 248 special? What is so unusual about those numbers?

It's a bit like wanting to list all possible shapes for a brick, and finding an answer something like this:

 

• Oblong blocks of size 1, 2, 3, 4, . . .

• Cubes of size 1, 2, 3, 4, . . .

• Slabs of size 1, 2, 3, 4, . . .

• Pyramids of size 1, 2, 3, 4, . . .

 

Which would be very neat and tidy, except that the list continues:

 

• A tetrahedron of size 14.

• An octahedron of size 52.

• A dodecahedron of size 78.

• A dodecahedron of size 133.

• A dodecahedron of size 248.

 

And that's it, there's nothing else.

Why do bricks with these strange shapes and sizes exist? What are they
for?

It seemed completely mad.

It seemed so mad, in fact, that Killing was rather upset that the exceptional groups existed, and for a time he hoped they were a mistake that he could eradicate. They spoiled the elegance of his classification. But they were
there
, and we are finally beginning to understand
why
they are there. In many ways, the five exceptional Lie groups now look much more interesting than the four infinite families. They seem to be important in particle physics, as we will see; they are definitely important in mathematics. And they have a secret unity, not yet fully uncovered, relating them all to Hamilton's quaternions and an even more curious generalization, the octonions. Of which more, in due course.

It's a wonderful series of ideas, and Killing had all of them. To be sure, his work included a few mistakes—some proofs that didn't quite work. But the mistakes were all repaired long ago.

That is how the greatest mathematical paper of all time went. What did Killing's contemporaries think of it?

Not a lot. It didn't help that Lie poured derision on Killing's magnum opus. He had fallen out with Killing for unknown reasons, and as far as he was concerned, Killing would never do anything important. Worse, of course, this was a theorem that Lie himself would have dearly loved to prove. Having been beaten to the punch, he resorted to the age-old technique of sour grapes. Anything in the area not done by Lie, said Lie, was rubbish. Though he wasn't quite that blatant.

It helped even less that Killing underestimated the value of his own theorem. To him it was a pale shadow of something far more important, which he had failed to achieve: classifying all Lie groups. Killing was a modest man, and Lie did his best to make him more so.

In any case, Killing was ahead of his time. Very few mathematicians saw how important Lie theory was going to become. To most, it was a rather technical branch of geometry associated with differential equations.

Finally, Killing was a staunch Catholic with a strong sense of duty and humility. He took St. Francis of Assisi as his model, and at the age of 39 he and his wife entered the Third Order of the Franciscans. He seems to have been a thoroughly decent man who worked tirelessly on behalf of his students. He was a conservative and a patriot, greatly saddened by the extreme social dissolution of Germany after World War I. His feelings were made worse by the deaths of his two sons in 1910 and 1918.

The true worth of Killing's researches became apparent in 1894, when Élie Cartan rederived the whole theory in his PhD thesis, and took it a big step further by classifying not just the simple Lie algebras but their representations in terms of matrices. Cartan was scrupulous in giving credit to Killing for nearly all of the ideas; he just tidied everything up, plugged a few gaps (some serious), and modernized the terminology. But a myth quickly grew up to the effect that Killing's work was riddled with holes and the real credit should go to Cartan. Mathematicians are seldom good historians, and they tend to cite work that they know rather than the earlier work that led up to it. So Cartan's name became attached to many of Killing's ideas.

Anyone who reads Killing's papers quickly discovers that the myth is just that. The ideas are clear and well formed, the proofs are perhaps old-fashioned but nearly all correct. Most importantly, the overall sweep of the ideas is beautifully chosen to produce the desired result. It is mathematics of the highest order, and it is not anyone else's.

Unfortunately, hardly anyone read Killing's papers. They read Cartan and ignored the credit he gave to Killing. But eventually, Killing's work began to achieve proper recognition. In 1900 he won the Lobachevsky Prize of the Kazan Physico-Mathematical Society. This was the second time the prize had been awarded: the first one went to Lie.

Killing died in 1923. Even today, his name is not as well known as it deserves to be. He was one of the greatest mathematicians who ever lived. His legacy, at least, is immortal.

B
y the beginning of the twentieth century, groups were starting to show up in fundamental physics, a field they would transform just as radically as they had transformed mathematics.

In the golden year of 1905, the man who would become the most iconic scientist of his time published three papers, each of which revolutionized a separate branch of physics. He was not at that time a professional scientist. He had studied at university but had not been able to obtain a teaching position and was working as a clerical official in the patent office in Bern, Switzerland. His name, of course, was Albert Einstein.

If any one person can symbolize modern physics, it is Einstein. To many, he also symbolizes mathematical genius, but in fact he was merely a competent mathematician, not a creative one on the level of Galois or Killing. Einstein's creativity lay not in producing new mathematics but in an extraordinarily rigorous intuition about the physical world, which he was able to express through remarkable uses of existing mathematics. Einstein also had a flair for the right philosophical standpoint. He drew radical theories from the simplest of principles and was guided by a sense of elegance rather than a wide knowledge of experimental facts. The important observations, he believed, could always be distilled into a few key principles. The gateway to truth was beauty.

Acres of print and many lifetimes of scholarly study have been devoted to Einstein's life and works. A single chapter cannot hope to compete in either completeness or erudition. But he is a key figure in the history of symmetry: it was Einstein, above all others, who set in motion the web of events that turned the mathematics of symmetry into fundamental physics. I don't think Einstein saw it that way: to him, the mathematics
was a servant of physics—often a rather disobedient one. Only later, following the trail that Einstein had blazed and tidying up the tangled, broken vegetation that his pioneering efforts had strewn across the path, did another generation uncover the elegant and deep mathematical concepts upon which his work was based.

So we must retell the main outlines of the astonishing rise to fame of this minor patent clerk—technical expert third class, to be precise, and on a trial basis at that. Since he is but one part of our story, I will select only the relevant events. If you who want a more comprehensive, unbiased assessment of Einstein's career, you should read Abraham Pais's
Subtle Is the Lord.

Subtle, yes—but not, as Einstein once remarked, malicious.

Einstein, who had little interest in religion, devoted his life to the principle that the universe is comprehensible and that it runs along mathematical lines. Many of his most famous sayings invoke the deity, but as a symbol of the orderliness of the universe, not as a supernatural being with a personal interest in human affairs. He worshipped no god and practiced no religious rituals.

Einstein is generally seen as the natural successor to Newton. Earlier scientists had made additions to Newton's “system of the world,” as his
Mathematical Principles of Natural Philosophy
was subtitled, but Einstein was the first to make significant changes to that vision. The most important of the earlier theorists was James Clerk Maxwell, whose equations for electromagnetism brought magnetic and electric phenomena, especially light, within the Newtonian purview. Einstein went much further, making major changes. Ironically, the changes that led to a revised theory of gravity came about as consequences of the Maxwellian theory of electromagnetic waves—light and its relatives. Even more ironically, a fundamental feature of that theory, the wave nature of light, played a key role, yet Newton denied that light could be a wave. To cap it all, one of the most elegant experiments now used to demonstrate that light
is
a wave was first carried out by Newton.

Scientific interest in light goes back at least to Aristotle, who, though really a philosopher, asked the kind of question that scientists would find natural.
How do we see?
Aristotle suggested that when we look at some object, that object affects the medium between itself and the onlooking eye.
(We now call this medium “air.”) The eye then detects this change in the medium, and the result is the sensation of sight.

In medieval times this explanation was reversed. It was thought that our eyes emitted some kind of ray, which illuminated whatever we looked at. Instead of the object transmitting signals to the eye, the eye left eye-tracks all over the object.

Eventually, it was understood that we see objects by means of reflected light, and that in daily life the main source of light is the Sun. Experiments showed that light travels in straight lines, forming “rays.” Reflection occurs when a ray bounces off a suitable surface. So the Sun sends light rays to everything that is not shadowed by something else, the rays bounce all over the place, some enter an observer's eye, the eye receives a signal from that direction, the brain processes the incoming information from the eye, and we see whatever object the ray bounced off.

The main question was, what is light? Light does a number of puzzling things. Not only does it reflect; it can also refract—change direction abruptly at the interface between two different media, such as air and water. This is why a stick poked into a pond looks bent, and also why lenses work.

Even more puzzling is the phenomenon of diffraction. In 1664, the scientist and polymath Robert Hooke, whose career repeatedly clashed with Newton's, discovered that if he placed a lens on top of a flat mirror and then looked through the lens, he saw tiny concentric colored rings. These rings are now known as “Newton's rings” because Newton was the first person to analyze their formation. Today we consider this experiment a clear demonstration that light is a wave: the rings are interference fringes, where waves do or do not cancel each other out when they overlap. But Newton didn't believe light was a wave. Because light traveled in straight lines, he believed it had to be a stream of particles. According to his
Opticks
, completed in 1705, “Light is composed of tiny particles, or corpuscles, emitted by luminous bodies.” The particle theory could explain reflection very simply: the particles bounced when they hit a (reflecting) surface. It encountered difficulties explaining refraction, and pretty much fell apart when it came to diffraction.