Joe Rosen, Symmetry Discovered: Concepts and
Applications in Nature and Science (1975)
Mathematicians are subtle with language, but they have made a mess of the word “symmetry.” What ordinary people mean by “symmetry” is kindergarten shapes (such as Square, Triangle, etc.) and snowflakes.
At the center of what mathematicians mean by the word “symmetry” is the word “always.”
A square always looks the same when you rotate it through a quarter-turn, or half a turn, or three-quarters of a turn, or turn it backside up; or do any combination of two, five, ten, or 743 of these things. It is symmetrical with respect to these operations.
Triangle always looks the same when you manhandle it through a third of a turn, two-thirds of a turn, flip it bloomer-side and rotate it as many times as you like, or keep yourself churchlike and Sunday-mannered and pass it by altogether. It is symmetrical with respect to it all.
If you can phrase an idea—any idea, on any subject—to include the word “always,” then you’re within whistling distance of what mathematicians call “symmetry.”
Simon always fills his bath too full.
I always get cross and kick shut the Excavation door when Simon cooks packet rice with Chinaman in it.
E always equals mc2.
The Monster always looks the same (although no one yet knows what that look is like) when you perform on it one of the 808,017,424,794,512,875,886,459,904,961, 710,757,005,754,368,000,000,000 symmetry operations in its Group Table.
When drunk, Jim, our next-door neighbor, always leaps on his motorbike at 2 a.m. and starts revving the engine.
Each of these cases makes an appeal to symmetry. Symmetry is the invariability of some object, circumstance or relationship to a specified set of changes. Put Simon in the bath in our house or on a planet in Alpha Centauri; in the middle of a nuclear war; when he was five, seventy-five or upside down—it makes no difference. As far as the water in the bath is concerned, it’ll always dribble over the edge, seep under the skirting, ooze along the basement wall and cause toilets and stair treads to collapse. It’s a knack Simon has.
In the starched language of squares and triangles, “rotate” Simon-in-a-bath among any of these different situations, and the result always looks the same:
However many times, in whatever order, we perform these operations, it will still be splish-splash Norton in the bath:
The anthropologist Claude Lévi-Strauss noted that certain types of marriage among Australian tribes are always considered incestuous and therefore forbidden. But he couldn’t figure out the underlying structure behind these taboos, so he went around New York (where he was living at the time) banging on the doors of mathematicians. The first was dismissive: “Mathematics has four operations, and marriage is not one of them.” But the second was the young and brilliant André Weil, brother of the philosopher Simone. “When in doubt,” cried Mr. Weil, “look for the group!” and he bustled Lévi-Strauss off the street into his study. Within a few days, Weil had solved the problem. Not only had he used the theory of symmetry to explain why certain marriages were taboo among Australian Aboriginals, but he’d also discovered that, by mathematics alone, it was possible to investigate a tribe today and determine if at some time in the past it had been in fact two different tribes, which had since met and settled together. There would be an imperfection in the “symmetry” of the incest taboos—the “alwaysness” of their rules—that would give the game away. You didn’t need to know the history of the tribe, or have any further anthropological knowledge than an understanding of the current restrictions on sex: all you needed was to pick up a glint of asymmetry, the murmur of a “sometimes.” With mathematics, you could see 1,000 years into the past.
I’ve charged about my study for hours trying to understand how going to bed with your sister is a mathematical statement, and failed, but it is.
(“I don’t understand it either,” grumbles Simon. “In fact, I’m not sure I believe in it.”)
The word “taboo” suggests always forbidden, so symmetry must be involved somehow. “Rotate” any one of the banned couplings in front of Australian tribal elders and there will always be uproar.
This is why symmetry is important: it’s everywhere. Symmetry does not mean just preschool cut-out shapes like triangles and squares, it speaks for anything—from quantum fluctuations inside your eyelash, to the songs of the Beluga whale, to the despotic delusions of madmen in North Korea—that involves, in some respect, however far-fetched, “immunity from change.” It is the study of imperturbability. If, under certain conditions, a situation always appears the same despite the fact that things are actually being done to it, then that’s symmetry.
This tells us something about Simon’s Monster. It is not a raging thing. Its power lies in its unchangingness. It is a monster of calm.
And here’s a fifteenth way to say it: a symmetry operation is an act that leaves a thing “unfazed.” Mess and boring bus journeys, for example, leave Simon unfazed. Mess and buses are a part of Simon’s symmetry.
“I don’t think that’s an analogy at all,” interrupts Simon, looking up, tongue clamped between teeth.
“Loosely, ‘being unfazed’ is an analogy to being symmetrical,” I pursue.
“No, not loosely. The two ideas have nothing to do with each other. A symmetry operation in mathematics is something that operates on something else. It needs to do something to be interesting, as in a rotation operating on a square. If that rotation makes the square appear different than it did before, it is not a symmetry operation; if the same, it is.”
“Exactly. Operate on you with mess, and your character and mood don’t change.”
“What do you mean, ‘operate’ with mess? Operate how? I don’t understand that at all. The character of a human has nothing to do with operations. It’s just to do with what a human is. It’s nonsense to say they’re analogous. I don’t even know what that would mean, to talk about the mathematical character of humans. I really don’t want to talk about this anymore.”
Mess, buses and nitpicking are part of Simon’s symmetry.
18
There are types of authors who are not O.K., names it is O.K. to pitch into. It is alright to pitch into:
Any author who has written a book about dogs.
Any author who has written a book on natural history, illustrated with woodcuts.
Any author who has written a life of Napoleon, Byron, or Dr. Johnson, without footnotes or bibliography.
Any author of a life of anybody not yet dead.
Any author of a book on Sussex.
Stephen Potter, Some Notes on Lifemanship (1953)
“Did I tell you,” declaimed Simon in a buffalo voice, “when I was a boy I could pee for longer than anyone else?”
Our train was heading through the chalklands of Sussex toward Ashdown House Junior School. It was one of those over-eager electric services that shouts at you to stop smoking and refuses to let you be in charge of the lavatory door.
Me? I was over-eager too. I’d had a biographical smash hit! The third person sitting with us, a tall, thick-set man with fingers chunky as carrots, was Malcolm Russell (not his real name—he wants to keep that secret), one of Simon’s old school bullies. He’d Googled Simon, then emailed him. Tormented by what he’d done half a century ago, he wanted to…to what? He wasn’t quite sure. To apologize? To make amends? To understand? Not knowing which, he’d proposed this trip back to the school instead.
Already, his influence had transformed Simon. Ever since East Croydon station, Grunter had become Babbler. It was astounding. It was miraculous.
“I could stand further back from the urinal too,” Simon hollered again, filling the carriage with triumph.
It was embarrassing.
I turned to look out the window in shame. Purple and white crocuses clustered along the banks, and pastures were puddled with water. Oak trees had begun to bud in the cold, sun-startled air; rabbits, t
rembling with winter, crouched among daffodils. Everything was precarious and young, and two months early, chancing on there being no frost in April.
Malcolm runs a fertilizer store in Shropshire. After Ashdown, he’d gone to Radley, a public school next to Oxford. Memory has scraped that hellhole deep into his conscience. “Other boys had mental breakdowns while I was there. One committed suicide.”
Malcolm is constantly searching his pockets for not-too-discomforting recollections of childhood.
“The urinals!” he roared now with delight, sinking his hand into a packet of salt-and-vinegar crisps, exploding the bag. “Yes, I remember the school urinals! I can see the loo. I remember it exactly. I really hope they haven’t pulled them down. We’ll have to go there. We’ll have to see everything! And the Spanish teacher, Mr. Juh—, Juh—”
“Jabuzi!” cried Simon.1
“Yes! Did he put his hand up your shorts too?”
The school announces it is “set in rolling hills,” but it isn’t. The Downs roll. Devonshire rolls. North Sussex wobbles. William Cobbett called the glorious Sussex countryside around Ashdown “the most villainously ugly spot I ever saw in England.”
Seed-speckled haze; woodpeckers scooping between the spruce trees; cuckoo cadence lilting in the breeze: Ashdown House is set at the end of a half-mile drive in forty acres of rhododendrons. After the wood (which starts to the right of the school and is all that remains of the ancient Ashdown Forest) comes the high, scratchy heath, then a stretch of gentleness much more to Cobbett’s farmhouse taste, followed at last by the escarpment to the South Downs.
The school charges £5,400 a term for non-boarders, and unmentionable sums for boarders. It has alumni who are mayors of London (Boris Johnson), have been sent to prison for shooting poachers on their estates in Kenya (The Honorable Thomas Patrick Gilbert Cholmondeley of Delamere) and write novels about Notting Hill Yummy Mummies (Boris’s sister Rachel). As a schoolgirl, Rachel found a “live, plump maggot wriggling” in her shepherd’s pie. She bore the plate “with its live cargo” across the dining hall and pointed out the ghastly object to Clive Williams, the headmaster.
“Clive peered at it expressionlessly. ‘Yes, very nutritious, I expect,’” he said, and dismissed her.2
In the cattle-farming language of the wealthy, Ashdown is known as a “feeder” prep school for Eton. That’s why Simon’s father (or was it his mother?) wanted him to go there. It’s the best school for getting boys into Eton. Eton is the best school for getting boys into Trinity College, Cambridge. Trinity is the best college for providing genius mathematicians with a pleasant life.
The main school building was designed by the architect Benjamin Latrobe. After completing it he migrated to Virginia, where he designed the State Penitentiary, taking a particular interest in solitary confinement.
“To be honest,” admitted Malcolm quietly as we walked up the drive among the cuckoos and seeds, Simon yards in front of us, “when I first emailed Simon last month I half thought he’d be banging his head in an institution or have died, though not suicide—car accident, I expected.
“There was a kind of sweetness to him when you sat down and paid attention to him. There was a sense of him being defenseless in the world. You really felt him soften, he liked friendliness so much. If you asked him who the Prime Minister was, he might say someone from the First World War: ‘Asquith.’
“He certainly didn’t make you envious of genius,” he added. “To be like Simon seemed a high price to pay.”
A cluster of pigtailed girls poking out their tongues and waving hockey sticks rushed from the school gates ahead, pushed us off the verge of the drive and, hopping over a hump in the road, surged toward Simon.
“Once—do you remember this, Simon?” Malcolm called to Simon, who was making miserable efforts to regain his balance following the pigtail attack. “—There was a school quiz and you were asked to spell ‘bikini.’”
Simon chuckled contentedly.
“Who else in the world but Simon would you know who spelled it ‘Becquigny’? The only thing he could think of that sounded like ‘bikini’ was the Treaty of Becquigny between England and France in 1340.”
“Pecquigny,” interjected Simon, shocked by the scent of a false fact.
“Becquigny,” repeated Malcolm.
“Huunh! No! Pecquigny. August 29th, 1475. Louis XI paid Edward IV 75,000 crowns and had to ransom his wife, Margaret of Anjou.”
Malcolm shrugged his shoulders, stepped back onto the drive and sighed happily. “I’ve talked about him a lot over the years,” he confessed.
A car zipped past us and coughed to a stop in the car park.
“Excuse me, can I help?” exclaimed the driver, bounding out onto the gravel and striding over. He gave a vicarish peer.
“Well! If it isn’t Simon Norton! Ashdown’s greatest scholar!”
Clive Williams, ex-headmaster, had come back for the day. A former Ashdown boy, he’d spent five decades at the school and, when he’d got wind that Simon was visiting, had raced back to it from his house in Lewes to say hello. Even after forty-two years, he recognized in a second his old fellow pupil.
I’d recently interviewed Clive about Simon’s time at Ashdown: four hours sitting in Clive’s cozy kitchen, going through the school bulletin, him gazing at me and the middle distance while he remembered the golden schooldays of the 1950s. But I am not an Ashdown boy. I am, in the kindest sense, dust to a man like Clive. After greeting Malcolm with more cries of happiness, he now turned to me and held out his hand:
“And who,” he said pleasantly, “are you?”
It’s the subjects that Simon couldn’t do that are interesting. In mathematics, of course, he understood everything he was taught, and divined the rest.
In Greek and Latin (as so often with mathematicians) he was a vacuum cleaner.
To a certain extent, this is down to memory. A mathematician who can’t remember obscure facts is, like a detective who can’t remember clues, impossible. Mathematical ability depends on being able to make links between remote ideas, and people such as Simon never do this by the Inspector Plod method of dully applying and reapplying the rules of school algebra. They sense the correct approach, sniff it out like a dog. This doesn’t mean they’ve just got more sprightly than normal nerve endings in the mathematical parts of the brain; it means they have a superb memory for certain types of details, can use it to draw comparisons with other mathematical discoveries they’ve made or read about elsewhere, and therefore can exploit tiny and hidden analogies of argument.
If Simon had not become a mathematician, he would have made a very good scholar of dead grammars, publishing compendious books that were instantly outdated.
Yet in history, a subject you’d think would appeal to his excellent memory and obsession with fact gathering and numbers, Simon was at sea. “Sublimely indifferent,” according to his school reports, his essay on Warwick “extremely feeble”:
“I could never understand what history was about,” says Simon. “Why were they always fighting over a field?”
“But you remember the Treaty of Pecquigny.”
Religious education was the same, despite the fact that Simon’s father was president of the Jewish Reform Synagogue, and the Ashdown headmaster—a brilliant and charismatic scholar called Billy Williamson—used to pace through the classrooms each morning, Bible in hand, reading passages of gore and genealogy in a booming voice that carried across the entire school.
Written after his pen had fallen nib-first on the floor.
Billy Williamson operated a five-year plan. By the end of each five-year stretch—the length of time a boy was at the school—he had declaimed the whole of both Testaments. If you arrived for your first year when he was starting Genesis, you left for Eton or Harrow at the end of Revelation. If you appeared at the Second Book of Kings, with the little boys who persecuted Elisha being eaten by bears, you left, sixty months later, as the headmaster read from the First Book of Kin
gs about randy King Solomon and his 700 wives and 300 concubines.
Williamson did his best to make the subject engaging to odd pupils such as Simon who didn’t love the usual bloodshed and despicable behavior of our God in the Old Testament. In an attempt to nudge Simon at least a step or two into an interest in the history of his own people, the headmaster included this question in an end-of-term test paper:
If you complete the following sum correctly you will arrive at a critical date in Jewish History. What was it? Multiply the number of Jacob’s children by the number of Zeruiah’s sons and then by the number of brothers left by Dives. Then subtract the date of the destruction of the Samaritan Temple on Mt. Gerizim by John Hyrcanus, plus the number of tribes of Judah multiplied by the number of tribes of Israel after Rehoboam’s reign.3
Simon was no better at this than he was at the Earl of Warwick and his wretched king making. Yet what Billy Williamson may have done successfully was to encourage Simon’s mental playfulness. He used to let him lie about on the floor of his study for hours, doing sums and laughing and enjoying being teased and playing word games.
“Simon,” wrote Williamson in one of his delightful end-of-year reports, “goes on his way rejoicing, and I rejoice with him.”
It barely needs saying: sport was Simon’s worst subject. This will be a short section.
“During a game of cricket he spends his time counting blades of grass or calculating angles. He takes about as much interest in the proceedings as Archimedes did during the siege of Syracuse.” In football “he flutters about like a butterfly.” During swimming, he sank; for rugby, he didn’t move.
“I used to stand on the pitch,” admits Simon, “and eat the sorrel.”
* * *
In other subjects, Simon’s failure required more effort. It takes determination for a boy to be at the bottom of the class in geography. Simon sank there like sedimentary graphite. Incapable of disguising boredom, he’d rather endure hours of disapproval than five minutes of homework about why rivers wiggle or the post-war output of a Dagenham car factory. This is a limitation that would crop up later, in his work as a mathematician: if he doesn’t take to a subject and feel excited about it and want to dominate it immediately, he can’t be bothered with the thing. Intellectually, he is extremely lazy.