He scuttled across to the crisp racks and snatched up three party packs of onion rings.
“No, I don’t think I did. Aaaah, no. I don’t think that is a safe assumption.”
He flung all the packets into his basket, and darted through a crowd of students to crouch in front of “Condiments.” This was not a supper food run. These were bus-jaunt supplies, to stash in his duffel.
“Who did hang them, then?”
“Hnnnh, aah…perhaps it was my brother. He and his wife I remember came up one afternoon and decorated for me—they must have hung them then.”
“When was that?”
Simon looked up, adopted a calculating expression, then dropped a jar of pickled beetroot into his basket, among the onion rings.
“Twenty-six years ago.”
31 The Monster
* * *
Groups can have Subgroups—smaller symmetries hiding in the larger symmetry. By looking carefully at the Group Table for the rotations of Square, it’s possible to spot a Subgroup skulking inside:
The Group Table for Triangle does not contain any Subgroups. It is an atom of symmetry. In the language of Group Theory, it is called a “Simple Group.”
There are quidzillions of these Simple Groups, all sharing this fundamental quality: they have no normal Subgroups.1 The last of these Simple Groups, the final piece in the periodic table of all the universe’s finite symmetries, wasn’t discovered until 1973 (not by Simon).
* * *
1 The word “normal” is essential, but too big a subject to discuss here.
* * *
* * *
It is the Monster.
But what is the Monster? What set of symmetries does this gargantuan object represent? Is it just spinnings and flippings of another flat shape like Triangle or Square, only, instead of three or four edges, it has
808,017,424,794,512,875,886,459,904,961,710,
757,005,754,368,000,000,000?
* * *
This number, 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000, is the number of columns and rows in the Monster Group’s Table. But no one knows what strange object might have these symmetries. It might be that many different objects have them. Three pieces of garbage kicked about in a bag turned out to have the same symmetries as the rotated and flipped Triangle. Perhaps it is the same in the case of the Monster Group: there are lots of different ways in which this Group of symmetries can appear in the universe, we just haven’t found any of them yet. How Simon can understand the symmetries of an object before he’s discovered what object might possess these symmetries I have no idea. To understand that, you have to study mathematics until your brain turns green.
One thing he does know about any object with the symmetries of the Monster Group is that it can’t exist in fewer than 196,883 dimensions.
I don’t know how he discovered that either. I don’t even know what it means. How can something exist in four dimensions, let alone 196,883?
Simon has tried to explain the answer to me, and a little has sunk in. In a certain sense, people live and work in more than three dimensions all the time.
We taste food in five dimensions: sweet, sour, salty, bitter and umami. These are not spatial dimensions, of course, but they are still dimensions of a sort. To appreciate the full savor of Mackerel Norton, Simon’s tastebuds therefore have to evaluate this mush in five distinct ways that form the basis of each tasting, from the “salt” of the headless fish, to the “umami” (one of Simon’s favorites) of the monosodium glutamate in boil-the-Chinaman-in-the-bag packet rice, to the “sweet” of the flecks of freeze-dried red peppers. A complete flavor description requires five variables, which is the same as saying it exists in five taste dimensions.
We are sensual in ten dimensions. There are the five senses everybody knows about (taste, smell, sight, touch, hearing); but if you strip a human apart and study its cell structure, it turns out we have at least five other senses as well: a sense dedicated to pain, another specifically attuned to balance, a third for joint motion and acceleration, a fourth that focuses on temperature differences, and the fifth newly discovered type of sensory cell is devoted to detecting time.
A complete sensual description of Simon in the act of eating Mackerel Norton therefore needs at least fourteen dimensions (five of taste, plus the nine other senses), because each sense will have a value during the masticating process, and only by taking them all into account can the biographer reach full understanding of Simon’s mesmeric joy.
This doesn’t explain what it means to say an object exists in 196,883 spatial dimensions, but it starts to give the mystery a little structure. The idea that, at least in non-spatial terms, I calmly deal every day with situations taking place in four, five or ten dimensions lessens the quaking in my shoes.
For the last thirty years Simon has (in 196,883 dimensions) chased the object that has Monster Group symmetries in and out of learned journals, across campuses in England, the United States, Canada, Germany, up whiteboards, down blackboards, jumping at it over seats at international conferences, shouting at it on the phone, pestering it around his computer keyboard and down Internet wires, into cyberspace. Even in bed Simon doesn’t relax: he bounds after the terrifying thing through symbol-speckled dreams.
When Simon does not come home at night, it is because he has not gone to sleep at all. He has felt the tremor of the Monster’s shadow outside his office door and jumped up to sprint after it down the Mathematics Faculty’s miles of neon-lit corridors.
Or it is because he is on a bus to Pratts Bottom, not thinking about it at all.
Each time Simon snatches a piece of this prey—a clip of whisker, a splice of scale, a stray slash of mathematical eyebrow—he marks up the result in pinched handwriting that looks as if ants have crept under the surface of the page. He believes that exposing what object/objects have the Monster Group’s Table of Symmetries will be among the most important discoveries of modern times.
Here’s another way to illustrate Simon’s puzzle with the Monster:
Simon has never heard of the man who discovered the Four Rules of Groups. These are four very simple descriptions of what every Group Table must possess in order to be a Group Table. Even a person who has failed every level of school mathematics can learn them. They are the mathematical definition of a Group. They are the equivalent of observing that a book must have pages, a spine, and two boards (material irrelevant) to make a cover, or else it is not a book. It is a scroll or a tablet or a manuscript. You’d think that a portrait of the man who first wrote these rules down would be tattooed on the forehead of every Group Theorist in the world.
“He’s not important,” says Simon. “I will do my best to forget his name as soon as possible.”
“Not important! He codified your subject! Imagine a physicist who looked as empty-headed every time you mentioned Newton. Or a philosopher: ‘Aristotle? Who he? Oh, that geezer who married Kennedy’s wife?’”
“Hunnnhh, you mean widow,” corrects Simon.
To me, for Simon not to know the name of the person who chipped the rules of his subject into stone is unthinkable. And it gets worse.
“I don’t think these rules help you to understand Groups. They don’t mean anything to me. Can we change the subject now, please? I don’t like to think about it.”
What Simon means when he says he doesn’t “understand” the four trivial rules of Group Theory is that they give him no sense for the potential and feel of the subject. When he gets excited about these themes, essential to every good mathematician, I see Simon as the one-year-old again, playing with his pink and blue bricks, plucking patterns out of chaos.
To understand why Groups exist, what their beauty is, and how they can be turned into Monsters, forget rules, cries Simon, concentrate on your senses. Think of the subject aesthetically, develop empathy for it, use your intuition. All the touchy-feely language we would emp
loy to characterize a good artist, Simon uses to describe good mathematical ability.
Discussing the four simple rules of Group Theory is, to Simon’s mind, a waste of time, because they have no interpretative element to them. All they do is tell you something tiresome and mechanical: any Group Table that satisfies these four simple rules is a Group.
If so, bingo! The elements of such a table will form a Group and therefore describe a type of symmetry. But what type of symmetry? What kind of object has this symmetrical set of operations? The rules don’t tell you that.
And this is Simon’s difficulty with the Monster Group.
He has somehow acquired the Group Table, and can prove it is a Group because it satisfies the four simple rules. But he can’t figure out what object or objects might have this set of symmetries. It is as though he’s watching a complex swirl of light made by something dancing madly with a fluorescent tube in a blackened room. He can document all the twists and leaps the fluorescent tube makes, and therefore work out what the object holding it can and cannot do. But ask him to describe what this invisible contortionist is, and he looks as blank as a lake of cream.
(“I’m not sure I want this comparison to cream. Cream is one of the foods I strongly dislike. I also dislike slicing bread.”)
How can that be? He’s got
808,017,424,794,512,875,886,459,904,961,710,
757,005,754,368,000,000,000
facts about the Monster’s basic symmetrical properties, and
808,017,424,794,512,875,886,459,904,961,710,
757,005,754,368,000,000,000
multiplied by
808,017,424,794,512,875,886,459,904,961,710,
757,005,754,368,000,000,000
= God-only-knows-what-number
details about how these properties interact with each other (i.e., its Group Table).
How many more pieces of information does the greedy man need?
Perhaps just one.
32 Atlas
What do you mean, my genius vanished? That’s the first I heard of it.
Simon
The office where they worked was called Atlantis.
Magnolia paint frothed damply on the brick wall; the middle of the room was punctured by a pillar; the window frames were made of metal and shook in the wind. Atlantis was housed on the second floor (or was it the third? Simon’s uncertain) in a converted book warehouse. From the south-west corner of this building you could just see down to the river and punts, and, in a slice of park, girls stretching themselves along the grass, looking firmly away from mathematics.
Conway, Curtis, Parker, Wilson and Simon: Atlantis threatened at any moment to sink under the weight of paper these five mathematicians generated. Articles, books, abstracts, treatises, napkins, paper tablecloths, backs of envelopes, torn-off corners of cardboard boxes, reams of sprocket-holed computer printouts—they soused the floor, suffocated the three office tables, bubbled up the window, drenched the door. Every day, more paper poured in—postcards from India, summaries from Novosibirsk, cuttings and marginalia from Honolulu, MIT, UCL, NYU, Beijing, St. Petersburg, a carbon-copy manuscript from the University of Birmingham (always good for mathematics), pensées from São Paulo, scribbled notes of phone calls to Rome.
Conway, Curtis, Parker, Wilson and Simon were producing an almanac of Groups without normal Subgroups. Conway had had the idea in 1970: to gather together all known information about the different atoms of symmetry—a book of foundational wisdom. An atlas of symmetry.
It would take him, he thought, until 1973.
Conway would have to confirm every piece of known information, fill in thousands of gaps in the record, and dismiss all idiocies that had crept in when he hadn’t been in charge.
1974, at the outside.
In the end the project covered a fraction of the original idea, involved hundreds of mathematicians sending in contributions from around the world, and took fifteen years.
Conway, Curtis, Parker, Wilson and Simon were so different in character that nothing except mathematics could have kept them sitting together in that dreary, oppressive room.
Richard Borcherds, winner of the most prestigious award in mathematics, the Fields Medal, remembers standing in the faculty tearoom one day, discussing with another mathematician an ornate calculation he’d been working on for several days. At one point, Simon knocked past and overheard the conversation. Before he was out of earshot, he’d solved the problem and called back the answer over his shoulder.
Among a select group of mathematicians, Simon Phillips Norton’s status as a solver of long calculations of filigree delicacy is mythological. This is not the same as the conjuring tricks you read about in the papers, in which a schoolboy splurts out the answer to 987,654,678 × 770,645,321 in two seconds, or recites π to 30,000 places before you can tie your shoelaces. Those are unimportant, mechanical skills. Simon’s genius also isn’t the wild, Picasso-like brilliance of the world’s greatest mathematicians, who bolt together ideas that no one had previously imagined could be united. His ability is a precise, rigidly circumscribed, top-hat-and-cravat sort of genius. It’s a Nicholas Hilliard, exquisite miniaturist talent, lying a long way from the paint-by-numbers clunking of the π chanters, but still at an angle to the chaos of frontier mathematics. It’s as though Simon shares his mind with a god who has a passion for making lace.
Solutions frequently appear to Simon without thought or questions about correctness. They appear—insofar as he can explain it at all—in the same way that hunger or lust or revulsion might appear to the rest of us.
Curtis, Wilson and Parker are all good mathematicians, but not in the same league.
Conway was Simon’s equal—and almost his opposite. They have contradictory temperaments. Conway is ebullient, teasing, garrulous, full of joyful and moving anecdotes, effortlessly eloquent, with a fine poetic sensibility (it was he who came up with the name Monster), and as arrogant, and eager with women, as a peacock.
Larissa Queen, a mathematician from Volgograd who became Conway’s second wife, remembers Simon for his modesty, good looks…
“Good looks!” I exclaim.
“Very good-looking. Several of the wives commented on it.”
…and total lack of sexuality. He always had “this enigmatic expression which I described as ‘I know, but can you guess?’”
I just smiled back, and that was it! I quickly realised that just because he smiles it doesn’t mean that we are going to have a conversation. There was a certain gentleness about him which I could describe only as intellectual kindness. He didn’t make you feel stupid, and he didn’t ever make you aware that you are wrong. He would just, kind of, you know, not do anything.
And another thing I remember! Whenever somebody entered the room he’d say, “Here comes my mother!” It didn’t matter whether it was a man or woman, and it was the Maths Department, so it was usually a man. You would hear several times a day: “Here comes my mother.”
“Simon, OK already!” people would say. “I’ve had enough, could you please shut up!” Oh, he said it loud enough for a group of ten people around him to hear…You could be as odd as you wanted to be in that Maths Department, and some people sometimes matched Simon’s oddness, but he was the most consistently, reliably odd. He maintained his permission to be odd.
And some people were not odd, and that was OK too.
Larissa often used to play Simon at the Child’s Game, a form of backgammon Simon had introduced. “More difficult, but with simpler rules,”
and very bright people, very good strategists could never beat him. He told me that he used to play it with his mother. I really got hooked, and practised and practised. I was determined to beat him, because for years he had this reputation in the Maths Faculty that nobody could beat him. At last I did beat him. And, you know, he reacted with joy. After that, he often suggested we play. And every time I beat him and he lost, he reacted with joy.
“I tried Nola’s hymen!”
/> Another favorite faculty game played in the Common Room was making up anagrams, especially for Cambridge mathematicians. Nola’s defloration was an anagram of “Miles Anthony Reid,” now a Professor at Warwick.
“Oil thy rim, Neasden!” Parker shouted.
“Slime ride, if you miss off the ‘Anthony,’” exclaimed Conway.
Simon was by far the best in the department at this sort of thing. His answers were immediate, relevant, and you didn’t have to glance around to see if there were any children in the room before he said them. Miles Anthony Reid was a geometer; he had recently been to Russia.
“Earthly dimension!” brayed Simon, shaking his glass of lemon squash, because he refused to drink tea or coffee. “Lenin made history,” he added a second later.
Another of Simon’s anagram triumphs, which Conway still remembers with amazement, was for “phoneboxes,” which Simon solved “as if with no thought” before you can reach the answer given in this footnote.1
Simon’s attitude to mathematical problems was the same as it was to board games—what delighted him was the clever defeat of a puzzle. He was never interested in who got the credit for it:
He had the same smile when he came up with some brilliant solution. People would be working on a problem and he would just say a number, some long number. And people would continue talking, and maybe two or three hours later they would realize that this number explained the phenomenon they’d been puzzling over. Sometimes it would be hours later. They suddenly would realize what Simon meant—this number which is very brilliant and requires several very deep steps to arrive at—a whole sequence of extremely complicated thinking that other people who are the best mathematicians in the world and paid a lot of struggle and several hours to arrive at—and Simon just said it. And that would be it! He would never say “I told you so,” or “Aha!” He would have this joy of recognition; he would be silently pleased. And if without reason they didn’t remember what he’d said, he might repeat it. And add an additional clue. And the clue could be something else really deep—or it could be just, “Here comes my mother!”
Simon Page 18