“Simon, your brother told me about David Elton, who you used to play with as a child in the summer holidays, when your family went to the coast, where you went swimming. Did you know that he became a murderer?”
But Simon has a knack of treating queries as if they are the end of a conversation. His brain simply substitutes a period in place of the question mark, and his face goes on smiling with no adjustment.
He sat back on the tour-boat bench with a sigh of happiness.
“With a champagne bottle?” I pestered. “His wife: he bludgeoned her over the head, then drowned himself. Simon!”
Simon’s nose was above the porthole again. A small village rippled past, barely more than a tarnish of houses among the waterside rocks. Against one of the corrugated-iron walls, a baked moose skull, stuck on spikes.
“What does it feel like,” I continued eagerly, “to have gone swimming with a murderer?”
“Aaaaah…I didn’t like swimming.”
Puffin Island is a motorway diner for birds. The sea serves up huge fish suppers all summer long to thirty-five different types of seabird that react by screaming and fighting and killing each other as they jostle for table space and turn the rocks white with excrement. The puffins are so tubby that they land on water by bumping into little waves, bouncing their plump breasts from crest to crest until they stop, and sink.
On the way back from Puffin Island, an awful, high-pitched noise came from behind my left ear. I turned to see Simon, bleating foully—“humming,” he pronounced it.
“Beethoven’s Sonata No. 19. A prime number!” he declared joyously.
The ecstasy of the birds had driven him to music.
Night above the Arctic Circle is not like ordinary daylight. It is autumnal light, a bit egg-yolk-colored, but also gray. Warm, cushioned, soft, it makes you feel slightly sick as you move north up the coast; but each night I tried to stay up on the boat, to make sure the sun truly didn’t ever pop off for a snooze, I fell asleep before the miracle happened: somehow, between 4 a.m. and 6 a.m., the day is replenished with freshness. Strawberries and cherries grow in north Norway, dense with the sort of sweetness you normally read about only in children’s books, force-fed by twenty-four hours of sun. In winter, said a banker from Tromsø, the day isn’t really quite black either, but blue as if just drowned.
Once, 2 a.m., the ship hit a storm. The prow tossed up and belly-flopped into the waves. The rain slashed in under the overhanging lifeboats, then appeared suddenly to be defeated, leaving a narrow sheltered section beneath their keels, under one of which I stood on deck, feet spread against the pitching of the sea, and drank my first can of beer of the morning in a state of pleasant gloominess.
“Opening only by authorised personnel!” read a sign on a gate in the deck railing. “Danger of life!” It unbolted onto the Beowulf waves.
Three years ago, the cook on one of these ships sneaked out, dropped a small blow-up lifeboat into the sea, clambered over this sign and vanished off to the islands with one of the waitresses. I understood his sentiments. It took the police fifteen weeks to find them, radiant among the moss and puffins.
The next morning—or perhaps it was evening—it might even have been the middle of the night—there was an important bridge coming: a famous wonder of the North, soaring like a ghostly arc of paint high above the water. But Simon (standing on the deck) had lost his place on the map: he had to find where the bridge was on it before he could do anything else. For a while I watched him desperately trying to outwit the wind that rushed frigidly down the walkways of the boat and buffeted the page.
“Oh dear, oh dear!”
“Quick, Simon, quick!” I goaded. “You’re going to miss it!”
A gust again snapped the map from his grip; he bounced comically about the deck grabbing at the air to retrieve it, even as the shadow of this gloriously precise creation of bridge engineering—a mathematical astonishment—clutched on to the prow and started relentlessly to slide down the sides of the hull.
High above, on the top of the arc, I could see a small crowd gathered. We didn’t wave to each other. We were just humans, protected by roughly equal masses of steel and concrete in the silly vastness of the North—one set of us suspended in the air, the other, on water—neither where humans were supposed to be.
A quarter of a mile farther on, Simon at last got his paper under control, pinpointed the spot, “Stawr…sigh…sun-det Bridge,” breathed a sigh of relief and looked back.
“It’s…really good!” he barked, and tittered with merriment.
Simon has two further points to make about his brilliance.
The first is that everyone is mistaken—he never was a great brain, just a very quick one. He reached the peak of his ability, lickety-split, in ten years. At five, he could do the mathematics of a twelve-year-old; at twelve, he was reading university textbooks on complex numbers; at fifteen, he was better than many university research students; by twenty, the equal of a professor, only his reading was not as broad. Then Simon’s brain stopped developing. The alchemical process fizzled out. Other people began to catch up. They wouldn’t reach him for a further five years, but what they call his “loss of genius” is actually their arrival. So used to thinking of Simon as miles ahead, a speck dancing on the distant mathematical horizon, they didn’t know how to appreciate him when they were finally alongside, celebrating the subject as equals. They mistook equality for Simon’s decline, and declared that he’d suffered a catastrophic intellectual failure.
What else, argues Simon, could explain the fact that despite his infamous “collapse” he is doing math today that is as good as, if not better than, any he has ever done before? Witness his performance in Montreal. Witness his great discovery of “the appearance of Conway Group in the projective plane presentation of the Monster” (Simon: “I don’t think I can make it any more comprehensible to your readers than that”), done two years after Conway had left for America, long after Simon’s supposed “first mistake” in the Atlantis office. Witness his paper on socks.
Simon’s second explanation of his loss of mathematical direction is heartbreaking. Now that Conway has fled to America, there is no one in the mathematical world who will work with him. They say he is too peculiar, too shabby, too old. His interests are fixed in mathematics that has had its day. His brilliance is frigid. His talent, perfectly suited to an extraordinary moment in algebraic history (the symmetry work at Cambridge during the 1970s and 1980s), is out of fashion.
“I sometimes think that I would not have been capable of doing outstanding work in any field other than what I worked in. In due course I had worked out the field which I was expert in, and the cast of my mind was not amenable to diversifying.”
He is, somehow, both too meticulous and too playful for modern theoreticians.
Too meticulous, because he is like a man at the airport check-in desk, madly searching through his luggage for his ticket long after everyone else has boarded the plane: Simon continues to be obsessed with exposing errors and patterns in the Group Table of the Monster, believing that the secret of the universe is hidden there, even though everyone else has taken the hunt elsewhere.
Too playful, because…well…because he is equally likely to write a paper about socks.
At 3 a.m. I left Simon on deck, in a state of vertical trance: joyous, windswept and in possession of all he saw.
Simon is not heartbroken.
At least, not about anything to do with his life in mathematics.
About the state of our public-transport services, it is a different matter:
I’d say that you ought to treat me as if I was currently watching the great love of my life being slowly murdered, torn between my desire to save her and expose her murderers and my wish to spend as much time as I can with her while she’s still alive.
35 Moonshine
With the Moonshine Conjectures, which Simon played such a large part in formulating, it was as though Simon opened the door between
two very strange areas of maths but decided not to go through it.
Umar Salam
* * *
The symmetries of Triangle, and the way they interact with each other, can be written out as a Group Table with three columns and three rows:
The symmetries of Square have a Group Table with four columns and four rows. The symmetries of a garbage-bag-being-kicked-about-with-three-pieces-of-rubbish-inside result in the six-by-six Garbage Bag Group Table. But what is the monster whose symmetries are documented in the Monster Group Table, with its
808,017,424,794,512,875,886,459,904,961,710,757,005,754,
368,000,000,000
columns and
808,017,424,794,512,875,886,459,904,961,710,757,005,754,
368,000,000,000 rows?
Simon doesn’t know.
For reasons inexplicable to biographers, he knows that it lives in 196,883 dimensions.
But he can’t explain why.
In 1979, Simon and Professor Conway discovered another thing about this mysterious monstrous object. The result was so shocking, so astounding, so contrary to what anybody expected, that they called the discovery Monstrous Moonshine.
* * *
But Simon can’t explain what Monstrous Moonshine is either.
I think how it goes is like this. In 1979, John McKay, a mathematician in Canada, whose office is so messy that he once lost his baby son among the papers, discovered a remarkable coincidence. The set of numbers that explains why any object that has the Monster Group of symmetries must live in at least 196,883 dimensions cropped up in another, entirely unrelated, area of mathematics.
“Huugh, aaah, no. Alex, you’ve got that not quite right. What McKay discovered was that the Fourier expansion of the j-function, j(tau), where tau is the half period ratio, has as its first coefficient 196,884…”
“196,884,” I point out with admirable speed. “Not 196,883, then?”
“Exactly. That’s the point. 196,883 + 1 = 196,884, so that shows that in 196,883 dimensions…”
John McKay sent Conway a postcard explaining his discovery. Fortunately for Conway, when the card arrived Simon was away on a two-week jaunt around the railways of England. By the time Simon got back, Conway had made significant progress toward explaining the astonishing coincidences, otherwise the Monstrous Moonshine Conjectures might have been entirely Simon’s work.
“Hnnnnhh, no. That’s not correct either. No one has explained it satisfactorily! What Conway and I did, as I say…aaah, let me see…perhaps…I should begin by explaining what hyperbolic geometry is. If, uuugh, I put this map on the ground to represent aaaah, huunh…Euclidean geometry…”
“Thank God I had that two-week head start on Simon,” Conway is quoted as saying in Finding Moonshine, Marcus du Sautoy’s book on the subject, “else I wouldn’t have got a look-in!”
For me, the familiar panic sets in at this point, and instead of having the sense to make Simon go back over anything I don’t understand, I narrow my eyes, try to look as if I’m savoring each new theoretical observation with the enjoyment of a sage, and feel the earth slip away.
“I think of myself,” says Simon proudly, “as a fixer.”
His campaigning letters to the local member of Parliament, badgering of government ministers about timetabling, demonstrations in Cambridge concerning the new guided-bus scheme, protest meetings about traffic-easing schemes for cars on the A14 that could be just as readily achieved by providing three wiggly buses from Cambourne plus a double-decker to Six Mile Bottom—in all these cases, Simon sees his role as that of a person who spots flaws in the way public-transport numbers and schedules have been totted up, and “brings them to the attention of the best people available to have them corrected.”
“You could also say I am a hub,” Simon adds. “I pass the information out along the spokes. The people who make the actual changes and see that the machine works better as a consequence of what I have shown them are the rim of the wheel. What you call my jaunts are part of this campaigning, because how else can I report on what is going on, except by experiencing it myself?”
“And with the Monster and Monstrous Moonshine…? Is that what you think of yourself doing there too? You are still going over all the facts and calculations involved, spotting errors in the Group Table and reporting them to other mathematicians, being the fixer? You have, so to speak, not stepped through the door and got on with further profound mathematics, because you got distracted by the door frame?”
“Yes, aaah, huunnh, I suppose so. As I say, I never understand you when you are philosophical. I don’t think you make sense. Incidentally, when I went to buy a new bag and hiking boots, the shop I usually use was a hole in the ground. But I have not thrown out the old pair. I’m keeping them to hand, in case a nail comes up.”
“Thank you, Simon.”
Sitting in our cabin one night on our trip to Lapland, I tried another approach to get the subject back to understanding the Monster and Moonshine. I started talking about primes. Simple Groups such as the Monster—these atoms of symmetry—are often compared to primes because primes are numbers not divisible by any other number except one and themselves: primes are the “atoms” or building blocks of numbers. 2, 3, 5, 7, 11, 13…are all primes. 4, 6, 8, 9, 10, 12…are not, because each is divisible by other numbers. E.g., 12 can be divided by 2, 3, 4 and 6. Every number in the infinite universe of numbers can be reached by multiplying primes together, but it can also be easily shown that there are an infinite number of primes…
“Not necessarily,” interrupted Simon. “Not if it’s a pring.”
“What’s a pring?”
“A ring. I invented prings.”
“A pring is a ring?”
Simon nodded. “With power.”
“So what’s a ring?”
“Uuugh, let’s see…if you have an, uugh…” he began, with the same bright-eyed optimism with which he starts all his attempts at mathematical explanation with me, although for some reason the hesitancy had come back. “No, don’t interrupt, the quickest way to make progress with this is for you not to talk…if you have a mapping from R cross R into R, in the case of, a multiplicative and…no, I mean, given a binary operation on an algebraic structure that is a homomorphism…”
When Simon was awarded a mathematical fellowship at Cambridge it was (according to rumor) specifically added into the job description that he could have the post only if he promised never to teach.
This much I do know: Monstrous Moonshine links the Monster to distant mathematics and the structure of space in ways that are as awe-inspiring to a man like Simon as it would be to an astronaut to step out of his space machine on Jupiter, and find a Sainsbury’s bag floating past. That’s why it’s called “Moonshine,” because mathematicians can even now hardly believe it.
“I think,” said Simon, standing up from his berth and shaking crumbs and clotted blobs of oil and fish off his T-shirt onto the covers, “I can explain to you what Moonshine is in one sentence.”
When he really tries, Simon can be a model of clarity.
“It is,” he said, “the voice of God.”
* * *
Gold flakes bouncing in thick liquid are stars. Ultraviolet light makes blue shadows glow moonishly. Flashes of ginger and green are asteroids.
To ease the pain of another day of failing to understand Simon’s Moonshine Conjectures, I have invented a cocktail called Monstrous Moonshine. Two swigs induce a state of contented idiocy that’s similar to the look Simon has when he’s back from a week on the buses.
Monstrous Moonshine contains, in whatever proportions suit the torment of your mood (the given quantities below are a starting suggestion):
Recipe for Monstrous Moonshine
2 × Absinthe
1 × Blue curaçao
3 × Goldwasser
(“Isn’t that a euphemism for urine?”—Simon)
3 × Soda water
Lime juice, mint leaves and fresh
ginger
A peeled lychee
Dry ice
Mix the first seven ingredients, drop in the lychee, add dry ice (the drink will instantly start to boil with great violence) and drink in ultraviolet light through bared teeth.
Important Note for Historians of Cocktails: The absinthe is to represent moonlight—wormwood, the most important ingredient in absinthe, glows in ultraviolet light. Blue curaçao also glows in UV, and the color suggests night. Goldwasser is the oldest liqueur in the world, invented in 1598, and tastes disgusting. You’d think that after 400 years of tinkering Goldwasser would be drinkable, but it isn’t. However, it contains flecks of real gold, which represent the stars. Lime juice is to offset the filthy taste of the Goldwasser.
The other trouble with Goldwasser is that the flakes don’t stay afloat. They sink to the bottom. Soda water makes them bounce about.
The lychee is, of course, the moon.
Dry ice is for monstrosity. I get mine from a Professor of Chemistry. You mustn’t use dry ice that comes from a nightclub machine. That’s toxic. The type you want comes in small, hazy pellets and will freeze a hole through your stomach if you swallow it.
* * *
36 Discovery!
Simon is quite likely to be the person who understands the Monster in the end. Why? Because Simon’s Simon.
Professor John Conway
I’ve seen a mathematical discovery! I witnessed it start! 6:37 a.m., at the Hurtigruten breakfast buffet counter, Deck Three.