The Sciences

Mathemagician

(Original title: Impressions of Conway)

          "Have I done this to you yet?" He grabbed my hand and held it out in front of him, palm down. Before I could react, he pulled a rubber stamp out of his pocket, and my hand suddenly was emblazoned with big red letters. "John H. Conway's Seal of Grudging Approval." Within seconds, it had smeared to three red lines that wouldn't wash off for several days. Still grasping my hand, he pulled me toward his office. Brightly colored polyhedra hung in disarray from a network of strings dangling from the ceiling. The dim outline of a computer terminal was visible through a pile of Rubik's cubes and wooden toroids. "We'll be better off in the undergraduate lounge. The doctor says I should rest, and I can lie down over there." He pulled me across the hall, into a room where tinkertoys buried the shelves and tables but didn't cover the chairs. He lay down on a sofa, crossed his legs, and put his hands behind his head. "You didn't comment on my shirt," he said. It depicted a boldly colored Escher print, where fishes transformed into birds, then boats, back to fish again, and finally into horses before disappearing under his belt. He didn't wait for a reply. "Well, off we go! I was born in Liverpool, England. Do you want to know how old I am?"
    "If you don't mind," I said, desperately trying to keep up.
    "I was born on December the twenty-sixth, 1937. No, I don't mind. I actually did mind once ... I'll tell you about it some time if you like."
    "O.K. Do you ..."
    "When I was first married, my first wife was seven years older than I was," he continued. "She was very, very much worried by this. I'm afraid I wasn't particularly sympathetic. Every time she passed a decade, she would be terribly, terribly depressed. And then a few years ago when I turned fifty - oh, my God, it was absolutely awful!" He grinned, slightly embarrassed. "I suddenly had a wave of sympathy for her."

          As Conway stroked his bushy beard, he resembled the unemployed younger brother of Santa Claus. His hair and clothing were in disarray, yet he looked distinguished. It was an odd combination, yet perfectly suited to a mathematician.

          "Anyhow, I enjoyed living in England, but it was really hard growing up during wartime. I remember I saw a banana, once." It was a really small one, and it wasn't yet ripe. Every day, Conway would check to see whether it was ready to be eaten. "And then the great day came. My mother divided it, and I got a piece - and I didn't like it very much, after all that fuss." Once, when he was five or six, he went to a birthday party, and each guest received a balloon. "Of course I had seen one, but I had never actually owned one." Conway frowned. "When it burst two days later, I was absolutely inconsolable." His parents talked to the family who gave him the balloon, but they didn't have any more. There were none to be found anywhere, due to the wartime shortage of materials. "Times were hard. My father was almost a teacher of Chemistry, he was actually a laboratory assistant officially. Mathematics wasn't his specialty, but he certainly knew about it."

          Conway reached into his pocket and pulled out a well-worn box of cards. "Have I shown you this one?" He took the pack out of the box and started pulling cards off of the top and putting them at the bottom. "A, C, E," he said, pulling off one card per letter. He then flipped over the next card, which happened to be the ace. "T, W, O," he continued, then flipped over the two. "Would you like to try?" He handed the deck to me. "T, H, R, E, E," I said, then flipped over the next card. The joker. "No, silly," he cried. "Here's how you do it: T, H, R, E, E." He flipped over the three. "Try again." "F, O, U, R." Joker. The deck was passed back and forth - I always got jokers while he got the correct card. At last, he finished. Carefully rearranging the deck into the necessary order for the trick, he put it back in his pocket, ready for the next victim.

          "Math was always there for me," Conway said, settling back on the sofa. "My mother said that when I was three or four, I knew the powers of two. When I was about eleven, I changed schools, as do all English children. I remember that at an interview for my new school, I told the interviewer that I wanted to be a mathematician at Cambridge. I don't remember what could have sparked such interest." Conway was at the top of the class in almost everything, until he reached high school, when it ceased. However, he was always the best student in math. "I liked math above all other subjects, but I used to go through phases. I still do. I was very, very interested in fossils for a time, then I liked spiders. But most of all, I loved astronomy." Suddenly, he sat up and leaned toward me. "Do you know that two years ago I became interested in astronomy again? I learned the names of all the visible stars in the northern hemisphere. Here's Orion." Jabbing his finger in the air to denote the location of each star, he recited all of their names. "... Betelgeuse, Mintaka, and Saiph. I like knowing things. When a constellation is covered up by a cloud, I predict where the stars will be when the cloud moves away. I really get a feeling of power - knowing what will happen is almost like making it happen."

          A graduate student wandered into the lounge. "Nice shirt!" Conway looked at his belly. "Do you really like it?" He leaped off of the couch. "I'll be back in a moment." He stepped outside with the student, rummaged around in his hopelessly cluttered office, and finally handed over a pile of papers. "That's going to my book, you see," he said, grinning. "Well, not really mine. This person decided to write up some of my lectures, and I've finally gotten around to approving his writeups. That's what this silly stamp is for," he said, pulling it from his pocket and waving it. I hid my hands behind my back. "The book will be called 'The Sensual Form,' as I describe how to 'see' a quadratic form - a second- degree polynomial - and 'hear' and 'feel' it, after a fashion. Now, where were we ... oh, yes, astronomy. Oh! Have I shown you this yet? Name a day."
    "Tomorrow."
    "Waning gibbous. Now give me a harder one." I chose another day, and within five seconds, he told me the phase of the moon on that day. "It's a very simple formula, actually, but it looks really impressive. I just like knowing these things, and showing off. I like to impress people." He smiled. "You know, Cambridge had these lovely, lovely gardens. That's one of the things I miss most about England. And I used to walk through the gardens and look at the flowers - and I knew why the petals of the daffodil were arranged the way they were." Conway stroked his beard as he thought. "When I was a student at Cambridge, I used to go to a small coffee shop and do anagram crosswords. One day a procession came in, and one woman was carrying a fake daffodil - and it only had five petals. There have to be six! It made me rather annoyed; I guess that it shouldn't have." He smiled. "You know, there is a beauty in nature that is too subtle for man. It really bothers me that when artists paint a pineapple, they always make the lines symmetric. They aren't symmetric, and that's the real beauty of the thing. The asymmetry is related to the Fibonacci sequence - there might be eight grooves in one direction, and thirteen in the other, so the lines don't meet symmetrically. And artists ignore this. Another thing that nobody notices is a brick wall." Six months ago, he had gone through a 'brick wall' phase, which culminated in his giving a lecture called 'How to stare at a brick wall.' He described, in detail, the enormous variety of intricate patterns that can be found in brick walls. "And nobody looks at these damn things!"

          With no warning, Conway leaned forward and whipped out his pack of cards again. "Numbers or suits," he asked me. "Suits," I replied. After shuffling the deck a few times, he flipped four cards off of the top. One was from each suit. He went through the deck, flipping over the cards, in groups of four, and each time, one was from each suit. He gathered up the cards, carefully preserving the order. "Now, numbers." He counted off twelve cards. "Stop! What's missing?" He looked at the flipped cards, and noticed that ace through queen were all there. "This one must be the king." He turned over the next card from the deck; "Behold! The king."

          Putting the cards back in his pocket, Conway sank back into the sofa. "Game theory wasn't my first specialty, you know. In Cambridge, I got started in number theory." His advisor, Harold Davenport, was an eminent number theorist. "He gave me a very difficult problem - proving a conjecture that said every integer can be written as the sum of thirty seven numbers, each raised to the fifth power. When I told him that I had solved it, he didn't believe me." But the proof was correct. After carefully going through Conway's solution, and finding no major flaws, Davenport said, "Well, now, Mr. Conway, what we have here is a poor Ph.D. thesis." Though the comment annoyed Conway at first, he realized that it meant that he was free to do whatever he liked. He went on to study set theory, especially transfinite numbers.

          If you ask most people about infinity, they will probably respond something like, "Well, you can count one, two, three - just keep going forever, and that's infinity." In the realm of transfinite numbers, that is only the lowest level of infinity, dubbed w (the Greek letter omega.) Conway explained, "Omega is a transfinite number. These numbers have some very bizarre properties. For instance, w + 1 does not equal 1 + w; in fact, 1 + w equals w. In the same way, 2w also equals w, while w2 does not. So you can see that adding and multiplying transfinite numbers together can give you some very odd results - but they are consistent, and they are very, very interesting." The mathematics of adding and multiplying numbers greater than infinity is very strange, but there is a beauty to it. Instead of counting one, two, three ... and finally getting stuck at infinity, you can keep going -- w + 1, w + 2, up to w2, w3, then go to w2, w3, and keep going. There is no reason to stop - you can go to omega to the omega power, omega to the omega to the omega power, and so on. When you have done that omega times, just add one, and you have a new creature. There are infinite infinities in the realm of transfinite numbers.

          Conway stared at the ceiling. "I didn't stay in set theory, however." Upon graduation, Conway got a job at Cambridge as a mathematical logician. "In my late twenties, I became very depressed. I felt that I wasn't doing real mathematics; I hadn't published, and I was feeling very guilty because of that." He was studying the symmetries of a certain lattice, and all of a sudden Conway stumbled upon a very large group which nobody had seen before. A group is a set (like the integers) associated with an operation (like addition), which follows certain very restrictive rules. Finding a new group was so rare and important that Conway was made a Fellow of the Royal Society for his discovery. "You know, when they make you a Fellow, you get to sign a book which has the signatures of all members of the Royal Society - since 1660, when it was founded. It was a great thrill to flip through the pages and see the signatures of I. Newton, C. Wren, and A. Einstein. And I got to sign J. Conway in that same book. It gave me a warm feeling."

          "Now what was I going to say?" he asked abruptly, stroking his beard. "I have a very odd sort of memory. I can remember the most useless, obscure details, but when it comes to things that other people think are important, I can't recall them for the life of me. When I was at Cambridge, I never learned the names of some of my colleagues - and I worked with them for twenty years!" Conway smiled a bit, embarrassed. "But I must have good memory. I know pi to one thousand places." He started rattling off digits as fast as he could catch his breath. "3.1415 92653 58979 32384 62643 38327 95028..." I joined in. "... 84197 16939 93751 05820 ..." He stopped abruptly. "How many digits do you know?" he asked, surprised. "Oh, only about seventy- five. I had nothing else to do in computer science class, so I tried to memorize pi." Conway nodded. "I had a similar experience. When I was an undergraduate, I had a summer job at a biscuit factory. I had to clean the ceiling of the oven room - and it was completely black with soot. We worked on a scaffolding which was fifty feet high, and we scrubbed the ceiling. It soon became apparent that it was futile; after an hour of hard work, the ceiling changed color from a black matte to a black matte with a sheen." Conway and his friends would play poker on the scaffolding, and every so often, climb down, move the scaffolding a few feet, climb up, and resume their game. Soon he tired of poker, so Conway decided to memorize pi. "I learned it to seven hundred and seven places - which was the extent to which it was known back then. A few years later, I learned it up to one thousand." He sat up on the couch, and fluffed up the pillow behind his head. "I convinced my wife to learn it, too. In fact, every Sunday, we took a romantic little walk to Grantchester, a lovely, lovely little town near Cambridge, and we ate lunch at a pub there. We would stroll along the road, reciting pi to each other; she would do twenty places, then I would do twenty, and so forth."

          Conway paused for a moment, and his bushy eyebrows furrowed. "Yes, I must really have a tremendous memory. As you know, I crossed the Atlantic in 1985 or 1986, and became a professor at Princeton. Several years later, when what's-his-name ... Harold Shapiro became President of the university, he invited some of the faculty to a dinner party each week. There were about eight or ten guests, and Shapiro asked each of us to say a few words about ourselves. I didn't like that one bit. It reminded me of the recitations of poetry we had to do in elementary school. So I recited a little poem about elves and goblins that I learned when I was, oh, about six years old. I hadn't thought about since then, and I was able to recall it at an instant. Well, I wasn't invited to a dinner party again. But I don't worry about that; I guess it looks as if I have an irresponsible attitude. However, to do good work in math, you have to be somewhat irresponsible. I only started doing real mathematics after I found the Conway group. I got a much-needed ego boost - obviously I don't need one anymore. Anyhow, after I made my name, I could do what I like, even if it was totally trivial. When I want to play backgammon instead of doing math, I play backgammon. If the people at Princeton don't feel that they're getting their money's worth out of me, that's their problem. They bought me!"

          After finding his group, Conway continued his work in group theory and published an "atlas" of groups. In fifteen years, Conway and his colleagues collected all the "interesting" groups, classified them, described their properties, and put them into one volume. "The work of producing the book was quite heavy, especially in the last year. It turned me off of group theory and algebra in general. At about the same time I started the atlas, I was trying to come up with a mathematical understanding of the game of Go. There were two very strong players at Cambridge, one of whom was the British champion. I noticed that near the end of a match, the whole game looked like a sum of a lot of little games. And, to my great surprise, I discovered that certain games behaved like numbers." By analyzing the properties of these games, Conway discovered a whole new set of numbers; they were soon dubbed "surreal." "The theory was a real big shock to me. It was bizarre ... crazy, but it was true! It was like climbing to the top of the beanstalk, and there was the enchanted castle. I had no idea what to expect. The rules have all changed, like magic. It's like a new world. Exploring that world took me some time." Soon it was a fully-fledged theory. "You can view it as an extension of the real numbers, but it also has a surreal quality. New numbers were appearing - and nobody had seen them before. But they exist. Old numbers fell straight out of the definition - all the reals, and even the transfinite ordinals like w. By the way, I didn't coin the term 'surreal numbers.' Donald Knuth did, in his book which should be around here." Conway jumped off of the sofa. After steadying himself for a moment, he shuffled toward his office. He looked through the cluttered bookshelves for his copy. He found a trashy-looking novel, Soho Madonna, which sported a scantily clad woman on its cover. "I don't remember buying this." He tossed it aside. "Oh, well. In any case, Knuth's book was in the form of a novel. I was God. There were two main characters, Alice and Bill, and a third character named C speaks from the sky. In the beginning, they find a stone inscribed 'And Conway created numbers.'" After a frantic search, Conway gave up looking for the book, and trundled back to the lounge. "I daydreamed for weeks about these wonderful numbers. I often do that. I have what I call a 'white hot period' which lasts for a few days. I can't sleep, and I am completely absorbed with a problem. This used to get my wife terribly, terribly upset. After the white hot phase, I enter a daydreaming phase for a few weeks." He frowned slightly. "These phases are becoming less and less frequent."

          Conway led me back to the lounge. "I discovered surreal numbers a long time ago. I guess that it was around 1970. At about the same time I invented the game of Life." The roots of the idea originated with the famous mathematician Von Neumann. He wanted to create a model of a "universal constructor," a machine that could build any other machine if correctly programmed; it would even be able to make a copy of itself. He imagined an infinite board ruled into squares, and assigned one of twenty-nine values, or states, to each square. There were a complex set of rules to determine how each square's state affected its neighbors' states. He succeeded in creating a universal constructor, but his system was so complicated that it was very unappealing. Conway changed the rules so that each square had only two states - living or dead. He had only three rules. The first, the "birth rule," was that a dead cell becomes alive if it has three neighbors which are alive. The second, the "isolation rule," was that a live cell dies if fewer than two of its neighbors are alive. The third was the "crowding rule"; a live cell dies if it has four or more live neighbors. "I decided to observe this simple system, rather than build in the universal constructor property like Von Neumann did." Almost anyone who owns a computer is familiar with the game of Life - it has become a popular screen saver, because of the unpredictable and interesting patterns that flicker across the screen as cells are born and die. "You can embed any mathematical question into the game of Life - for instance, with a tremendous amount of effort, you can plug in Fermat's last theorem and see whether the program terminates." He stroked his beard and smiled. "You know, I occupy the John von Neumann chair at Princeton. He was interested in transfinite numbers and game theory. I think it was just a coincidence; I wouldn't have been interested in building bombs. I don't think I would even have liked the man. But our interests do have a great deal of overlap."

          Conway propped up the pillow behind his head and grinned. "I like showing off. When I make a new discovery, and I really like telling people about it. I guess I'm not so much a mathematician as a teacher. In America, kids aren't supposed to like mathematics. It's so sad." Conway sat up suddenly. "Most people think that mathematics is cold. But it's not at all! For me, the whole damn thing is sensual and exciting. I like what it looks like, and I get a hell of a lot more pleasure out of math than most people do out of art!" He relaxed slightly, and he lowered his voice. "I feel like an artist. I like beautiful things - they're there already; man doesn't have to create it. I don't believe in God, but I believe that nature is unbelievably subtle and clever. In physics, for instance, the real answer to a problem is usually so subtle and surprising that it wasn't even considered in the first place. That the speed of light is a constant - impossible! Nobody even thought about it. And quantum mechanics is even worse, but it's so beautiful, and it works!" Conway grinned. "I really do enjoy the beauty of nature - and math is natural. Nobody could have invented the mathematical universe. It was there, waiting to be discovered, and it's crazy, it's bizarre. Math explains why the petals of the daffodil are arranged the way they are, and I think that I get more pleasure by looking at the daffodil than others do because I understand this. I guess I'm a Sybarite. I like beauty, and I like to eat and drink." He patted his belly. "I used to anyway. My heart attack changed that somewhat." There was an awkward pause. Conway stood up and walked toward his office. He pointed to the polyhedra that hung from the ceiling. "I'm getting interested in group theory again. Each of these represents a different type of symmetry, and group theory is really the study of symmetries. Unfortunately, all my groupy friends dispersed, and Princeton is a wasteland for group theory. I'm somewhat interested in two and three dimensional stuff. It's not very serious, I'm afraid."

          I looked around the office, and noticed seven long sheets of paper with footprints on them. "They represent the different types of symmetry in two dimensions," he explained. "One day, I was walking along, trying to think of an example of translation linked with reflection. All of a sudden, I realized that walking was just what I was looking for! I xeroxed my feet, and made up these pictures. Each one represented a type of symmetry. And each symmetry has a polyhedron associated with it." Conway sank down into a chair. "I'm not really doing mathematics right now." He looked at the ceiling. "I guess you can say that I'm expanding it. Instead of trying to prove new theorems, I'm trying to fill in the holes that other people have left behind. I want to have a better understanding of what we already know. I want a more visual, more intuitive feel for math, like my book on the quadratic form." He stopped short, and looked a little stunned. "I guess you can say that I've almost ceased being a mathematician.

          All of a sudden, his eyes lit up. "Oh, I haven't shown you this! Do you have any pennies?" I fumbled through my pockets and came up empty. A graduate student walked by. Conway leapt up and grabbed his arm. "Pennies?" he cried, and the startled student looked through his jacket and came up with six. "I guess that it will have to do. I am going to change your luck forever, for the better." He pulled me toward a table that was only mildly cluttered. "Heads or tails."
    "Uh ... heads."
    "Good. Now help me." He began balancing the pennies on their edges. By the time that I had finally gotten one to stay standing, he had done four. "Come on, come on." He took the last one out of my hands, and with a swift, well practiced motion, he had it balanced on its edge. He made sure that no two were facing in the same direction. "Heads, you said?" I nodded. "Well, here goes!" He knocked the bottom of the table gently, and all six pennies fell. All heads. "Want to see it again?" Without waiting for my answer, he quickly set up the pennies again. "You try it." I tapped the bottom of the table, and they fell. Heads. He smiled. "It's really much more impressive when you have twenty pennies or so. Now, heads or tails?"
    "Tails. But what ..."
    "Wait." He held his finger up. "Help me spin these." In a flash, he had four pennies skittering along the table. "Tails, heads, tails, tails. Come on!" I was finally able to get one up to speed. Heads. He spun four more. "Heads, tails, tails, tails. You know, if you spin pennies this way, roughly two out of three are tails."
    "Why?"
    "It's very simple, really. Balance the penny on the table and look at it very closely. You will notice that it doesn't stand up exactly straight." I didn't notice, but I nodded anyhow. "That's because the sides aren't flat like the edges of a disc. They're angled like a slice of a cone - and the tails side is narrower than the heads side. That's a result of the minting process. So, if you balance them on their edges, they will lean toward the tails side, so when you disturb them gently, they will fall heads up."
    "But when you spin them ..."
    "The center of mass likes to be above the point of contact. If you think about it for a little, you will see that when the penny spins, the tails side is facing up." Conway grinned as he shuffled back to his office. I was impressed.

[Copyright (c) 1994, The Sciences]

[This is actually a pre-edited version, and differs slightly from the published article.]