I stood on the pavement watching the car disappear into the darkness. In my pocket, together with my room key, I had a key to the side door of the Institute and the swipe card for getting into the library out of hours. I decided that it was too early to go to bed, so I walked to the Institute in the yellowish glow of the street lights. The streets were empty; the only signs of movement I saw were in Observatory Street, through the window of a tandoori restaurant: two waiters were placing chairs on tables and a woman in a sari was closing the curtains. St Giles too was deserted, but there were lights in a few windows at the Institute and a couple of cars in the car park. Some mathematicians worked only at night, and others had to come back to check on the running of a long program.
I went upstairs to the library. The lights were on and, as I entered, I heard footsteps-someone was walking quietly among the bookshelves. I went to the History of Mathematics Section, and ran a finger along the titles. One book was jutting out, as if someone had looked at it recently and placed it back carelessly. The books were packed in tightly, so I had to pull it out with both hands. The illustration on the cover showed a pyramid consisting of ten points surrounded by fire. The title-The Pythagorean Brotherhood- was only just out of reach of the flames. From close up, the points were actually small shaven heads, as if they were monks seen from above. So perhaps, rather than being vaguely symbolic of the inflamed passions that geometry could arouse, the flames alluded specifically to the horrific fire which destroyed the sect.
I carried the book to one of the tables and opened it under the lamp. I didn’t have to turn more than a couple of pages. There it was: It had been there all along, in all its overwhelming simplicity. The most ancient and elementary mathematical concepts, not yet quite divested of mysticism. The representation of numbers in the Pythagorean doctrine as the archetypal principles of divine powers. The circle was One, unity in all its perfection, the monad, the beginning of everything, enclosed and complete within its own line. Two was the symbol of multiplicity, of all opposition and duality, of bringing into being. It was formed by intersecting two circles, and the oval-like an almond-enclosed at its centre, was called ‘Vesica Piscis, the belly of the fish. Three, the triad, was the union of two extremes, the possibility of bringing order and harmony to differences. It was the spirit that embraced the mortal and immortal within a single whole.
But also, One was the point, Two was the straight line joining two points, Three was the triangle and, at the same time, the plane. One, two, three: that was all, the series was simply the sequence of natural numbers. I turned the page to examine the symbol for Four. It was the tetraktys, the pyramid of ten points that was on the book’s cover, the emblem and sacred figure of the sect. The ten points were the sum of one, plus two, plus three, plus four. It represented matter and the four elements. The Pythagoreans believed that all of mathematics was encoded in the symbol. It was both three-dimensional space and the music of the celestial spheres, and it contained in rudimentary form the combinatorial numbers of chance and the numbers of the multiplication of life that Fibonacci rediscovered centuries later.
I heard footsteps again, much closer. I looked up and to my surprise saw Podorov, my Russian room-mate, emerge from among the shelves. On seeing me at the table, he approached with an intrigued smile. It was strange how different he looked there, quite at home. I imagined he liked having the library to himself at night. He was holding a cigarette and he tapped it gently on the glass table top before lighting it.
“Yes,” he said, “I come here at night so that I can smoke in peace.”
He gave me a wry but friendly smile and flipped over the book’s cover to see the title. He was unshaven and his eyes were hard and shining.
“Ah, The Pythagorean Brotherhood…This has something to do with the symbols you drew on the board in our office, doesn’t it? The circle, the fish…If I remember rightly, they’re the sect’s first symbolic numbers, aren’t they?” He thought for a moment and recited, as if showing off his memory: “The third one is the triangle, the fourth is the tetraktys.”
I looked at the man, amazed. I realised that Podorov, who’d seen me studying the two symbols on the blackboard, hadn’t even considered that it might be anything other than a strange mathematical problem. Podorov, who obviously knew nothing about the murders, could, all along, have simply got up from his desk and drawn the continuation of the series on the board for me.
“Is it a problem that Arthur Seldom has set you?” he asked. “It was from him that I first heard about these symbols, in a lecture he gave at a conference on Fermat’s last theorem. You know, of course, that Fermat’s theorem is simply an extension of the problem of the Pythagorean triples, the sect’s best-kept secret.”
“When was that?” I asked. “Not recently, surely.”
“No, no, it was years ago,” he said. “So long ago that, as far as I can tell, Seldom doesn’t remember me. Of course, he was already the great Seldom then, while I was just an obscure graduate student from the small Russian town in which the conference was held. I showed him my work on Fermat’s theorem-it was all I thought about at the time-and I asked him to pass it on to the Number Theory group in Cambridge, but they were apparently too busy to read it. Well, not all of them,” he said. “One of Seldom’s students read my work, corrected my faulty English and published it under his own name. He was awarded the Fields Medal for the most important contribution of the decade to solving the problem. Now Wiles is about to take the final step thanks to those theorems. When I wrote to Seldom, he answered that there was an error in my work and that his student had corrected it.” Podorov laughed drily and exhaled a puff of smoke forcefully upwards. “My only mistake,” he said, “was that I wasn’t English.”
I wished I had the power to make him stop talking. I felt again, as I had during my walk in the University Parks, that I was on the point of seeing something and that perhaps, if I were alone, the piece of the puzzle that had eluded me once would fall into place. I got up, murmuring a vague excuse, and quickly filled out a card so that I could borrow the book. I wanted to be outside, far away, in the night, away from everything. I rushed downstairs and, as I was heading out the door, I almost collided with a black-clad figure entering from the car park. It was Seldom, now wearing a raincoat over his dinner jacket. I suddenly realised it was raining.
“You’ll get your book wet,” he said, and held out his hand to see what it was. “So you’ve found it. And I can see from your face that you’ve discovered something else, haven’t you? That’s why I wanted you to try to find it yourself.”
“I bumped into my room-mate, Podorov. He said he met you once years ago.”
“Viktor Podorov, yes. I wonder what he told you. I’d forgotten all about him until Inspector Petersen gave me the list of all the mathematicians at the Institute. I wouldn’t have recognised him anyway: I remember him as a rather troubled young man with a pointed beard, who thought he had a proof of Fermat’s theorem. It was only much later that I remembered that I’d given a talk about Pythagorean numbers at that conference. I didn’t want to mention it to Inspector Petersen. I always felt a little guilty about Podorov. I heard that he tried to commit suicide when my student received the Fields Medal.”
“But it couldn’t have been him, could it?” I asked. “He was here in the library this evening.”
“No, I never really thought it was him, but I knew he was probably the only person who would see immediately how the series continued.”
“Yes,” I said, “he remembered your lecture perfectly.”
We were standing beneath the semicircular awning at the entrance, getting splashed by the rain that was blowing in on gusts of wind.
“Let’s make our way to the pub,” said Seldom.
I followed him, shielding the book from the rain. The pub seemed to be the only place open in all of Oxford. It was full of people talking in booming voices and laughing, with the exhilaration and slightly artificial cheerfulness that the English only seemed to achieve after a lot of beer. We sat at a table, the wood marked with wet rings.
“I’m sorry,” said the landlady from the bar, as if there was nothing she could do for us, “you’ve missed last orders.”
“We can’t stay here long,” said Seldom. “I just wanted to know what you think, now that you know what the series is.”
“It’s much simpler than anything a mathematician would have devised, isn’t it? Maybe that’s what’s ingenious about it, but it’s still a little disappointing. After all, it’s just one, two, three, four, like the series of symmetrical figures you showed me the first day. But maybe it isn’t a kind of puzzle, as we thought, but simply his way of enumerating the murders: first, second, third.”
“Yes,” said Seldom, “that would be the worst-case scenario, because he could go on murdering indefinitely. But I still have hopes that the symbols are the challenge and that he’ll stop if we show him that we know what the series is. Inspector Petersen just called me from his office. He’s got an idea he thinks might be worth trying and apparently he has the psychologist’s approval. He’s changing tack regarding what appears in the papers: he’s going to let the Oxford Times run an article about the third murder on its front page tomorrow, with a picture of the triangle and an interview in which he’ll mention the first two symbols. The interview questions are going to be carefully prepared so as to make Petersen appear baffled by the murders and outwitted by the murderer. According to the psychologist, that’ll provide our man with the sense of triumph he craves.
“The short note about the tetraktys that I wrote for Petersen will appear, with my name, in Thursday’s edition, in the same section in which they published the chapter on serial murders from my book. That should be enough to show him that I know and can predict the symbol for the next murder. That keeps it on the level of the almost personal challenge that he seemed to lay down at the beginning.”
“But supposing it works,” I said, slightly taken aback, “supposing, with luck, he reads your note in Thursday’s paper and that, with a lot more luck, it stops him, how is Inspector Petersen going to catch him?”
“Petersen thinks it’s only a matter of time. I think he’s hoping that eventually a name will emerge from the list of those attending the concert. Anyway, he seems determined to try anything to avoid a fourth murder.”
“The interesting thing is that we’ve now got everything we need to predict the next step. I mean, we’ve got the three symbols, like in one of Frank Kalman’s series, so we should be able to infer something about the fourth murder, to link the tetraktys…but to what? We still don’t know anything about the link between the deaths and the symbols. But I’ve been thinking about what that doctor, Sanders, said, and I’ve found a recurring theme: in all three cases the victims were, in a way, living on borrowed time, longer than expected.”
“Yes, that’s true,” said Seldom, “I hadn’t noticed…” His gaze became lost in the distance for a moment, as if he felt suddenly tired, overwhelmed by the constant ramifications of the case. “I’m sorry,” he said, unsure of how long his mind had been elsewhere. “I’ve got a bad feeling about this. I’d thought it a good idea to publicise the series. But perhaps there’s too much time between tomorrow and Thursday.”