by Miriam Sved
Finally Tibor spoke to an old colleague from the engineering department – a gentile, but a decent one – who agreed to have a look through the library collections, and made a copy for Tibor of a slim research paper on trigonometric sums. Not being of a strong mathematical bent, Tibor’s friend didn’t know what to look for and picked a paper that was quite flimsy, only deriving a few inequalities for double trigonometric sums and using them to play around with powers of rational numbers; even I could follow it. We drilled away at it, picking at this or that bit of notation to look for something interesting, some enticement for Pali.
I should say that I was quite happy with all this distraction, keeping us engaged in busywork at a time when I felt intensely self-conscious whenever we met as a group. Ildiko, you see, now knew about my Pali problem; I was no longer safely cocooned with my secret.
I won’t tell you too much of the halting, embarrassed, shame-faced scene at my home in which I communicated my problem to Ildiko. I had watched Pali for so long unobserved and unsuspected that giving voice to it felt like a kind of violence, thrusting into the world something that had been sheltered and inviolable. It also had the unexpected effect of making the situation somehow real. I began to understand that I would have to act, I could not go thoughtlessly forward this way, marry Tibor and spend the rest of my life watching Pali. I would have to choose. (Although, to be clear, I could only imagine that choosing Pali would mean a monkish existence of celibate devotion, Pali being barely a corporeal being in my eyes and not a possible romantic subject.)
The whole thing was excruciating when viewed through Ildiko’s steady gaze. I have had my share of adolescent infatuations and Ildiko, before this, had known of them all: our divinely ascetic English professor at the gymnasium, for instance; not to mention every single Jókai hero. But this was different. Our group, the five of us, was a solid structure. Sometimes I felt that our friendship was the only true bedrock in our world. We seemed so unshakable, and yet here I was undercutting the very conditions that kept us so. I knew, intuitively, that if I left Tibor for Pali everything would fall apart, the centre would not hold and we would all be flung into chaos because of my perverse desires.
Another painful thing to consider was Ildiko’s friendship with Tibor. They had known each other almost as long as Ildiko and me, although when we were at our respective segregated schools their friendship was mostly confined to correspondence. The two of them were bound to know each other because of the school mathematics journal, their pictures appearing alongside one another in the list of high-achieving students at the end of each year, and the yearly interschool competitions, at which they always competed for first and second prizes.
They wrote to each other, I believe, quite often, although Ildiko was somewhat secretive about the contents of their correspondence. I used to tease her about it; I felt sure there must be a romantic entanglement. Their photographs beside one another in the journals seemed like a kind of union, both unsmiling, Ildiko beautiful and Tibor handsome in a thick-browed, earnest way. I did not meet him myself until we all started university, at which point I still expected Tibor and Ildiko to end up together. No-one was more surprised than me when he instead showed interest in her plainer, stupider friend.
So Ildiko and Tibor had never been romantically involved, but I was sure they had a strong friendship: one that I perhaps did not quite understand the nature of – sometimes they seemed almost like brother and sister, competitive with each other and prone to biting criticisms, but fierce in mutual protection against any external threat. Ildiko respected Tibor, whereas I sometimes felt she was a little contemptuous of Pali, not as impressed by his genius as the rest of us and impatient of his unworldliness and eccentricities. I was afraid of her judgement: how, I expected her to ask, could I think to throw away a man such as Tibor for one who could not tie his own shoes?
But whatever she thought, she kept it mostly behind her eyes. She questioned me closely about my Pali problem – when had it started and how had it progressed and, most importantly, what did I plan to do about it? She offered no judgement, and at the end of it she gave a little shrug and a half smile, and reached out for my hand for a moment, so I knew at least that she didn’t hate me. I could have wept. Instead I turned the conversation to other things. And those other things – Vienna, the professor – had continued to hold our attention until the afternoon when Ildiko brought me the upper limit problem.
It was a warm day, the last of the snow dribbling into rivulets in the corners of the courtyard. The five of us had had an unproductive meeting dominated by Tibor wanting to discuss Vienna and Pali wanting to discuss prime numbers, and at the end of it Ildiko said to me, ‘You stay, Eszti. I want to talk to you about the dress pattern we discussed the other day.’
She was looking at me somewhat intently but she needn’t have; I knew that there was something of significance she wanted to talk to me about alone. Ildiko sewed many of her own clothes out of necessity, but it was not the sort of thing we discussed. There had been no dress pattern. I assumed that she wanted, finally, to take me to task over my Pali problem. And I was a little afraid, to be honest, that if I evaded her she might throw me over out of loyalty to Tibor. She might tell him my secret. So I sat meekly like a child awaiting punishment as the men ambled off in different directions – Pali with his hands moving compulsively in the air as he went back to chewing over the sieve problem he was preoccupied by; Levi turning to look back at us before he was out of sight. Tibor did not amble: he struck out across the park with long deliberate strides.
When they were all out of earshot Ildiko turned to me and said, ‘You know the boy I tutor named Lotz? The one with the unfortunate halitosis?’
Ildiko is usually extremely direct. She and Tibor and Pali, the brilliant ones in our group, might all be great mathematicians one day, but their brilliance is of quite different types. Pali’s is the magic of quicksilver, moving fast and unpredictably between subjects and ideas, while Tibor’s mind is like a base metal, unvarnished but sturdy, chipping away at problems until they begin to give. But while Ildiko is so much smaller in stature than the two men, who perhaps tended to dominate, I have never underestimated her mind. Some people make assumptions about Ildiko because of the way she looks. Men especially approach her softly, obliquely, as though preparing to pay homage to some artwork that is there to be admired, expecting her to strike a pose like the women who parade around on the promenade, always arranging themselves for the eye even if they are only buying liver from the butcher. Then these would-be votaries, these connoisseurs of beauty, are surprised to find the art looking back at them with pointed comprehension, defying anyone to take her as a thing to be merely viewed. I have digressed, but I hope it might help you to understand Ildiko a little better. All I mean to say is that she is, above all, direct. So I was very surprised that day at the statue when she started things off with what seemed to me like a prevarication.
Lotz, Ildiko’s student: I knew the one, she had taught him for years up in his family’s villa near the citadel; his parents, convinced of his potential, were set on him attaining one of the precious places for Jewish students at the university.
Ildiko went on, ‘I was coaching him in calculus yesterday – in the afternoon, when the weather was so lovely outside.’
‘Oh dear.’
‘Yes. I’m afraid I was quite impatient. He was slogging through his workbook, labouring over the same problem again and again, and I started tinkering on some loose paper while he worked, just to keep myself sane.’ She reached into her satchel and said, ‘I want to see what you think of something. I’m not sure yet what to make of it.’ She pulled out a few loose sheets of paper and lay them on the bench between us.
The one on top appeared to be an asymmetrical set of geometric points. They looked rather like stars strewn in a night sky. My eye automatically looked for patterns, any recurring sets or shapes, but they seemed quite rand
om.
‘I was mostly just amusing myself at that point,’ Ildiko said. ‘Turn to the next page.’
On the next sheet of paper was a similar array of points, though this time some of them joined together into shapes, mostly quadrilaterals. I looked up at Ildiko.
‘Look at this one,’ she said, indicating with her pencil a simple structure near the centre of the page: four points, joined into a rough quadrilateral, and one extra point alone in the middle.
Of course I thought of the five of us: Ildiko, Levi, Tibor and I joined in a protective flank around the central point of Pali. I nodded.
‘I started playing with the points,’ she said. ‘At first just randomly, moving that central one around, migrating it outside the structure.’ She indicated with her pencil a version of the five-pointed constellation with the Pali point outside the quadrilateral.
‘A basic pentagon formation. But here’s what I’m thinking about: it shares a common property with the previous set, that four and only four of the axes have a direct connection with each other.’ She used her pencil to join up the four points:
‘Do you see?’ She was looking at me intently.
‘You’re talking about an upper limit problem?’
She smiled, nodded.
I looked back down at the work. ‘What about a triangle with two points in the middle?’
She made a small flourish with her pencil. ‘Behold, mademoiselle.’ On an empty corner of the paper she drew the configuration:
Then she cut a line through the two central points:
‘You just bifurcate it,’ she said, sketching, ‘and, voilà!’
‘The same pattern emerges.’
I saw straight away why this interested her, the potential of it. ‘It’s a different kind of problem. A combination of geometry and combinatorics. Has anybody else done that before?’
‘I don’t know.’ She leaned towards me, conspiratorial. ‘Not that I know of.’
‘I suppose the next thing is to try it for a convex pentagon?’
‘Exactly. I’ve made a start, in here somewhere.’ She rifled through the pages. ‘Here, this one.’ She pulled out a sheet with larger constellations on it, more chaotic in appearance, with pentagons lurking in their midst. ‘It might be eight,’ she said. ‘That’s what I have so far, seven or eight points to generate a convex pentagon.’
I pulled my own workbook and a pencil from my satchel and started drawing sets of seven and eight points, looking for the conditions that might guarantee a five-sided structure, Ildiko leaning over and suggesting the connections, both of us playing with different formations.
I suppose I want you to know about this problem partly because it came to seem like such a catalyst for our group, splitting us apart and re-forming us into a different order, almost like the problem itself was externalising its intrinsic tension between chaos and pattern, the beautiful essence of it. (But here I am in danger of presenting things as though the upper limit problem itself had some kind of agency or will beyond me and my actions, as though I was not to blame. If only I could sustain such a fantasy, but it will not hold. I will come to my blameworthiness soon enough.)
I suppose I am also telling you this out of pride. Although it was not my work to begin with, when Ildiko generously opened it out before me I quickly caught the whiff of the infinite, and I hope I helped her a little. At that stage, when we were at the statue searching for the conditions for a convex pentagon to emerge, there was only the vague, sly leaning towards what we both hoped was hidden in this problem: if we could find a generalisation and a proof, can you see? We might be able to prove that complete disorder is impossible, given the right conditions, a large enough set. Something that looks like randomness – a cluster of stars or of cells, some chaotic nucleus – might need one more variable only and here we are at life, at the universe, meaning and order. You remove that one necessary variable and back we tumble into chaos.
I have kept the pages we were working on that day. We played with the pentagon formation until all the warmth of the afternoon had drained away, neither of us wanting to admit defeat until we had the five-sided structure. We eliminated a seven-point limit easily enough. For a while every variation of eight points we could think of yielded the convex pentagon, until I came up with this structure:
No possibility of a pentagon.
‘You are a genius!’ Ildiko declared, and I basked shamelessly in her praise.
One more point then, nine points, and every variation yielded the structure.
Ildiko sat back against the statue and surveyed our work, scattered across loose sheets on the bench between us.
‘This is good, Eszti,’ she said. ‘This will be good for us, I can feel it.’
‘Nine points for the five-sided figure, five for the quadrilateral. Do you think it will be enough?’
‘It will be enough to get us started, even just a hypothesis for the generalisation and we can work backwards, apply it to more complex figures.’
I looked at my breakthrough with the eight-point structure. ‘We could show the others what we have at the next meeting. If Pali can leave his primes alone for long enough he might help us along.’
I felt a little frisson when I said Pali’s name, a scuttling of secret feeling beneath the skin. Or not so secret anymore.
‘No,’ Ildiko said. ‘Let’s work on this together, just you and me. We work well together.’ And then, looking up from the work with her devious lopsided smile, ‘The way we smote the mighty Snakeman.’
She was referring to the day we met, her first day at the Jewish gymnasium. Snakeman was the name we gave to our mathematics teacher, who was instrumental in forging our friendship, even though he tried hard to tear us apart. The funny thing was that I had been thinking of that day as well: a day so cold that the pipes had all frozen and half the girls stayed home. It was in Snakeman’s class that Ildiko first came to my attention. A new girl arriving part way through the term – she stood out even more in the diminished class, although she was pretty enough that she would have stood out anyway in our Jewish girls’ academy, with her dusty blonde hair and green eyes. I watched her without any special interest as she walked into the mathematics classroom. I thought I knew the sort of girls she would end up being friendly with: the priggish, perfect ones. I made this assumption despite a certain forward-leaning directness of her walk, and despite the fact that she must have been a scholarship student from the citizens’ school (almost all new arrivals in later grades were in that situation, unless they were foreigners or had moved from the country, and she did not have the appearance of either). She sat at the empty desk beside mine and opened her workbook, aligned two pencils just so beside it and then straightened up and faced the front, apparently oblivious to the little stir she was generating in the classroom. I wondered if she knew what she was doing in the advanced mathematics class and (to my shame) how long she would last. Mathematics was not a popular choice at our all-girls school.
Dr Antal arrived and launched immediately into the work. I remember that by way of introducing the new girl he had her come to the blackboard to write out some formulae. He had started writing them in a high corner of the board so that she had to reach up to continue his notation. It was a teaching strategy he often used with the prettier girls in the class, while he stood by his desk surveying their work.
Most of the class was uneventful and I forgot about the new girl – Dr Antal was so particular about everyone transcribing his lectures verbatim, and my handwriting not quick enough to keep up with that dry stream of words. It is a wonder, really, that my enjoyment in the subject survived and even flourished in those years, a testament to Pólya’s book of problems and the school mathematics journal, so much more alive with interesting ideas than Dr Antal’s classes.
He was lecturing that day about functions. He had just given an example of a discontinuous one,
and at first I didn’t notice anything wrong. When my transcription caught up with what he was saying I stopped writing and looked up in surprise. I had never known our teacher to make an error. Immediately I met the eyes of the new girl sitting to my right, the only other student in the small class not frantically scribbling. A little thrill of recognition passed between us and her mouth twitched in a quick smile that stripped her face of its coldness. It was a great relief to have company in that moment; the fact of Dr Antal making an error, a fairly basic error, felt like a shift in the tectonic plates of reality. But then she turned back to the front of the room and, as composed as Empress Elisabeth, raised her arm in the air. I shook my head to try to divert her, thinking that she didn’t know what she was about, she hadn’t seen enough of our teacher to know the peril of this course, but her face was set with a steely look. (How well I would come to recognise this look – I am thinking of Ildiko at the university, her shoulders squared and that defiant set of her jaw as she stood in front of Levi on the stairwell, projectiles and insults flying towards her.)
