The Fibonacci Sequence
A sunny morning in late September found Phineas striding down Trumpington Street, the busy, bustling main street of Cambridge. The Muggles milling around were giving him strange looks as his travelling cloak and dark work robes swirled in the wind, but he affixed a sneer to his face and increased his pace. He understood that Merlin College had to be approached discreetly, on foot, but why did that mean he had parade himself in front of these scum?
Thankfully, he soon turned down a quieter lane, and after following it for a few minutes he came quite suddenly upon the little River Cam, threading its way between the green meadows belonging to various ostentatious Muggle colleges. Here Phineas faltered a little. His acceptance letter had informed him that the entrance to Merlin College was to be found here, between Garrett Hostel Bridge and the grounds of Trinity College, but there was nothing to be seen here but a few loutish Muggle students drifting down the river in a strange, flat boat that, for some reason, they were pushing with a long pole. Certainly, there was no evidence of any secret magical college.
Just when he was beginning to panic, wondering if this was all some sort of elaborate joke set up by Julius Malfoy and his ilk, a voice behind him said, "Black, is it?"
Phineas turned, but there was nobody there. "Yes, that is I," he said. "Phineas Nigellus Black."
"Indeed," said the voice. It was the voice of a wizard, perhaps in his late twenties, and distinctly unimpressed. It seemed as though Phineas' surname would not be quite the passport here that it had been at Hogwarts. "Follow me."
"You're invisible," said Phineas, feeling a fool and hating it.
The man snorted. "No student at Merlin College needs their eyes to track another," he said. "And for the love of Circe, get up a Disillusionment Charm." His voice was already growing more distant.
Phineas swallowed his discomfiture and Disillusioned himself. "Er — Homenum Revelio." Thankfully there were no Muggle students within casting range, for his wand began to give the faint sensation of being dragged in the direction which the voice had been fading in. If he recast the spell often enough, he could use it to detect whether the wizard was heading. Which was all very well, but right now his wand was pointing straight towards the river.
"Um," said Phineas. Then he took a deep breath, gathered his cloak up and waded into the water, gritting his teeth against the cold. He had not taken more than four shivering steps before he walked straight into a solid mass – the invisible wizard had waited for him.
"Bravo," the voice said drily. "Well, do keep up."
"I'm trying," Phineas said crossly, but he was once again ignored. He recast the Detection Charm, following the invisible wizard into the very middle of the river. His sodden clothes were beginning to drag behind him, and to distract himself from the cold he closed his eyes for a moment, daring to imagine being warm and dry in the autumn sunshine. When he opened them again he found that it was so; he was standing at one end of a large courtyard, lined with stone buildings and with a quaint little flower garden in the middle.
To his side was a tall man, wearing smart robes in the College's plum and copper and an unmistakable sardonic expression. "Congratulations," said the now visible wizard. "Perhaps you aren't a complete dunderhead after all, Black. My name is Magnus Wycke, and I have the dubious honour of being your Director of Studies during your time at Cambridge. You may call me Master Wycke when you have the privilege of addressing me. Questions?"
For a fleeting moment Phineas was eleven again, standing awkwardly in a carriage on the Hogwarts Express and trying to adjust to a whole new world of rules and expectations; then he found his tongue and his composure. "Yes, Master Wycke," he said crisply. "When do classes begin, and how shall I enrol? I'm particularly drawn to Numerology—"
"You'll be given that information in due course," Wycke interrupted, apparently bored. "Your room is in Staircase J. Settle in, and I will speak to you and the other Arithmancer matriculating this month tomorrow, or the day after."
"Wait, there's another Arithmancer—?" But Wycke had turned on his heel and strode off across the courtyard.
Lifting his chin, Phineas began to make his meandering way around the courtyard, checking each doorway set into the stone walls until he found the one marked with a J. A narrow stone staircase led him up two flights to a bare little attic room, furnished solely with a bed, desk and wooden chair. It was barely a third the size of his bedroom in Grimmauld Place, but, all the same, it was a place to call his own, and that made it instantly an improvement on at least the opulent dormitory at school.
He set down his sleek monogrammed trunk – worlds away from the battered old one he had carried onto the Hogwarts Express seven years ago – and dared to smile.
-B-
Merlin College was one of Cambridge's smallest, with only eighty full-time students and twenty professors, and also one of its oldest. Early in the fourteenth century, a dispute between Hogwarts' Headmaster and his subordinates had coincided with a wide-ranging ban by the Wizards' Council on the study of 'forbidden' magics, ranging from the mechanics of love potions to the mysteries of immortality. This being some hundred years before the formation of the Department of Mysteries, disgruntled scholars had instead joined forces with the dissident faction of Hogwarts teachers to found a college deep in the River Cam. The little island colloquially known as Garrett Hostel Greene had soon disappeared off Muggle maps as stronger magical protections grew up around it, although some argued that these barriers were sufficient proof that the College existed not in Cambridge but on a different plane of reality entirely.
"What is place, anyway?" asked one such philosopher, as the students who had recently matriculated gathered in someone's rooms to pass the evening in idle chatter. They had emptied a few of the bottles of Firewhiskey by that stage. "One cannot see the College from either shore of the river. Nor can we see Cambridge itself from here. Tricks of navigation — compasses, stars — seem not to work here. We must have been displaced from it entirely."
"You might as well claim that Platform Nine-And-Three-Quarters isn't in King's Cross Station at all!" someone else cried passionately. People grew very passionate in debate here.
"No, I agree," a third person — a Seer, Phineas thought — said dreamily. "The very idea of location is flawed; to consider merely the three-dimensional coordinates of the College absurd. The question lends itself to a deeper examination of the metamagical concept of spacetime—" And so on.
Wherever its true location, the College had become a reputed academic institution, and for many years had been the foremost destination for those witches and wizards whose curiosity had not been sated by a Hogwarts education. Merlin himself had been one of the independent-minded scholars who had founded the College, and after the great wizard's death the governing body had unanimously voted for the College to carry his name. These days, of course, most Hogwarts graduates interested in academia would apply to the Unspeakables, or hope to become schoolteachers themselves, but Merlin College had always stayed slightly outside these regulated territories.
There were rumours, too, of a sister College in Oxford, aptly named Morgana College and with a yet more arcane and secretive application process. There was a supposedly friendly rivalry between the two Colleges, but after a few weeks in Cambridge Phineas, always a quick learner, realised that both Colleges were highly tribal and relations were at best suspicious if not openly hostile. "If you see anyone on our grounds wearing Morgana robes – they're red – curse first, ask questions later," a worldly second year told him at one point.
"It's rather like House rivalry back in Hogwarts, isn't it?" Phineas mused. "With the different colours and the mindless loyalty?"
The second year looked affronted. "Hardly," he sniffed, and forever after would not interact with Phineas beyond a chilly nod in passing. Apparently, even in these hallowed halls of learning, a healthy dose of cynicism was unwelcome. Well, Phineas could live with that. He was accustomed by now to filing down his sharp edges in order to fit the role assigned to him; he could become whoever they wanted him to be, so long as he had the unshakeable voice of superiority in his head: You are better than all of them.
-B-
Phineas had always been clever, and it would have been easy enough for him to coast through Hogwarts as so many of his classmates had done, coming out with a respectable mix of Outstandings and Exceeds Expectations grades in his N.E.W.T.s. Thankfully, however, some perverse mixture of pride, tenacity and genuine pleasure in his studies had taught him the habits of hard work and accustomed him to late nights in the common room, perfecting essays and going over spells. He was thankful for this now as he had been for precious few things in his twenty years, because studying at Cambridge was hard. Every week, he was assigned a long problem set in each of the three core Arithmantic fields of Numerology, Statistical Divination and Functional Magical Calculus; he attended tutorials dissecting these problems in minute detail, and in the remaining hours found time to work on his dissertation. Or, more accurately, he sat in the ancient College library — whose high ceilings were filled, cathedral-like, with natural light twenty-four hours a day — and flipped through textbooks when he should have been sleeping, trying to settle upon a topic for his dissertation.
His classes were taught for the most part by two professors: Master Wycke, his sardonic and disdainful Director of Studies, and Master Fawley, the College's Head of Arithmancy. Some classes were taken by up to six or seven students — many of the first-year Transfiguration students needed to understand the basics of Functional Magical Calculus, and the more numerically-minded Seers would often drop in on Statistical Divination lectures. In Numerology, however, there were only two students: Phineas and his fellow first-year Arithmancer, Simon Murphy.
Murphy, a year or so older than Phineas, had slicked-black blond hair and wore formal robes to every class. He had started out the term with a set of ostentatious flamingo-feather quills, but by the third week he was already down to just a single one, having given them away to every hard-luck case who'd rolled into lectures five minutes after they started and forgotten their own writing implements. He was clever like a Ravenclaw, the kind of clever that held up classes to ask impenetrable questions about mathematical formalism and sat in the front row of the College's small lecture theatre as though he had something to prove, the kind of clever that demanded to be seen; for this reason, he would have made a terrible politician, coming loud and blundering into an echelon that demanded subtlety. For Murphy did not belong in so many small ways, but they all came back to one inescapable root: he was Muggle-born.
Phineas had not needed to ask this — Murphy's heritage, or lack of it, was made painfully clear by his overcompensating attitude, his strange fashion sense and most obviously by his common, unmagical surname. In any case, he had overheard Murphy declare his blood status in a loud conversation in the courtyard, and since then had resolved to keep well away from the boy. Merlin College was undoubtedly academically selective, but the cleanliness of students' bloodlines was considerably less satisfactory. Phineas shuddered to think how Julius Malfoy would sneer if he heard that Phineas was rubbing shoulders like equals with a Mudblood.
Conscious of Master Wycke's suspicious eyes upon him — for Wycke was, he learned very quickly, no friend of the old pure-blood families — Phineas did not, as his peers might have expected of him, outright demand that he and Murphy be taught on separate timetables; nor did he threaten to leave the College if Murphy were not immediately expelled. That would probably have been the radical, political stance to take. He would have made the front page of the Daily Prophet and ignited a national debate about the place of Muggle-borns in respectable pure-blood society. But the spectacle involved all seemed rather embarrassing, so he kept his head down and maintained a chilly, unapproachable distance from Murphy instead.
For this reason, his surprise on glancing up from his textbook during yet another late-night library session and finding Murphy standing in front of his desk was considerable. In fact, he spent some moments gaping instead of saying anything, and by doing so lost control of the situation, because Murphy seemed to take his silence as an opportunity to begin a conversation. "Black," he said formally. "May I sit?"
Instead of saying No, or even If you must with the least welcoming tone possible, Phineas inclined his head, and Murphy pulled out the chair opposite his own and sat down. Under his arm he was carrying a heavy book, which he set down on the table. He nodded at Phineas' own textbook. "Searching for a dissertation topic?"
The assumption that he hadn't already chosen a topic, while correct, was additional irritation. "What do you want, Murphy?"
"Well, to talk with you," said Murphy, with a gaze so earnest it could melt cauldrons. "We haven't spoken a great deal, and I consider it important to build close relations with my colleagues."
Phineas raised an eyebrow. "Oh, you do?"
"Of course," said Murphy. Was he oblivious to Phineas' frosty demeanour, or ignoring it deliberately? "We are the only two Arithmancers in our year, after all. I think it could be worth both our whiles to share ideas sometimes."
Phineas bristled. "I don't need any help from you," he said sharply. "And our dissertations are supposed to be independent research, not collaborations."
Murphy gave him a shrewd look. "I'm hardly suggesting we become the best of friends, Black," he said. "I know what you think of people like me. But I respect your intelligence, and I suspect you think rather more independently than old sticks like Fawley would particularly like."
"You flatter me," Phineas said sarcastically. Somewhat mollified nonetheless, he added, "Fawley is… quite the traditionalist, certainly."
"He's a bigot," said Murphy. "Which would endear him to you and your kind, I suppose, but his mathematical views are just as rigid as his social ones, which makes him a threat twice over to my academic career. He would never approve a dissertation proposal on something even slightly off the beaten path."
"True," Phineas conceded. "His own research is in the magical properties of the normal distribution, which is…"
"Pointless," said Murphy. "Those were extensively categorised thirty years ago. But if you're feeling particularly adventurous, Black, it's possible he'll consider a proposal to study the properties of the Poisson distribution instead."
Phineas rolled his eyes. "I'm not a Seer."
"Nor am I," said Murphy. "In fact, I do suspect we both prefer the same field." He nodded at the copy of Euclid on Phineas' desk. "Numerology."
Phineas tensed. "Indeed."
"And," Murphy pressed, "I've seen you in here night after night, trying to choose a topic. In my opinion, we could both do with the other's help. What are you thinking of?"
Phineas hesitated. "I've always been interested in integer sequences," he said.
Murphy nodded knowingly. "Primes and the like?"
"Well, primes are interesting," said Phineas. "But they're also all anyone talks about when you mention integer sequences, you know? There are whole books on their uses in curse-breaking and the odd places they appear in Magizoology. There must be so many other overlooked sequences. But I can't find much discussion of them in any of the textbooks." He gestured at the pile of books on the desk. "Every Numerologist knows that there isn't a more fundamental text than Eigen and Kernel. Do you know how many pages they have on primes? A hundred and fifty. And one page on other integer sequences. It's such a rich area for research, if only I could find something to research."
"So why the Euclid?" Murphy asked. "Hardly cutting-edge modern Arithmancy, after all."
"Exactly," said Phineas. "I was hoping he might have… well, a fresher perspective, without all our modern biases. But it's no use. They hadn't identified most of the sequences that Eigen and Kernel deign to mention in the Classical era, so Euclid is really just a curiosity."
"But you thought to look," Murphy mused. "You stepped outside the orthodoxy. Fawley wouldn't be pleased."
Phineas didn't entirely know whether he was being praised or mocked — nor, indeed, which he would prefer from this upstart Mudblood who seemed to love Numerology as much as he did. "I make that a habit," he said at last.
Murphy nodded. "Well, that settles it," he said. He rose, and, smirking slightly, added, "G'night, Black," before striding out of the deserted library.
Phineas checked his watch. It was two o'clock in the morning, he still had no dissertation proposal, and as he packed up his books he was left with the unsettling feeling that he had just become Simon Murphy's pet project.
-B-
He had a lecture in Functional Magical Calculus at nine o'clock the next morning, so, having gulped down a coffee in the College's Hall, a smaller reproduction of the Great Hall at Hogwarts, Phineas endured a sleep-deprived hour of frantically scribbling down equations. Master Wycke's teaching style was quite as dry as his wit, and he seemed to regard any request to slow down slightly as a personal affront. Today, many students seemed to have decided the lecture would be a dubious reward for braving the brisk November morning; there were only three attendees: one of the few witches studying Transfiguration, whose name Phineas didn't know, Phineas himself, and, of course, Murphy.
Unnerved by the degree of openness he had shown in the library the night before, Phineas resolved to keep away from Murphy now; but here he was faced with a will perhaps equal to his own. As he was heading out of the lecture, marginally more awake than when he had entered and with a head (and roll of parchment) filled with the odd, rune-like symbols of vector calculus, his classmate caught up to him. "Good morning," he said cheerily.
Phineas groaned. "You're a morning person, are you, Murphy?"
"All the most successful people are," Murphy said, and while Phineas was still bristling indignantly he produced from his satchel a heavy textbook bound in supple black leather. "Here you are."
Phineas frowned at the gold letters embossed on the book's cover. "Axioms of Cardinal Arithmetic? I've never even heard of this."
"You won't have, you're astoundingly narrow-minded for someone so intelligent," said Murphy. "Consider this an attempt at rectifying that. It's a Muggle text."
Phineas recoiled. "What? Don't be absurd! I don't want this!"
"Really, Black? Turning down knowledge just because it comes from a source outside your dogma? Fawley would certainly approve."
The goad stung, as Murphy had certainly known it must. Phineas retained his hold on the book's spine, and said, "What can Muggles possibly know about Numerology?"
"Very little," said Murphy, "but they do understand number theory. Four whole chapters on integer sequences, Black. Aren't you tempted?"
"I'll wager it's the most simplistic stuff imaginable," Phineas said, but the argument sounded weak even to his own ears.
Murphy rolled his eyes. "Then read it and give it back to me, Black. You aren't legally bound to cite it. Or I'll take it now: it's my only copy."
Phineas paused. "I'll read it," he said stiffly. "When do you want it back?"
"Take your time," Murphy said. "I was up late last night looking through it after our conversation, and I believe I've chosen a research topic."
"In what?" Phineas asked, interested despite himself.
"You rather intrigued me with all your talk on integer sequences," Murphy said. "I'm going to examine triangle numbers, specifically their potential uses in magical architecture. I'm curious to know if there were wizards behind the building of the Pyramids of Giza; they're one of the great wonders of the Ancient World, after all. My work could end having extraordinary historical significance."
"I see you're a modest fellow," said Phineas. "And integer sequences were my idea."
"You don't own them, Black," said Murphy. "But I do owe you for the inspiration, so I thought about the sequences that might hold some more interest for you as well. Have a look at page nine-hundred and eighty onwards. I'm sure you'll find plenty of material to work with."
"Well," Phineas said gruffly. He knew what he needed to say, but his pride could not quite let him surmount that final hurdle.
Murphy gave him a lopsided grin. "Thank me later, Black. I'll see you in class."
-B-
He attempted to wait until the end of the day before opening the book, just to prove to himself that he was not beholden to Simon Murphy or to anyone else; but curiosity overcame stubbornness by noon, and he spent his lunch hour in his room with Axioms of Cardinal Arithmetic open on his knee. There were, as advertised, four chapters on integer sequences and their classifications and properties. Phineas began by reading from the first of them, but almost of their own accord his fingers kept turning the pages until he had reached page nine-hundred and eighty.
We now turn our attention, he read, to the most marvellous and beautiful of all integer sequences, quite rightly characterised by the number known as the golden ratio. Examine the family tree of the humble honeybee, the seed pods of a pinecone, even the segments of your own fingers, and the same numbers recur in every situation.
Phineas read the next part aloud. "Zero. One. One. Two. Three. Five. Eight. Thirteen. Twenty-one…"
The famous sequence itself was not unfamiliar to him. Nor was its name. Even so, there was a deep, almost visceral satisfaction in reading the six pages of the enormous textbook dedicated to it, the feeling that something at last had settled into place.
Phineas had found the topic of his dissertation. He set the book aside and picked up a clean roll of parchment. It would serve for making notes on a second read-through of the book. In his precise, deliberate script, he wrote out a title at its head: The Fibonacci Sequence.
