"Hermione?"
Hermione looked up to see Anthony Goldstein looking down at her. He was looking down at her books now, though, spread all over the library table.
"What are you researching?" he asked, fascinated. "You've got quite the eclectic collection here – medical books, Alchemy texts, Arithmancy…"
"Ah, it was two different projects," Hermione said hastily, sweeping her notes into her bag. "I was looking up a theory about – never mind. How can I help you?"
Anthony grinned at her.
"I wanted to ask you about Arithmancy, actually," he said. "Professor Vector's been losing it recently, and I'm struggling with the homework. You're the only one who seems to get it lately."
Hermione laughed.
"Alright," she said. "But this is going to seem odd – I've been breaking down the equations to muggle maths to help."
Teaching Anthony Goldstein how to complete the square for quadratic equations was not what Hermione had expected to spend her afternoon doing, but it was nice to be teaching someone something again. She'd been ostracized by the younger students who used to ask her for help because of the rumors about her coven– it gave her a warm feeling, to be asked for help once more.
"So the answers are where the parabola passes through the x-axis," she said, pointing to the graph she'd drawn.
"But what is it?" Anthony wanted to know. "This is just a line. What does it mean?"
"They're the two different outcomes you're trying to predict for," Hermione explained. "With numbers, this is where they cross the line. When you plug runes into the equations, it helps determine what's going to occur."
"This is a nightmare," Anthony groaned. "How do you even figure this out?"
Hermione shrugged. "Carefully? I don't know. Here, let's do a simple example."
She withdrew a new piece of paper, writing across the top Will Anthony pass the test?
"There are two possible outcomes here, yes?" she said. "Either you pass, or you fail."
"I mean, there's a grading scale," Anthony protested, but Hermione shushed him.
"That's more specific than we have information for. So we just want to know if you will pass or fail." She sketched out a new grid. "If we abstract this, x² is the pass/fail. For the x, we'll use Dagaz, for your intuition and light within, and for the integer, we'll use Kanaz, for knowledge." She wrote the equation at the top. "To understand the answers you're getting, you have to abstract this. Watch."
She touched just the two runes at the top of the sheet, making them glow and fly off the sheet. A moment later, they returned, splashing -4x and -5 on the parchment.
"Now we have just numbers. So if we plug these in, and we complete the square…"
"What do those numbers mean, though?" Anthony protested. "I have negative five of what?"
"Units of non-knowledge? I'm not sure, I'm abstracting. But watch—"
As Hermione finished, she was left with two solutions: -1 and 5.
"So you have roughly 5:1 odds of passing the test," she told Anthony. "We can sketch this on the graph, too, to visualize it. The x-axis is going to represent the possible outcomes, and the y-axis is the pass/fail divide."
As Hermione drew out the parabola, Anthony stared at her.
"You're brilliant, you know," he said abruptly.
"You're too kind," Hermione said, smiling lightly. "I just have a head start with this – my parents gave me muggle maths books to help with Arithmancy. And Vector hasn't really been very clear lately—"
"No, you're a genius, muggle Arithmancy be cursed," Anthony said, laughing. "No one else, Hermione, would be able to combine the two so easily as you do."
"I don't think that's quite true," Hermione protested, but Anthony was looking at her graph.
"Okay, so this is pass/fail possibilities," he said, picking up her quill. "What does it mean if the answer looks like this?"
He drew another curve, this time sitting on top of the x-axis on the 2.
"That would mean there's only one possible outcome," Hermione explained. "It's to the right of the y-axis, so it means you would pass the test."
"What's the 2 mean?"
"Nothing," Hermione said, grinning. "That's just where you drew the base of the curve."
"Okay," Anthony drawled. "What if I draw it like this?"
This time, he drew the curve in the upper-right quadrant, and Hermione frowned.
"I think this is the problem Professor Vector is getting," she admitted. "I think the problem is the solution to the equation uses imaginary numbers. I think that whenever that happened, the presumption was that the equation was flawed somehow, but I think now they've discovered that there might be some third influence on the situation that transforms the plane into a 3D space."
Anthony blinked. "…I have no idea what that means."
"I barely understand it myself," Hermione admitted. "I mean, I don't understand it. I just have a vague idea of what's happening."
"So long as it's not going to be on the test, I don't care," Anthony said, shaking his head with a slow smile. He looked at Hermione fondly. "You really do have Magic talking to you, don't you?"
Hermione laughed, hiding her unease. "Oh, I don't know if it's as literal as all that."
"Why not?" Anthony was grinning.
"Well, for one, magic is an energy more like light than anything else," Hermione said, smiling. "It wouldn't act like sound…"
She paused, frowning.
"Only – that's not quite right, is it?" she said slowly. "Light can be a wave and a particle, and sound is a wave. And when you visualize your magic, you can match frequencies of sound with it, as well as see it flowing out of you like a physical thing when you cast." Her eyes widened. "Does that mean magic is both a particle and a wave?"
Anthony blinked. "What?"
"Nothing. Never mind," Hermione said hastily, standing up and shoving her books hastily into her bag. "I just had an idea. I need to go!"
Hermione hurried from the library, though she took care not to run and try to still walk with one foot in front of the other.
"Go where?" Anthony called after her, earning an angry hiss from Madam Pince. Hermione turned to look back, and Anthony was grinning at her. "What's the big secret rush?"
She paused, giving him a slow smile in return.
"The Owlery," Hermione told him. "I need to get some more books."
The nature of magic had long since confused and fascinated Hermione. If she were to ask anyone born in the Wizarding World what magic was, she'd just get a blank look and a vague response of 'it's just magic, innit?' But the fact was, magic was something, and Hermione wanted to know what exactly it was.
She'd confused magic with light before, expecting her parents to be able to see spells just because she could. But now she was more convinced that magic was something else – some other kind of energy in the world that could be shaped and molded.
Hermione was morbidly curious about what magic was. If it was a particle – if wands literally sent out little beams of magic-dust – that brought a set of questions with it: could the particles be captured? Did they have a half-life? Where did they come from? What were they composed of? And if magic was a wave, that had implications all of its own – could wizards not cast in outer space, then? Or was it just sound waves that didn't work in space, and other waves still did? But also, could someone interfere with the wave somehow and effect the result of the spell? If machinery emitting certain frequencies were set up on a battlefield, would it result in changing the effects of any magic cast?
Hermione had no idea, and part of her realized she was fast getting out of her depth.
It was with this in mind that Hermione knocked on Professor Vector's office door one evening, pushing it open at a cheerful "come in!", and going inside.
Professor Vector's office was painted a soft brick red, with art of different fractals on the walls. Incredibly, they seemed to be wizarding art somehow – the fractals started, then continued splintering over and over down into nothingness so small they couldn't be seen, before starting over again. A large chalkboard with a massive prediction tree with equations scrawled all over it took up most of one wall, and the side of the room held a small sitting area, where Hermione was surprised to see Vector and Snape playing chess.
"Hello, Miss Granger," Professor Vector greeted her, not looking up. "How are you?"
"Ah, I'm doing well," Hermione said, hesitating. "How are you?"
"I'm getting absolutely destroyed," Vector said pleasantly, as Snape's rook demolished one of her bishops. "How can I help you?"
Hermione bit her lip.
"You mentioned at the beginning of the year that recent Arithmancy discoveries were indicating that quadratic equations weren't fully measuring or predicting certain systems," she said. "I have an experiment I want to do, on the nature of magic, and I think the results might explain the weirdness that happens in those systems."
Vector paused, pushed away from the table, and looked at Hermione with astonishment.
"I'm sorry, but to speak frankly: you know nothing about Arithmancy," Vector told her. "We've barely covered algebra, to say nothing of advanced calculus and the trigonometry of spell angles—"
"I don't know much about Arithmancy, comparatively speaking," Hermione interrupted, cutting her professor off. "The Arithmancy would kind of be a side effect of my experiment. I want to do an experiment to determine the nature of magic. And depending on what I find, there might be Arithmantic implications."
"What, exactly, is this experiment?" Snape drawled, joining the conversation with an incredulous eyebrow raised.
Hermione took a deep breath.
"I want to recreate the double-slit experiment, but with magic," Hermione said bravely. "I want to see what results we get."
Her dramatic pronouncement was met with confusion and quizzical looks from her teachers, and Hermione's jaw dropped.
"You don't know about the double-slit experiment?" she despaired. "Merlin, it's so cool – and if magic works the same way as other energy, the observation effect could totally explain your Arithmancy weirdness – anyway, though, I figured I might need help, and if it worked you could document it and get published in Arithmancy Today and I could get co-author credit maybe?"
She gave Vector a tentative smile, nervous, and Vector laughed.
"I have absolutely no idea what you are talking about, but that doesn't mean you're talking about nothing," Vector said. She smirked. "Alright. I'll oversee your experiment, Hermione. If nothing else, it will satisfy my own curiosity."
"I, too, will oversee your experiment," Snape added.
"That's really not necessary," Hermione protested. "There's no Potions implications—"
"You have just told me you want to do an experiment on the nature of magic, Miss Granger," Snape said silkily, eyes glinting. "You have a history of rushing in and doing magically advanced and dangerous things without fully considering your choices and the consequences. I will be present as well."
Hermione grinned.
"Alright, you have a point there," she conceded. "Can we do it next week? I need to get all the materials I'll need first, but I really want to do this—"
"Materials?" Snape repeated. "Miss Granger, what sort of—"
"A bunch of tin and copper, mostly," Hermione said hastily. "Err, some marbles as well, a pipe, a gramophone, and big reactive sheets of—of anything, really. Maybe I could just enchant the walls? But nothing dangerous, I swear."
Snape sighed, while Vector grinned.
"If you can get your materials together and write out your methodology so it can be used and repeated if a paper does get published, I would be happy to collaborate with you," Vector told her. She seemed amused. "If nothing else, I am curious to see what you're going to do with a bunch of marbles that has to do with magic."
"It'll make sense!" Hermione promised, her face coloring. "I'll show you then, the marbles are for a demonstration—"
Vector laughed and waved a hand, cutting her off.
"We'll target next Tuesday, yes?" she said. "Almost a week before the drama of the visiting schools begin. That should be good."
"Oh, really?" Hermione was intrigued. "They haven't told us when they're arriving yet."
Vector stood, moving over to the chalkboard, her robes sweeping behind her.
"The Beauxbatons and Durmstrang delegations are scheduled to arrive on the Saturday before Halloween," she said. She pointed to part of her prediction tree, neatly labeled 10/29. "It is almost immediately after that some drama begins, before much drama begins after the announcement of the champions."
"Can you tell who they're going to be?" Hermione asked, fascinated.
"Not at the moment, but I don't know who all is going to submit their name as a candidate for champion," Vector explained. "We'll have to wait and see, but right now, everything points to there being a lot of drama surrounding the champions chosen."
Hermione blinked.
"Professor," she said slowly, her eyes following the prediction tree. "If I'm not mistaken, there are four results from that nexus point."
"You're correct," Professor Vector said mildly. "And?"
"And it's the Tri-wizard tournament," Hermione said, a feeling of alarm rising in her. "Not Quad-wizard tournament."
Professor Vector raised an eyebrow and gave her a thin smile.
"Like I said, Miss Granger," she said. "A lot of drama."