The Zanna Function

by Daniel Wheatley

  Zanna’s stomach made an angry sound, and for the first time that morning, it had nothing to do with heights. “I don’t need your help,” she muttered. “So go stuff yourself.”

  “Now, now,” Dr. Fitzie said before Cedwick could reply, clapping her hands to silence the class. “You’ll have plenty of time to chit-chat with your new friends afterward. We have a very exciting day ahead of us, so let’s get to it!”

  The class grumbled and took out their notebooks.

  “I’ve said before that this is my favorite lecture,” Dr. Fitzie said, as wide-eyed as a child at Christmas. “Because today we are going to be discussing the very foundation of mathematics. In fact, I would say it’s the very foundation of the universe. That’s what you’re studying, mathematics, you know—the bones of the universe. All that physics and chemistry? It’s useful, I guess, but without bones, all you’d have is goop. So let’s talk about the function.”

  She tapped at her lips, as if a thought had just occurred to her. “What a strange word that is. Function. It means purpose, doesn’t it? Like, this bit of metal’s function is to be a handle and open the door. We use it to sum up what an object does. If that bit of metal breaks, it no longer functions as a door handle. We melt it down and reshape it into something completely new. We have changed its function completely.”

  Dr. Fitzie paused to let her words sink in before continuing. “Everything has a function. Every atom. Every mote of dust. Every planet and every star. They’re massive, whirling, magnificent confusions of functions, all competing and working together to make something exist.” Her voice dropped. “But you know what the best part of it all is? Those functions can be changed. If you know how.”

  Zanna caught herself leaning forward, drinking in every word. It sounded like raving madness, but then again, the last two days had been nonstop raving madness, so Dr. Fitzie’s lecture was in good company. In fact, when Zanna glanced around the classroom, the only person who wasn’t absolutely fixed on Dr. Fitzie was Cedwick. Even Nora and Amir, who must have already known all of this, were listening intently.

  “I’m getting quite ahead of myself,” Dr. Fitzie said. “We certainly aren’t going to start fiddling around with solar systems and constellations today. Let’s look at something a wee bit simpler.”

  She lifted a finger, as if touching something in the air in front of her. When she lowered her hand, a black dot remained stuck there. “Can everyone see that? In the back, can you see?”

  “Not really,” a girl said.

  Dr. Fitzie made a white square appear behind her, as if she had pulled a curtain across the back of the stage. Now the black dot stood out on the background. “Better?”

  A few of the students nodded.

  “Fantastic!” Dr. Fitzie grabbed the black dot and slid it right in front of her, letting it hang on an invisible plane that stretched from one side of the stage to the other. “Let’s talk about dimensions, shall we? I find this to be the easiest way to understand functions and how they operate. You’re all familiar with the concept of dimensions, yes? This dot has none. No height, no length, and no width. Well, of course the dot does have a minuscule height and length and width, because otherwise you wouldn’t be able to see it, but let’s ignore that, shall we? Truly dimensionless objects are rather tricky to fiddle around with, and this is just for demonstration purposes. It’ll be our little secret.”

  She winked and grabbed the dot, sliding it back and forth. “Here’s where it gets exciting. When things start moving. Because then we get a dimension added to our object and with it, a function.” Two intersecting lines appeared on the invisible plane—the X and Y axis. Dr. Fitzie slid the dot straight across, leaving a perfectly flat line in its wake. “That is the simplest function we have. X=1. It has always equaled 1. It will always equal 1. It goes nowhere else, does nothing else. It is just a dot chugging along at the same level day in and day out, not even worrying about all the different dimensions and possibilities out there. It is flat and boring, and it is perfectly content with that.”

  Dr. Fitzie put her hands on her hips, frowning at the line she had drawn as if it had disappointed her. “But I’m not content,” she said after a moment. “There’s just so much more to do!”

  With two deft strokes, she cut the front and rear of the line so it was just a segment about a foot in length. “Now, this is the same line; I’m just using a piece of it because that’s a lot easier to work with. Let’s go back to our friend, that dimensionless dot. His movement is what made that first dimension, right? Because a line is nothing more than a traveling dot. Just like drawing a line on a piece of paper. The tip of your pencil is the dot and it leaves a line behind it. But what if we took that line and moved it? What would that leave behind?”

  She swiped the line segment across the plane, and it became a square. “You ever try writing with one of those markers that never seems to dry properly? You draw a line, and then your big, clumsy hand comes back across it, and suddenly, you’ve got this ugly black smear across your page. That’s what two-dimensional objects are. Big, ugly, smeared lines.”

  With a few more flicks of her wrist, she drew line segments and dragged them out into shapes. Some shrank and turned into triangles. Others ballooned into hexagons and octagons. “It’s the same principle as before. A line is a traveling dot. A shape is a traveling line. You see where I’m going with this.”

  She cleared away all the other shapes, leaving only the original square hanging in the air. “Now, what happens when that shape decides to start traveling?”

  The intersecting axis of the graph disappeared, and like a magician pulling a rabbit from her hat, Dr. Fitzie stretched the square into a cube. “Three dimensions,” she said. “The world we know and love. A dot traveling into a line traveling into a shape traveling into an object. That’s quite a bit of movement for something so simple, isn’t it? And this cube here isn’t even real.” She plucked the cube from the air and batted it around for a bit, like a cat playing with a toy. “It’s an idealized construct. But I’m getting ahead of myself once again, aren’t I?”

  With a grin, she tossed the cube to Beatrice, who caught it with a small yelp of surprise. “It won’t bite; it’s a good little cube,” Dr. Fitzie said as Beatrice turned it over and then passed it to Cedwick, who immediately handed it off to Zanna without so much as a look. It weighed absolutely nothing, and Zanna would have sworn she was just cupping air, were it not for the thin black lines that made up its outline. She gave it a gentle toss from hand to hand, like Dr. Fitzie had, and then passed it on.

  “Let’s take some time to try this out ourselves, shall we?” Dr. Fitzie said. “Group up in your little triangles. Start with a dot and smear it into a cube. If you’re adventurous, maybe try your hand at some different forms! Not everything in the world is a boring old cube, you know. Use your imagination!”

  The murmur of work filled the air as the students turned to their groups. Zanna reached out and touched the air, just as Dr. Fitzie had done, but nothing happened. No dot appeared.

  “How’d she do that?” Beatrice asked, apparently having the same problem. The girl looked at her finger as if it had run out of ink.

  “Like this.”

  Cedwick held a cube just like Dr. Fitzie’s and a bored expression on his face. He balanced it on a corner and gave it a spin. “It’s painfully simple. Just hold the concept in your mind. Here—”

  “I can do it myself,” Zanna snapped before he could reach over to her. She hunched down and focused on Dr. Fitzie’s words. The dimensionless point at the end of her finger becoming a line. The line becoming a square. The square becoming a cube.

  “Oh!” Beatrice’s finger left a mark on the air, and she jumped a little in surprise. “I did it!”

  “Keep holding on to it,” Cedwick said, deftly making a couple of spheres. Zanna felt his gaze flicker over to her. “How are you doing?”
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  Her jaw clenched. “You’re breaking my concentration.”

  “Sorry, sorry,” Cedwick said. “Let me know when you want help.”

  “Look!” Beatrice said. A theoretical line like a pencil drawing on the air flowed out of her finger. “I’ve got it!”

  Heat flushed Zanna’s face, and she bit down hard on her tongue, squeezing her eyes shut to better focus on what Dr. Fitzie had said. If that strange woman had been trying to keep Zanna out of St. Pommeroy’s, this was the best way to get revenge. By proving that she belonged here. Zanna squeezed her eyes so tight they began to water. She pushed aside thoughts about her upset stomach and the strange woman and showing up Cedwick, focusing just on the idea of a function. She extended a finger.

  A thin black line appeared on the air.

  “Zanna’s got it, too!” Beatrice said. She clapped her hands in a tiny explosion of glee.

  Cedwick just nodded, curling off exotic free-formed shapes from his fingers like a man playing with the smoke off his cigar. “That’s just the start. Now you’ve got to take it up through the dimensions,” he said.

  But with the theory solid in her mind, the rest fell into place easily. With a grin on her face, Zanna twisted her hand, and a perfect theoretical sphere appeared. She gave it a poke, and it drifted lazily over to Cedwick, like a big geometric mote of dust in a sunbeam.

  “I think I can manage on my own,” she said smugly, leaning back with her hands behind her head in a mimic of his own posture.

  “Well, look at this!” Dr. Fitzie came out of nowhere and gazed down at the triangle of students with her hands on her hips. “Seems like you all are getting along just fine.”

  “Yes, ma’am,” Beatrice said.

  Dr. Fitzie made a little gasp of discovery, as if she had just now noticed something. “Ah! The younger Hemmington! My, it seems like only yesterday it was your brother sitting where you are. We expect some great things from you!”

  A change came over Cedwick at the mention of his brother. His lips quivered and tightened, smiling in a way that had no joy in it. It was so slight and well-hidden that Zanna wouldn’t have noticed unless she already knew it well. She did the exact same thing anytime someone mentioned her father. “I’m looking forward to it,” he said thinly.

  “Excellent!” Dr. Fitzie’s enormously giddy eyes turned to Zanna. “And you, Ms. Mayfield? You’ve had quite the adventurous past few days. Did you receive the key we sent you?”

  “Yes,” Zanna said. “I got it just fine.”

  “Splendid,” Dr. Fitzie said. “I do hope you haven’t had to use it. A Weierstrass transform can be a tad . . . feisty. But it’s a surefire way to pick out an illusion, as long as whoever made it didn’t write it in discrete calculus. Which most people never bother with.” She tapped at the side of her nose. “Let’s see Physics and Chemistry claim that, hmm? I do hope you consider Mathematics for your specialization.” She gestured to the theoretical shapes bobbing around them. “You’re quite the natural. All of you.”

  Cedwick made a sound of disgust as soon as Dr. Fitzie was out of earshot. “As if. I’m going to specialize in physics.”

  “I think mathematics is quite fun,” Beatrice said softly, drawing out a tetrahedron.

  Cedwick rolled his eyes. “Fine, but you won’t get accepted into the Primers by specializing in mathematics.”

  “Oh no,” Zanna muttered sarcastically. From what she had seen of Lord Hemmington that morning, it hardly seemed like a sacrifice. Luckily, Cedwick didn’t feel compelled to further discuss his opinions on the proper specialization, leaving Zanna free to make geometric shapes with Beatrice until Dr. Fitzie had finished walking around the classroom and returned to the stage.

  “Now listen up, everyone,” the teacher said. “You’re all doing quite well, which is absolutely marvelous for the first day. Sadly, I’m going to have to let you go soon, but I’m just going to leave you with this to ponder over.”

  The spyglass hanging from her hip jostled and then shot up out of its holster. Dr. Fitzie sketched another theoretical cube and made her spyglass crumple and reform into a cube of gleaming brass, with circular glass windows on each of its six sides.

  “Two cubes,” she said, holding them out for everyone to see. “What is the difference between them? What are all the differences between them? It is, in a way, what you have come here to study. What makes this cube”—and she lifted the theoretical one—“different from this one?” She lifted the real one.

  The class was silent, and she winked. “I’ll give you a hint. Functions. Many, many functions.”

  Chapter Six

  By the time lunch rolled around, Zanna felt like her head was going to burst. After Mathematics was Chemistry, which was taught by a hapless and scatterbrained man named Dr. Simon Piccowitz. He had started off mumbling incomprehensibly, but as the lecture continued, he sped up and sped up until Zanna worried he was about to suffer an aneurysm. “Chemistry is the—the—the building block of the universe,” he gushed, summoning a collection of raw elements from the massive periodic table behind him and making them dance while he talked. “Everything you touch, everything you see, everything that keeps us warm and alive and breathing—it exists because of chemistry. The study of chemistry is the study of the fundamental existence of the universe—the elemental—the real stuff of it.”

  Not to be outdone, her Physics professor, Dr. Andrew Cheever, whose classroom was decorated with a gallery of grade-school art projects, gave his own sales pitch. “What ties us together?” he asked, spinning around with his arms out, his dreadlocks flying wildly. “What ties us to the Earth, the Earth to the sun? What keeps our atoms together? That’s physics. The glue of the universe. Without physics, chemistry would be nothing but a useless collection of fundamental particles bumping around aimlessly, and mathematics would have nothing to describe.”

  To prove his point, he turned off gravity and let the students float around. It was exhilarating, but at the same time, Zanna began to see a pattern underneath all the fun and games. All three classes—Mathematics, Chemistry, and Physics—when put together answered the question Dr. Fitzie had asked. The real cube had mathematical coordinates. The real cube had atoms. The real cube had gravity. Understand each of those parts, and she would understand the entire cube.

  “I just wish they’d skip the hard sell,” Libby muttered over lunch. The cafeteria was on the east side of the school and had a long stone patio that bordered an expansive, lush lawn and gardens. Squat, circular iron tables stolen from some Austrian café lined the patio, and the girls claimed one in the cool shade. “Reminds me of going to the used-car dealership with my dad.”

  “It’s only natural that everyone thinks their specialization is the best,” Nora said. She had laid her lunch out in precise rows and was just now beginning to go through it in an order known only to her. “I already know which one I’m going to choose in the spring. Mathematics.”

  “Does it really matter that much?” Beatrice asked.

  Nora looked shocked that Beatrice would ask such a thing. “Of course it matters! Without the right specialization, you won’t get into the right subfield you want to study! Without the right subfield, you won’t get into the right sub-subfield! Your entire future is at stake!”

  “Mhm. Sounds more like an excuse to stir up rivalry,” Libby said, her mouth full of cold beef brisket. “I think I’m going with physics. Yeah.”

  “You’re supposed to take the entire year to think about it,” Nora said, snapping a carrot stick in two. “It is not a decision to be made lightly.”

  “And yet you’ve apparently already made up your mind.”

  “I’m different,” Nora said with an air of superiority. “I’ve had a lot of time to think about this.”

  “Excuse me,” Beatrice said before Libby could respond. The small Italian girl pointed at the students out on the yard.
“I have a question. Why is everyone carrying around random things?”

  It was something that Zanna had noticed, as well. Every student at St. Pommeroy’s who wasn’t a freshman carried some kind of metal object with them. Some had simple items like hammers, buckets, washboards, bird cages, scissors, or axes. Others wore ornate pieces of jewelry. Still others had things that were downright strange—one red-haired boy out on the lawn carried a stop sign. But no matter how odd the items were, each one seemed to fit with its owner. Like how a dog and its master always managed to look like they belonged together.

  “That’s the Iron,” Nora said, switching gears at once. “Every Scientist picks one. It’s their personal tool. If you want to go somewhere, or send a Particle, or sit down, or watch some television, or protect yourself, you use your Iron. It’s sort of an extension of who you are.”

  “And we . . . pick one?” Zanna asked. “From where?”

  “From wherever,” Nora said. “But you have to pick correctly, because you’re stuck with it after this year. Everyone says you know it when you see it. I already know what mine is.” She set a metal ruler on the table with a self-satisfied smile. It was stainless steel and polished to an astounding shine, just as much a piece of art as a measuring instrument. “My parents bought me it for Christmas.”

  Libby rolled her eyes and stuffed more beef brisket into her mouth. As the girls tucked into their lunches, Zanna idly watched a senior girl out on the lawn with a neon-pink thermos transform it into a screen to show off pictures of her dog. Not two days ago, Zanna had been planning her year—and the rest of her life—safe on the assumption that she knew what the real world was. Now all of that had been thrown out the window. She had a stack of homework dealing with theoretical cubes, simple gravity functions, and elemental memorization; a specialization to choose; an Iron to find; and a madwoman out to kidnap her.

  The girls were looking at her. “You okay?” Beatrice asked, her eyes soft with real concern.