Episode 18: Projects

episode-18-projectsKen sat fiddling with the gears on his hoverchair arm as he waited for Jaxon to come out of Mrs McKinsey’s workroom. It wasn’t long before he walked unsteadily out.

“So, how are the new specs?”

“Whoa.” Jaxon held up his hand at arm’s length. “I can see my fingerprint ridges. It’s like somebody turned the resolution of everything way up.” He looked around the corridor. “And everything’s much closer than it should be. It’s a bit weird.”

Ken glanced at the wood-panelled walls and dark floorboards. “What do you mean, it’s closer than it should be?” Everything looked perfectly mundane to him.

“I guess I don’t expect to see this much detail unless things are right in front of my face,” Jaxon said. “I’ll get used to it. Come on, Nieminen is going to blow a fuse if we arrive late, after the war the girls have started.”

Ken frowned and started his hoverchair moving forward. “Yeah, I dunno how someone always manages to be late.”

“Because they don’t run when they have” — Jaxon consulted his watch — “four minutes to get from a lunchtime meeting to the lab.” He didn’t quite run, but the pace he set was fast enough that they arrived at the lab before Mathematician Nieminen did. “Made it on time!”

“Well done, goody two shoes,” Kelly Jean said. Ken figured she was still unhappy about the dressing down she’d received for being late the day before.

“And with glasses, huh?” Kelly Jean’s eyebrows shot up. “I suppose copying other people’s notes didn’t fit the new image.”

Jaxon screwed up his face in confusion and turned to Ken. “What?”

“You copied from her notes because you couldn’t read the board once, remember? Which is why you sit  next to me now, since I don’t mind if you copy my notes.”

“Oh, yeah. Wow, this is going to be great!” Jaxon flung his arms wide open, encompassing the rows of empty lab benches throughout the room. “I can see the board from anywhere! I can sit right at the back by myself if I want.”

Ken was relieved that he’d avoided the argument.

“Although I wouldn’t recommend that if you want to hear anything,” Melinda said. “It’s nice if you’re trying to work on a problem set, but when we’re getting a new project like today, it’s pretty annoying.” She dumped her bag onto one of the lab benches at the front and started redoing her ponytail.

“I forgot we were getting a new project today. This is von Rejk’s one, right?” Ken glid into place across the bench from Melinda.

Melinda stared at him. “How in the asteroids did you forget when we’ve all been frantically trying to get the measurement report finished for today?”

“Oh.” Ken shrugged. “I handed it in a couple of days ago.” He hadn’t expected the awkward feeling of being the only person ahead of the deadline to follow him to the Academy. He felt the blood rising to his face. “Probably I should have held onto it in case I missed something.”

“I shouldn’t even be surprised.” Melinda transferred a handful of pens from her bag into her lab coat pocket. “And there’s no way you left something out. You’re probably the most pedantic person I’ve ever met. No, hang on. What’s the word I mean?” Melinda asked Jaxon, who had joined them at the lab bench.

“If you think Ken’s anything like pedantic, you’ve never seen his sock drawer,” Jaxon said seriously.

Melinda burst into laughter.

“Hey, that’s not fair,” Ken protested, but he was grateful for the change of topic.

Bryony rushed in, braids swinging and sat down next to Melinda. “What’s so funny?” she demanded.

Melinda gestured towards the boys helplessly.

“She was up late last night, wasn’t she?” Ken asked Bryony. “It’s really not all that funny.”

“I was making fun of Ken’s sock drawer,” Jaxon said.

Melinda gulped down air. “Sorry,” she said, stifling a giggle. “I just had this image of Ken as one of those stereotypical Old Earthers with colour coded socks and a personal organiser thingy and–” She gulped again.

“Yeah, I don’t think she got enough sleep.” Bryony yawned. “Did anyone?”

“Ken handed in days ago, because he’s a machine,” Jaxon said.

Ken blushed.

“Good for you,” Bryony said. “You have an impressive work ethic.”

Ken shrugged. He wasn’t sure that putting less time into this project than everyone else seemed to have was really impressive. Then again, he’d been surprised at how much people had left to do last night, so maybe things balanced out. He wasn’t sorry when Mathematician Nieminen began talking and moved his thoughts to other topics.

“I’m glad to see that all of you have made it to the laboratory unscathed today. Try to keep it that way.

“By now, you’ve also all handed in your reports on measurement. I’ll be marking them over the next several days and we’ll have a debriefing and a discussion of uncertainty in due course. In the meantime, you’ll be working on a new modelling project with Mathematician von Rejk, which he will explain shortly.

“I expect today’s session to be mostly theoretical, but please remember that this is a laboratory and you do need to keep your safety gear in place. Especially, please be careful not to disturb the crystallisation experiment that’s set up on the back bench. That’s all from me.”

Ken turned to look at the experiment on the back bench, as did most of the class. There wasn’t much to see, beyond a collection of trays and beakers. Ken turned back to see Mathematician von Rejk waiting for attention. He wasn’t wearing a lab coat, but Ken figured that the heavy workshop gear probably served the same purpose.

“Alright, alright, you’ll get to look at it in due course, but try to pay attention for three minutes while I explain the project. As you should expect by now, you’re going to be applying mathematics pragmatically to get a method of describing some complex phenomenon or system. You’re going to choose a system, set up some mathematical equations to describe it, and then tell me where your model fails and where it falls short. Simple enough, yes?”

Ken frowned. The project was simple to describe, but it didn’t sound so simple to carry out. He wasn’t sure where you’d even start with making a model for most systems.

“Alright, some more details. You can choose any system you like, but you must get me to okay it. I recommend choosing the simplest thing you can think of and then simplifying it some more. It doesn’t need to be original, but I do expect you to be using at least one of the library and the laboratory to get the information you need.

“You have this afternoon to think and talk and come and discuss with me. I’m not going to set a final deadline for hand-in until I see the sorts of projects you’re choosing and how the research is coming along, but you can expect to have a few weeks to work on it. Make sure you’ve spoken to me by the end of the afternoon.” With that, Mathematician von Rejk gestured at them to begin and made his way off the platform at the front of the laboratory.

“Huh.” Ken stretched thoughtfully. “That’s quite a project. I’m not even sure where to start.”

“I didn’t make a plan at the beginning last time and I regretted it,” Jaxon said. “So I’m doing that before I even speak to von Rejk.”

“You already have something to make a plan of?” Ken was surprised that he’d come up with something so quickly.

“I want to do circuit diagrams and how they compare to real circuits. Like, you know how the length of the wire is irrelevant in a circuit diagram, but it does have a small effect actually. And you ignore the position in the diagram too, but I don’t know if that does affect anything. It’ll be really cool to measure that sort of thing and see.”

“Wow, that’s completely different to how I was thinking about it. Completely reasonable, I think, but completely different. I was thinking more of figuring out how to derive some kind of model.”

Jaxon was digging in his backpack. “I’m sure you can do that too. It sounds like a pretty open ended project.” He pulled out his pen and notebook and started making sketches.

Ken sighed. The girls on the other side of the bench were already deeply engaged in conversation about their topics, which left him on his own trying to figure out where to begin. Well, von Rejk had suggested starting with something simple. The most basic scientific thing Ken could think of was an atom. There must be some way of modelling the nucleus and electrons. By analogy with Jaxon’s circuit diagram being a model, he realised that the nucleus and electrons might already be a kind of model — but he would think about that later. It seemed reasonable to model how things would move inside an atom. There would be some kind of equation for the force between them. If it was an electromagnetic force, he would expect the positive nucleus and the negative electron to just run into each other. That didn’t sound quite right, but he reckoned figuring out why would give him something to learn. He headed off to talk to Mathematician von Rejk.

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Episode 17: Greater Than Zero

episode-17-greater-than-zeroBryony had decided that she did not have the temperament for being late or for being lost. In consequence, she was the first one into the Atkinson lecture theatre. She watched the rest of the class trickle in from breakfast.

Melinda and Buhle were talking excitedly as they arrived. “Mike must’ve put the posters up last night. I’m sure they weren’t there yesterday. And it’s barely a week’s warning!”

“Well, a week seems like enough warning for an audition, especially since we’ve been expecting it anyway. You worry too much.” Buhle shook her head.

“But worrying is half the fun. It’s not like there’s much else we can actually do at this point.”

Buhle laughed.

“What’s Melinda worrying about so happily?” Bryony asked.

“The robo-duelling auditions posters went up last night,” Buhle said.

“Really?” Kelly Jean had just walked in. “Why did nobody tell me? I still need to prepare!”

“It’s hardly our responsibility to keep you updated if you can’t read the news bulletins for yourself, Kelly Jean,” Bryony said. She rolled her eyes, wondering why Kelly Jean made everything about Kelly Jean.

“We were going to tell everyone,” Buhle added, “but Melinda only saw the posters on the dining hall noticeboard after breakfast. It’s not like we’ve had much chance.”

There was an awkward silence, broken by a group of new arrivals. “Hey, have you guys seen that the robo-duels audition announcement is out?” Ken asked.

“Oh dear stars and planets.” Bryony buried her face in her hands. “Is that all anybody can talk about?”

“It’s exciting for those of us taking part, silly.” Melinda sat down next to her. “You should come watch.”

Bryony snorted. “No thanks. I hope you guys have fun, but I somehow doubt your auditions will be anything as exciting as the City Hall matches.”

“Fair enough.” Melinda grinned and leaned down to grab her notebook as Mathematician Liang walked in at the front of the lecture theatre.

Mathematician Liang fiddled with the viewing screen for a few minutes. Once she had connected it to her notebook, a pair of graphs appeared and she turned back to the class. “Good morning, all of you. Today we will begin with the real substance of what we might call Real Analysis. So far, we have had a very hand waving overview.” She gestured dismissively. “I hope that the overview and your afternoon calculation exercises have helped you to become comfortable with these ideas of differentiation and so on. Today we will begin to properly define things, so that we can go on to proving the theorems we want.

“The graphs on the screen should be familiar to you. If we want to look at the difference between two points, we can think about connecting them with a line that forms a secant. If we bring those points closer and closer together, it seems intuitive that the secant will eventually become a tangent, so that it touches the curve at only one point. This is when the two points have become equal. Now, that’s a pleasant enough intuitive picture, but merely expecting something to happen is not particularly useful if we want to prove how things will actually behave. Perhaps you will offer me some thoughts on this.” She stopped and looked out over the class.

There was silence for a few moments and then Buhle said thoughtfully, “I think there was a thing on Ancient Earth when they didn’t really believe in gravity. I read somewhere that even after people made measurements, it was sort of controversial that objects would accelerate as they fell, because it wasn’t what people expected. I’m not sure if that’s exactly the same as a proof, though.”

Mathematician Liang smiled and nodded at Buhle. “We can relate the ideas, certainly. In pure mathematics we need proofs to make sure that what we assert is consistent with our starting definitions. In physics the most important thing is to be consistent with measurement. Either way, what is consistent is not necessarily what you would expect.

“More thoughts?”

Melinda said, “Once when I was little I thought I’d be able to double a number and get an odd answer. Of course, we know you can’t now, but I spent ages trying to do it before my brother showed me why you can’t. I think my expectations are a bit more sophisticated now, but the same idea still applies, I guess.”

It was the kind of thing Melinda would do, Bryony thought. She was always tinkering with numbers and patterns without any particular aim.

“Very nice, Melinda. I think you will all find that your expectations are challenged as you begin your studies of the Calculus. The infinite and the instantaneous are both quite outside of the everyday.”

The exercises they’d done so far hadn’t surprised Bryony, but Mathematician Liang had said that was the intuitive stuff. She wasn’t sure what infinity had to do with anything.

“Now, let’s move on to some definitions.” She turned a page in her notebook and the screen displayed, “Let ε > 0.”

“We’ll be using the symbol epsilon to represent a small, positive quantity throughout this course, as mathematicians have for centuries. Now, can epsilon equal zero?” Mathematician Liang turned to peer at Bryony.

“Uh, no, it’s greater than zero, right?”

“Good. Remember that in the weeks to come. We can make epsilon arbitrarily small, so that it comes as close to zero as you like, but from this definition, epsilon will never equal zero. Sometimes that turns out to be important. For instance, if we work out the slope of the line between two points, we divide by the distance between the points, yes?” The class nodded. “Now, in your exercises, you have been working out the slope of a tangent, which is the limit where the two points fall on top of one another.”

Bryony hadn’t thought about that. If the two points were actually the same point, dividing through by their difference would mean dividing by zero, which didn’t seem like a good idea.

“Some of you have already expressed some worry about this.” Mathematician Liang smiled. “That is an excellent attitude to cultivate. As your expectations become more sophisticated, you will become more aware of such issues. Epsilon greater than zero is a tool we will use to solve many of these inconsistencies.

“Now, one more question for you before I move on to more complicated definitions.”

Bryony hadn’t really thought “ε > 0” was a definition, but it did seem to be all the information they needed.

“If the absolute value of x is less than epsilon and we allow epsilon to have any value greater than zero, what can we conclude about x? I hope you remember that taking the absolute value just means throwing away the minus sign if there is one.”

Bryony wasn’t sorry for the reminder. That meant x should be a positive number that was smaller than any other positive number. No, that didn’t make sense, since you could divide any positive number by two and get a smaller positive number. Bryony wasn’t sure Mathematician Liang’s question made sense.

“It means x must be zero,” Quintessa said. “Otherwise you would be allowed to make epsilon equal to the absolute value of x divided by two, which would break the inequality.”

“Very good, Quintessa. Now, are all of you happy with this result? Quintessa has summed up the argument well.”

Bryony felt like she was squinting to make sense of it, but Quintessa’s logic was solid. The only way everything could be true at once was if x was zero. She guessed that was what Mathematician Liang meant about proofs making sure everything was consistent.

By the end of the lecture she had decided proving that the Calculus made sense was a lot harder than just using it. Her brain was tired. When Mathematician Liang had left, she stood up to stretch in an attempt at defogging her mind. She wasn’t the only one.

“That was great,” Melinda said.

“I think I’m going to be much happier about using all those rules once we prove them properly from the ground up,” Quintessa said.

“I was happy enough just using the rules, but it’s really fun seeing how to work them out properly like this.”

“You guys are crazy,” Bryony said as she came up from touching her toes. “I would be A-okay with just using the rules as long as somebody else has shown that they work.”

“I take it that Becky has begun her torture sessions.” Mathematician von Rejk had come in while they were talking.

Bryony nodded violently as Quintessa protested, “It was beautiful!”

Von Rejk laughed. “To each his own. There’s some use in understanding why the rules don’t always work, but I was as happy as the next engineer to drop Analysis in my second year. Daresay Becky feels the same about my sloppy modelling. Speaking of which-” He turned and began writing on the board.

Bryony sighed and sat down again. She was soon too absorbed in the lecture to worry about other classes.

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