Part Thirty-Four. The Mandelbrot Set

Some days Wheatley could not keep Caroline away from GLaDOS no matter how hard he tried.

"Sweetheart, mummy's working," he called out to her, but she only giggled and sped along the management rail in front of him. He could have stopped her forcibly, but he hated doing that. He only did it when she was in actual danger.

"I wanna visit her!" Caroline said cheerfully, ducking around a corner. "An' she wants a visit, anyways!"

"Really?" Wheatley asked, wondering if this was true, or just one of those things Caroline thought was true.

"Yeah!" Caroline said, nodding enthusiastically, and before Wheatley had a chance to ask her how she knew such a thing she had already gotten into GLaDOS's chamber. "Hiiii Momma!"

GLaDOS glanced up at the two of them, bent over a sheaf of paper and holding a long white pencil. "Hello."

"Sorry, luv," Wheatley said, shrugging apologetically. "She… she got away from me."

"I'm here for a visit, Momma!" Caroline announced, stopping next to GLaDOS and peering down at the paper. "Whatcha doin'?"

"Drawing blueprints."

"C'n I draw blueprints too, Momma?"

"I doubt it," GLaDOS answered dryly. "You lack the required technical knowledge."

"C'n I draw somethin' else?"

"Like what?"

"I dunno. I c'n find out if you gimme one a' those papers and one a' those… things…" She waved ncertainly at the pencil with her lower handle.

"A pencil?"

"Yeah!" She bounced up and down enthusiastically, and she honestly looked so excited about her idea that Wheatley forgot all about the fact that he was supposed to be watching her so GLaDOS could work. "C'n I have them, Momma?"

"You don't know how to use the maintenance arms yet," GLaDOS said, though to Wheatley's surprise she didn't sound that convinced. Maybe she really had wanted a visit.

"I'll learn! Please?" She was wiggling back and forth and giving GLaDOS her best pleading look. If it'd been Wheatley she'd been pleading with, he'd've broken a long time ago. But GLaDOS was a bit tougher than that.

"Ask Wheatley to show you how to use the maintenance arms, then, and we'll see."

"C'n you show me, Momma?"

"I'm working. Ask Wheatley."

"I don' wanna," Caroline said petulantly, staring sternly at GLaDOS, but she was no longer looking at the little construct.

"I guess you don't want a pencil or paper, then."

"I do but I wan' you t'show me, Momma!"

"And I want you to go ask Wheatley. And you're not doing it. Are you."

"Carrie," Wheatley interrupted, before GLaDOS got annoyed enough to say something she didn't mean, "come on, sweetheart. You've had your visit. I'll uh, I'll show you how to um, to use the arms, and then you can come back and use them with mummy. Okay?"

"No," Caroline said sulkily. "I wan' Momma to."

"Mummy's busy," he said as gently as he could. "Come on, princess. I'll show you, and then when we come back mummy won't be busy anymore. Okay?"

"You won' be busy when I come back, Momma?" Caroline asked hopefully, looking down at her, and GLaDOS lifted her core.

"I won't be busy."

"Yay!" Caroline ducked down and nudged GLaDOS, which GLaDOS ignored. Wheatley frowned, but decided not to say anything. "Bye, Momma!"

"Goodbye," GLaDOS said, almost automatically, and Caroline happily made her way over to Wheatley. He made a mental note to talk to GLaDOS about her behaviour and pushed Caroline out of the room. Okay, maybe she was busy. And maybe Wheatley had been asked to keep Caroline occupied for a while so she could work. But still. She'd been a bit rude.

For the rest of the day, Wheatley showed Caroline how to use the maintenance arms, and though she was not terribly good at it, she had a lot of fun. He didn't know if she was quite up to picking up a pencil, but she caught on quickly enough, and was soon able to stack Weighted Storage Cubes without too much trouble. Unfortunately for her, though, all of the excitement wore her right out and she fell asleep against Wheatley while he was helping her move onto gripping turrets. He stayed quietly like that for a while, listening to her sleep, and eventually took her in between his handles and moved them both back into GLaDOS's chamber. GLaDOS was still making blueprints, a stack of used paper on her left and a fresh sheet on her right. Wheatley left Caroline on GLaDOS's right side and came down in front of her. "Gladys."

"Mm."

"I need to talk to you."

GLaDOS looked up at him, though she did not put the pencil down. "Yes?"

"I need you to pay attention," he told her, shaking his lower handle at the pencil. "You've been at that all day. Give me a listen, will you?"

She laid the pencil down and put the maintenance arm away. "All right. What."

"You need to be patient with her," Wheatley said, trying to sound both stern and gentle. "She only wanted to spend time with you. She doesn't understand you can't do that and work at the same time."

"I know," GLaDOS said, looking over at her. She seemed to become suddenly despondent. "Why did I even turn her on, Wheatley? I'm no good at this. I keep telling myself I'll let the work sit, but then I get started and find myself unable to stop until I'm finished."

"You're doing fine," Wheatley told her in as encouraging a voice as he could. "If you weren't, she wouldn't try so hard to hang out with you. You need to work on that. All right?"

"All right."

"And Gladys," Wheatley said, before he forgot, "if your daughter tries to say goodbye to you, don't ignore her."

She looked away from him, chassis twisting uneasily, and Wheatley suddenly felt as though he'd taken it too far. "Gladys, I… I'm not trying to make you feel bad, I just… I'm trying to help you."

"I don't know what I'm doing," she said quietly. "And I do everything wrong."

"You don't," he insisted. "You just need to be a bit more… gentle. That's it. Other than that, you're doing splendid. Promise."

She carefully picked up the blueprints in one claw, as if trying hard not to wrinkle them, and removed them from the room. He wasn't quite sure what she was doing after that, but it didn't include him. She just appeared to be thinking, that was all, staring at the wall and quite studiously ignoring him, and that did hurt a little bit but he tried not to let it bother him. Maybe she was just working out how she'd talk to Caroline the next day. He hoped so.

As with the day previous, GLaDOS told Wheatley to take Caroline so she could get some work done, and he obliged quietly. Caroline wasn't pleased with this arrangement initially, which was normal, but once he gave her the maintenance arm and some turrets and Cubes to play with she forgot all about it. He watched her contentedly as she built some sort of weird structure with the Cubes, positioning the turrets in an altogether random pattern around it, squealing excitedly all the while. She went at it for so long Wheatley started to get bored, which he rarely did when he was watching her, and he felt a little bit guilty to be happy when she fell asleep. He only noticed this when one of her Cube stacks fell over, startling him out of some sort of trance. Somewhat sheepishly he took her back to GLaDOS's chamber, where she was, to his bafflement, still making blueprints. He set Caroline on her usual raised panel and moved toward GLaDOS. "Still working away on that, eh?"

GLaDOS twitched, glancing at him quickly in a way that he was equally surprised to see was a little bit panicked, turning the paper she was working on so that the clean side was showing. "… yes," she answered, though she'd have had a better result if she hadn't answered at all. Wheatley began to suspect something was up on the other side of that paper.

"C'n I see?"

"It's… nothing. Don't worry about it. I'll finish it later."

"All right," he said as casually as possible, trying to think quickly of a plan. Now that she was trying to hide what was on the page, he really wanted to know what it was. He had to hurry, though, because she could remove the papers and dock the maintenance arm before he could take a look. He started to ramble a little bit about Caroline and the Cubes, and while GLaDOS was distracted with that and the fact that mentioning Caroline made her want to check on the little core, Wheatley dropped down towards the floor and flipped the paper over.

"Wheatley, what are you –"

Wheatley frowned down at the paper. It was doubly confusing because it did look like a blueprint, but it was of a turret, which didn't make any sense. Why would she be drawing something she already had, and hadn't manufactured in years?

"Give me that." She snatched it out of his maintenance arm with her own, but Wheatley had already moved on to the stack of paper next to it. With every paper he looked at he only became more confounded. They were all exactly the same. He gave up after the seventh or eighth such paper and turned to look at her.

"What in blazes are you doing?" he asked in a bit of a helpless voice. This was, without a doubt, one of the stranger things she'd ever done.

She turned over the page on the top of the pile, chassis sinking in resignation. "I was trying to draw."

"Uh." He wasn't sure what to say to that, since it looked a heck of a lot like she was drawing turrets to him.

"They're…" She curled up the corner of the paper she was holding. "Those aren't drawings. They're blueprints. That's different."

"So you were trying to… to make art? That what you're saying?" That was a new one. Art wasn't science, and yet she was trying to do it anyway?

She sighed and dropped the paper, shoving the pile away from her and shaking her core. "I know it's stupid. But she wants to draw and I… can't do it with her."

Wheatley was left speechless for a good fifteen seconds.

"Well, uh, I'm sure she won't notice that um, that you're making blueprints and not um, not a painting, or something," he tried after he got his thoughts back in order.

"No. She won't notice. But the point is that I can't do it. No matter how hard I try, I continue to produce the exact same blueprint. It's frustrating. What's even worse is that I can do so many things, and yet this one thing that is literally one of the easiest things in the entire universe to do is something that escapes me. It's so stupid and yet I just… the mere fact that I can't do it grates on me more than I can describe."

"They're very nice blueprints." He was honestly impressed with the fact that they all looked exactly the same. He doubted he could draw a pair of squares, let alone anything technical.

"I know that. But they're not what I was trying to draw." GLaDOS seemed to have had enough of the papers and whisked them out of sight. He wondered if she was going to throw them in the incinerator or keep them. She didn't want or need them, that much was clear, but she liked to keep things for her records.

"What were you um, were you trying to draw?" Maybe he could help her in some way.

She shook her core again. "That's my problem. I couldn't think of anything to draw." She laughed shortly. "You know, it's actually kind of funny that I doodle blueprints. Not quite the conventional description of a doodle, but that's what just happened."

"Is this… because you haven't got an imagination?" he ventured, wondering if she was going to find that offensive. But she only gave a sort of half nod, half shrugging gesture and didn't answer.

He honestly didn't know what he was supposed to do. Okay, well, he wasn't supposed to do anything, but if your best friend that you loved very much was having a problem you'd better damn well try to do something about it. So he thought a little more and said, "Well, when Carrie wakes up we can uh, we can see if that um, that helps. 'cause she's not going to draw blueprints. And… and I won't, either."

"We'll see."

There was a bit of an awkward silence for a while. On Wheatley's end, anyway. He was sure GLaDOS had found something else to do while they waited for Caroline to wake up. When she finally did and GLaDOS told her she could have the pencil and paper she wanted, she jumped up and down for five minutes proclaiming how thrilled she was and seeming to forget entirely about actually doing it. Eventually GLaDOS just gave her the paper and an assortment of brightly coloured pencils, and she literally pounced on them, scattering them everywhere. GLaDOS gave a long-suffering sigh and gathered them back up again, Wheatley moving to join Caroline on the floor even though it still made him very uncomfortable.

It turned out that GLaDOS had given Caroline a pile of markers, not pencils, and Wheatley had to say that he liked the markers a lot better than the pencils. They had tips that were quite thick and sturdy, meaning they didn't break no matter how hard Wheatley accidentally pressed on the paper, and they didn't tear through it either. They didn't even snap in half, and the bright colours were far more lovely than the dully shimmering grey of lead. GLaDOS's pencils had white lead, but she never let Wheatley touch those.

GLaDOS did not join them, electing to watch (or perhaps feeling too discouraged to try again), and Wheatley had loads of fun, though he didn't really try to draw anything in particular. He just doodled shapes and occasionally drew faces on them. He spent most of the time watching Caroline, who was just scribbling all over the paper with the markers and not really drawing anything so far as he could tell. She was obviously having a whale of a time, though, so he didn't let it concern him. When she tired out a few hours later Wheatley went to gather up the papers, asking GLaDOS where she wanted them, but she merely glanced at them and told him to leave them be. He did, coming up next to her for their snuggle, and after a few minutes of silence he said softly, "You could've done that, couldn't you? She was just, just scribbling, luv, not drawing anything at all."

"No," GLaDOS murmured, sounding oddly tired. "I tried that. I end up drawing sine waves."

"Sign waves?"

"A graph that represents repetitive oscillation."

He stared at the floor and tried to figure out how to spell 'oscillation' so that he could look it up in his dictionary.

"It's a…" GLaDOS seemed to be having as much trouble trying to simplify it as he was trying to figure it out. "A… picture of… a pattern… and it looks like a... well, a scribble, but it's a lot more calculated than that."

"So… it's like a perfect scribble?"

She made a noise in consideration. "That's one way of looking at it, I suppose."

Wheatley was satisfied enough with that definition that he put it out of his mind.

"Daddy Daddy Daddy hey wake up c'mon I wanna go play with the Cubes again!"

"Not now, princess," Wheatley mumbled, even though he did not sleep the way GLaDOS and Caroline did and so was either awake or asleep, with nothing in between unless it was quite an extreme situation. Still. GLaDOS's core was warm and the coolness of her chamber was very tangible against his other side. He had no intention of moving anytime soon.

"Awww," Caroline pleaded, and she sounded so cute that Wheatley couldn't help opening his optic to see which face she was making this time. As he'd expected, she'd put on one of her near-irresistible pleading looks, and as soon as Wheatley saw it he knew he was doomed. He really wanted to know where she'd learned that, because he was fairly sure he never tried to convince GLaDOS like that, who was of course pretty hard to learn facial expressions from. Wheatley didn't think Caroline could read GLaDOS, but that was possibly because doing that involved taking her every bit of expression she used and combining them. Caroline was probably not good enough at taking in information to do that.

Reluctantly, Wheatley removed himself from his very comfortable position alongside GLaDOS and opened his shutters all the way as Caroline babbled, encouraging him to hurry up and follow her. When he went to do so he froze in shock.

"Princess," he said faintly, staring at the wall with an increasing feeling of trepidation, "what… when did you…"

"I waked up," Caroline declared, swinging back and forth. "'an I drew that 'cause I got bored. D'ya like it?"

She had scribbled all over the wall panels with the markers.

Wheatley had no idea what it was supposed to be, let alone whether he liked it or not. He was in a bit of a jam, because he didn't want to lie to her but didn't want to hurt her feelings either, so he asked as tactfully as he could, "Uh… why don't you show me um, y'know, how you drew it, so I can uh, can get a real idea of your uh… your… thought… process... that you were thinking when you… when you drew that."

Strangely she accepted that rather feeble request and began to point out to him just how she'd drawn the picture, or what he assumed was supposed to be a picture, and apparently she had drawn the three of them from yesterday, with her Cube towers off to the side and Atlas and P-body climbing a stack to put a turret on top of one of them. Wheatley was actually able to see it after she pointed out where Atlas was, but that wasn't the pressing matter. The pressing matter was that Caroline had scribbled all over GLaDOS's wall.

What was he supposed to do? How was he going to get rid of it without hurting Caroline's feelings before GLaDOS woke up?

"Hiiiiiiiii Momma!"

Too late.

Wheatley slowly turned around to see GLaDOS lifting herself out of the default position, and even as he tried to think of a plan he knew he didn't have time to implement it if he'd thought of one. He found himself unable to stop staring at her, and she seemed to be equally unable to stop staring at the wall. They all stared in silence for quite a while, and Wheatley really, really hoped she would be able to keep her temper. He began thinking up explanations to placate her, wondering if child psychology would work this soon after startup, when Caroline piped up. "D'ya like it, Momma?"

GLaDOS looked at her and back to the drawing. "It's very colourful," she answered noncommittally. Wheatley began to relax a little, against his better judgement. She was being surprisingly calm about it.

Wheatley, what in the hell is that.

Wheatley jumped and yelped a little by accident, causing the both of them to give him a sideways sort of look, and he quickly returned, It's uh, it's a drawing of us from um, from yesterday, with her Cubes and turrets and… I have no idea what Atlas and P-body are doing there. Honestly. They didn't uh, didn't go in there with us.

I don't see it.

"Momma? D'ya like it?" Caroline repeated, beginning to sound a little anxious. "I drew it for ya 'cause you must get tired of lookin' at the same wall all day long, don't'cha Momma?"

Wheatley knew for a fact that GLaDOS quite liked her uniform grey walls and prayed that she wasn't going to lie.

"This certainly is a change," GLaDOS told her, finally looking away from the colourful lines marking up her wall panels. "I'm afraid I'm going to have to move it, though."

"Huh?" Caroline moved forward, that pleading look back on her face. "But why, Momma?"

"Because I have to work in here," GLaDOS answered smoothly. "And I'm confident that would become very distracting. But don't worry. I'll be able to see it just fine no matter where it is."

"But do you like it?"

Wheatley got a little sad. From the way she was tilted a little bit towards the floor and the mournful note in her voice, he got the impression she already knew the answer to that question.

"I do like it." She reached over and gave Caroline a little nudge. "Thank you. But I can't keep it in here. I would never get anything done."

"Yay!" Caroline squealed, jumping onto GLaDOS's core and wiggling ecstatically, and GLaDOS indulged her for a few seconds before pulling back.

"Go play with Orange and Blue for a while. I think Wheatley wants to talk to me about something."

"Okay!" After sneaking in another little cuddle Caroline zipped along her merry way. Wheatley turned back to GLaDOS.

"You didn't lie to her, did you?"

"No." After she inspected the wall a little more, the panels began to recede, heading off to wherever she was putting them. Actually, Wheatley was pretty sure he already knew where that was and tried to make a note to check on that later. "I meant what I said."

"You're not mad that she uh, she drew on your wall?"

"My wall?" She tilted her core curiously. "If the panels didn't have a problem with it, why would I?"

Wheatley frowned. That actually made sense. Sort of. He'd never really thought that she considered the panels' feelings that much.

"It's almost time for my bi-monthly panel replacement anyway," GLaDOS went on, filling in the space. "I admit it irks me to be doing it three days early, but that's not nearly as bad as replacing those two now and all the rest in three days."

Wheatley was a bit embarrassed to admit to himself that he hadn't even realised there was a such thing as a bi-monthly panel replacement. He watched as she swapped all the panels in the room for new ones, having to manoeuvre a little so that she could get at the one he was using. When she'd finished he moved in, even though he was a little early, but she let him so he supposed she was alright with it. When the silence began to grate on him, he asked, "What're you thinking about?"

"I'm looking at the panels she drew on. I can't for the life of me see any of what you said was in there. I can't even find the Cubes."

"Hm," Wheatley mused, trying to think. "Well, I… how about um, how about we give it a go."

"Give what a… go."

"I'll help you draw something."

She moved so fast she almost pushed him off of her core. "What?"

He shrugged. "I could um, could tell you what to draw and then uh, and then you'd be able to make art, right? Because you wouldn't be able to make a blueprint if I said um, said 'draw a curve', now would you?"

"Don't second guess my ability to draw blueprints," GLaDOS remarked dryly. "But. That actually might work. Let's do it."

She brought the markers and paper back, and after a few seconds Wheatley thought of something to draw. So he outlined the steps to her, telling her to put a square here and a triangle there, and guiding her as to the proper colours as well. When he thought the subject was quite clear he told her so, and she nodded a little.

He'd told her what the shed in the middle of the wheat field looked like, in as basic terms as he could, and while it didn't quite match his memory it was still somewhat reminiscent of it. She continued to stare at it, for what reason he didn't know, and finally he asked tentatively, "Uh… are… what is it?"

"These curves," she answered a little distractedly, gesturing at the wheat with the brown marker she was still holding. "They look… like the top half of a tangent graph."

"A what?"

She pulled over a fresh sheet of paper and quickly sketched out a series of identical lines, all of which could be rotated and not change the look of the graph. "This is a tangent graph. It's calculated by dividing obtuse angles by acute ones. It's reminding me of something. I'll think of it in a moment."

"What's an obtuse angle, luv?" he asked, vaguely remembering the word 'obtuse' to be some sort of insult and not relating to angles at all, when all of a sudden she lifted her core and focused on the wall in front of her.

"It's a way to describe part of a triangle. But that's not important. What's important is the triangle itself." She moved away from him, dropping the marker, and Wheatley frowned.

"GLaDOS, you want to um, I dunno, explain where you're um, where you're going with that?"

"I'll show you." She brought out one of her monitors and began programming, as usual doing it so quickly Wheatley couldn't see the characters on the screen. "Give me a moment. I have to make this a bit more… presentable."

Wheatley waited patiently for her to finish, which she did in a minute or two, and when she ran whatever she'd just written a few windows popped up. One of them had a little grid with little triangles on it, and another had a chart with words he couldn't read and numbers next to them. "All right."

She changed some of the numbers in the chart, and after she did that a lovely little pattern appeared on the screen. It was actually so bright and colourful that he had to look away for a second. It was a sharp contrast to the dimness of her chamber.

"This is a fractal," GLaDOS told him, nodding at the screen. "A fractal is a repeating pattern made up of triangles, in this case, in which the pattern looks the same at any zoom level. Like this." She changed the screen so that the black outer edges were no longer showing, but other than that the picture seemed to remain unchanged. It was actually quite amazing.

"You can make an infinite number of patterns," she went on, adding a few triangles to her grid and moving two or three and changing the size of the others. "It is always retained, no matter how deeply you zoom into the resulting formation."

"That's amazing!" Wheatley exclaimed. "And that's all made with maths, is it?"

"Yes," she answered. "I'll show you something else. Hang on a minute. I have to synthesise it first."

He was so excited he didn't want to wait, but happily she didn't take too long and soon produced a very small container. "Quickly," she said, and with a bit of an unsettling yank he was looking through her optic. Inside the box was a tiny little white thing that sparkled a little, and as he watched she zoomed in her lens gradually. The little white thing was made up of tiny crystals that looked exactly like the thing had when he'd first seen it in the box! But it disappeared, which was sort of disappointing, and with another jolt he was looking through his own optic again.

"That was lovely," he gasped, trying to comprehend how such things were possible. "What was that thing, anyway?"

"A snowflake," GLaDOS answered, removing the little box. "You've never seen one? You're outside enough that you should have by now."

"Is that the uh, the stuff that falls out of the sky when it's really cold out?"

"Possibly. It's not always snow."

"I've seen it," Wheatley admitted, "but never like that. It just looks like, just seems to be white fluff from what I can see."

"It's far more than that. Listen… I understand this is probably not the most exciting thing in the world for you, but… would you like to see some more fractals?"

"Yes," Wheatley declared emphatically, parking himself close to her and looking at the screen. "C'n you change the colour, there?"

"I can change any part of it."

"Make a blue one!"

She laughed, and he was surprised at how genuine and light it sounded. Seemed as though saying yes meant a bit more to her than he would have thought. He smiled. Why shouldn't she be happy about that? She'd found a way to make art through science, and he wanted to share it with her. It all sounded like a very welcome end to the last couple of days.

She showed him all sorts of apparently famous patterns, some of which she built out of mathematical formulae and some by rearranging her little grid, then made little solar systems and cities and electronic pathways all out of triangles, for hours and hours into the night. He had no idea how long she did this for, only that he should have realised she was falling asleep by the way her chassis had been lowering gradually during that last hour, there. When she finally did nod off, Wheatley spent a few more minutes looking at the last pattern she'd been working on, which she said was supposed to be a reproduction of a plant that had a fractal pattern, but he couldn't quite make it out. There was some part of the pattern missing, a triangle or something, and the notion that she'd been so sleepy when she'd started it that she'd actually made an error in her calculations brought a smile onto his face. It was such a cute thought, it was.

Perhaps imagination was a nice thing to have when you were making art, but now Wheatley was convinced that being smart about it was just as important.

Author's note

What's this? An update? Oh my.

The bit about Caroline thinking that what she wants to be true actually is true… is true. Kids do that. One of my psych profs told us about kids who projected their wants onto objects (eg "My bike is sad because it's raining and I can't ride him). So because Caroline wants to visit GLaDOS, she believes GLaDOS wants a visit.

When I wrote the sentence "she liked to keep things for her records" I thought of GLaDOS being a hoarder and that thought is actually kind of funny. Like she just has boxes and boxes of files and filing cabinet s stacked on top of each other in the basement and Wheatley finds them and he's like "Luv, don't you have all of this 'lectronically?" and she's like "Yes, but I need those in case there is some event in which all of my hard drives are wiped" and he says "Um if you know that might happen why haven't you planned for it?" and she's like "I have. But I still need those. Just in case" even though she has like three backups of everything and there's no way she needs the paper files.

How'd you like the math lesson? Lol fractals are really cool. I was going to call this chapter 'the Art of Science' but 'the Mandelbrot Set' sounds cooler. And now I can save the other title for later. BTW the Mandelbrot Set is a fractal.

I may have discussed this already, but GLaDOS becomes tired when she keeps all her processes running for too long. So what happens when she falls asleep is that her system decides she's overworking herself and begins to suspend processes (because it needs to do eight hours of maintenance, which it can't do if she's active), which gradually makes her lose focus, the ability to calculate certain things, etc., and once enough of them are in suspension she goes into sleep mode.