Doodling in math: Dragons

So you're me and you're in math class, and you're supposed to be learning about logarithms, which you figure is probably something to do with avant-garde percussion performances. But every time you try to pay attention, you find that your teacher's explanations of logarithms inspires nothing but a bit of performance art involving drumming on wood. Anyway, your teacher gives you a dirty look. So you stop drumming and start doodling. You feel a need for motion, excitement, so you decide to do a flip book on the corner of your notebook. It's pretty easy because the paper is thin. You can trace the previous frame, modifying it just a little. And if you really want to get into the Zen zone of doodle bliss, you can make each new frame be modified according to some simple rule like, keep adding one new petal around a spiral, or add another dot in the next place in the spiral. Or how about, keep making the squiggle squigglier. But what does squigglier really mean? Like you could just increase the curvature of the squiggle, or you could make it squiggle more between the squiggles. You want to figure out an exact squiggle rule so that you can really get into the squiggle zone. So you discretize the squiggle into a zigzag line. On the next layer, you could just deepen the zag, or you could put a new zigzag on each old zig and zag. Or maybe zig's too good to zig zag, and zag's too good to zag zig. So the next would be, zigzag zag zig, zigza-- wait, no. That was a zag, and those get turned into zags zigs. So it should be zigzag, zag zig zag zig zig zag. And next, zig zag zag zigzag, zig zig zigzag, zag zig-- wait, was this a zig? Maybe if we put it all onto some sort of reference diagram, that generates each new pattern based on-- No! This was supposed to be about mindless zigging in the squiggle zone. This is unacceptable. So maybe just pretend each time you get to a new stage, the old path is all just zigzag zigzag zigzag zigzag. And to keep it all neat and orderly, you decide to try to make all the lines the same length, always with right angles between them. Here we go. [SINGING] Wait, how did this-- Huh. What if we tried starting with three lines? Zig zag zug. OK. So then each zig should get a zig zag zug, and each zag a zug zag zig? And then the zug gets, well, maybe it just goes back and forth, whether it goes on the inside or the outside. OK, wait now where did it start? And everything's running into itself, and getting bigger, and doesn't fit on the paper. OK, maybe if it were a little more open it wouldn't run into itself. Say like, half a hexagon, so you can keep all the angles perfect. And then trapezoids in and out. Yeah, this is totally going to work. Lookit. In fact, like this, you could even make it go inside first, and back and forth, and it wouldn't run into itself. Next, you could make it go even closer together here, and still not run into itself. Except that means you'd have to start on the outside this time. But that's OK, because next time you can go on the inside again. And now it's an easy pattern. Trapezoid in, trapezoid out, trapezoid in, trapezoid out. Back and forth until the last trapezoid in. And then the next time starts with trapezoid out. Although it's hard to tell what's in and what's out now that it's getting so squiggly. So you're just focusing on going on one side of the line, and then the other. And wait. This looks familiar. Is that, Sierpinski's triangle? The fractal you get when you put a triangle inside a triangle, and then triangles inside the new triangles you made, and then triangles inside those new triangles-- and why would filling triangles into triangles give you the same thing as a trapezoidal meta squiggle? Or is it the same thing? Which reminds me of another fractal made of triangles, where you make this snowflake by adding triangles to triangles. Each one in the middle third of each piece of edge, which you realize you could totally get with your tracing method. Start with one line with a bump, then trace each line to be a line with a bump, and so on. Which is interesting, because this time you're not squiggling back and forth, or forth and back, but bumping out the same way each time. Though, you could try doing it the other way. Which makes you really want to know what you get if you do the first thing. But instead of always starting with zigging out, you alternate starting zig out and zig in. And it's kind of bumping into itself. And this would definitely be more perfect with graph paper or something. OK, now it's starting to look like a triangle? On the one hand, not nearly as cool as a spirally thing you get when your zigs always go the same way. On the other hand, why would you get a triangle? And it's like a solid triangle too. If you kept doing this forever, would the triangle just fill up completely? These didn't do that. Although with this thing you have sections that are starting to fill up. Maybe here at some point that will happen, though it seems like it's just full of holes forever. You kind of wish you had a way to take some graph paper and skip all the way to what happens later in the sequence. Maybe if you had some sort of diagram, like-- this has a line with one right turn. The next one goes right, right, left. And then the next goes right, right, left, right, right, left, left. Is there a rule? Maybe it's just like the zig zag zag zigs. Or maybe not. OK, but there's probably some rule. Suddenly, a note lands on your desk from your friend Sam. Who writes, looks like you're concentrating pretty hard. Don't tell me you're actually doing math. As, if. You write back, no way. I'm just doodling this. And just to make extra clear it's not math, you turn it into a dragon, and name it the dragon curve. Yes. You don't want to crumple up your awesome dragon doodle, but you do have to throw it two rows over. So you neatly fold it into a note spear. Which just means you're folding it in half again and again, until it's easy to javelin across the room the moment the teachers back is turned-- Bam! Yes! Perfect landing. You watch as Sam unfolds it. And suddenly you feel like you see something familiar. Some sort of similarity between the paper and-- is that possible? You take your diagram, and fold it in half. And half again. And again. And wow, not only does it look like it's doing the same thing, probably, but it's also showing a new way to do it. Instead of keeping track of zigs and zags, you can just copy the old one, and add it 90 degrees from the other, which is totally traceable. Bam. Easy. Well, as long as you can keep track of what end to start from. And you don't even have to keep track of what order to draw the lines in. You just need to keep things roughly on a square grid so things line up. Easy. Until it gets too big for your paper and you have to dragon-ize it. It's funny, because one way it gets bigger and bigger. If you go on forever, it'll be infinitely big. But with the first way, it stays basically the same size. You just draw more details. Which means if you do it forever, the line will still get infinitely long, but the total size will stay the same. Will that even work? An infinitely long line all squiggled up into a finite area? And then with folding paper, the whole thing gets smaller and smaller until maybe it disappears entirely. Which you suppose makes sense because the edge of the paper stays the same length, no matter how you fold it. You can't make it longer and longer like copying it, where the length doubles each time. So instead, the whole thing gets smaller and smaller. Maybe this grid really would fill up until that infinitely long line is squiggled up into an actual solid two dimensional triangle with no empty space left in it. Maybe that would make sense. Or be crazy. You know lines are infinitely thin, but if you had infinitely much of it, maybe the infinities would cancel out or something. Like the line gets closer and closer to itself until it actually touches, but doesn't overlap, but with no space between. Which would make no sense, unless you do it everywhere at once, so who can tell the difference. Yeah, that totally sounds legit. Although, in this one, there's holes that never get filled at any point, so you'd still have an infinite line, but it doesn't fill up space, it's just all holes forever, but it also never overlaps, So where is all that infinite line going? Anyway, class is over, so you pack up and save the question. After all, you've got math class again tomorrow.