Doodling in math: Dragon scales

So you're back in math class again. It just-- it never stops. Day after day you find yourself trapped behind this desk with only your notebook for company, that pale mirror that reflects your thoughts on a comforting simulacrum of shared ideas. You're wasting another irreplaceable hour of your finite life not even pretending to listen to your teacher talk about logarithms. Or, at least, you think it's logarithms that she's trying to teach you. You haven't exactly been paying attention as you sit there casually, disposing of this only this moment you'll ever have, and-- oops, there goes another one. But either way, it's definitely logarithms in particular that you're not even pretending to learn, as opposed to, say, calculus, which your teacher wouldn't even expect you to be pretending to pay attention to. So instead, you're amusing yourself by doodling. Two days ago, you discovered the infinitely self-similar beast that is the fractal. And yesterday, you discovered that when you scale a two-dimensional thing up by two, it grows by a factor of four. And when you scale it by any amount, the area grows by that amount squared, unlike 1D where it's just that amount, and in 3D it's that amount cubed. And then 4D, it's that amount hyper-cubed. And in n dimensions, it's that amount to the n. So when you put it like that, you're actually making pretty good use of your limited time on this earth. And by limited time on this earth, I mean that we're all going to become immortal space robots. Anyway, you're continuing your plans for a fractal city of infinite dragon dungeons, triangles upon triangles upon triangles, each next set scaled down by a factor of 3. That's 1/9 the area. But there's four times as many. And the next set has four times as many as the last, but 1/9 times the size of the last. So the total weight of steel is 1 plus 4/9 plus 4 squared over 9 squared plus 4 to the 3 over 9 to the 3, dot, dot, dot, plus 40 to the n over 9 to the n. And maybe could learn how to add up an infinite series of numbers if your teacher would ever get past logarithms. But at least you know how to create the perfect fractal city, which is good, because understanding scale factors and city planning seems like the kind of thing that might come in handy if you want to help the species on our journey towards becoming immortal space robots. Really, the only thing that could make the city better is if it were twice as big. Or how about three times as big? So you can keep this part of the design and just draw the next iteration. Three times the scale in two dimensions means technically you'll need nine times as much steel. Though, as far as drawing these plans go, it's not nine times as difficult, but only four, since the hard part is the outside spiky part. And that's just copied four times, in order to scale up by three. OK, wait. Weird thing number one. Scaling up by three makes nine times as much steel. But it's the same thing copied four times, plus filling in this middle triangle. So it's also four times the steel plus 9. So if this mystery sum of an infinite series, total weight of steel were x, and 9x equals 4x plus 9, 5x equals 9, x equals 9/5 exactly. Take that, infinity! OK. Now weird thing number two. Look at just the edge, the actual Koch Curve not filled in. If it were a regular line, not an infinitely spiked one, scaling up by three would make it three times as much drawing, as expected. But if that spiky line were supposed to represent an infinitely spiked magical fortress city of dragon dungeon doom, by scaling it up in this way, you'd be losing details, making these long lines that should have had spiky bumps in them. Theoretically, no matter how much you scale up the city or no matter how finely you look at it, you'll never get any flat sections. This whole thing scaled down is the same as this section, which is the same as this section, which is the same as this. And so on. Three times as big is four times as much stuff. Not three, like if it were a normal 1D line, and certainly not nine, like that 2D area on the inside. Somehow, the infinity fractal-ness of the thing makes it behave differently from all 1D things and all 2D things. You convince yourself that all 1D things got twice as big when you make them twice as big, because you could think of them as broken up into straight line segments. And you know how line segments behave. And you convince yourself all 2D things scaled up by two get four times as much stuff, because 2D things can be thought of as being mad of squares, and you know how squares behave. But then there is this which has no straight lines in it. And there's no square areas in it, either. More than three to the one, less than three to the two. It behaves as if it's between one and two dimensions. You think back to Sierpinski's triangle. Maybe it can be thought of as being made out of straight line segments, though there's an infinite amount of them and they get infinitely small. When you make it twice as tall, if you just make all the lines of this drawing twice as long, you're missing detail again. But the tiny lines too small to draw are also twice as long and now visible. And so on, all the way down to the infinitely small line segments. Hm. You wonder if your similar line thing works on lines that don't actually have length. Wait. Lines that don't have length? Is that a thing? First, though, you figure out that when you make it twice as big, you get three times as much Sierpinsky triangle. Not two, like a 1D triangle outline. Not four, like a solid 2D triangle. But somewhere in between. And the in between-ness seems to be true, no matter which way you make it-- out of lines, or by subtracting 2D triangles, or with squiggles. They all end up the same. An object in fractional dimension. No longer 1D because of infinity infinitely small lines. Or no longer 2D because of subtracting out all the area. Or being an infinitely squiggled up line that's too infinate and squiggled to be a line anymore, but doesn't snuggle into itself enough to have any 2D area, either. Though in the dragon curve, it does seem to snuggle up into itself. Hm. If you pretend this is the complete dragon curve and iterate this way, there's twice as much stuff. That's what you'd expect from a 1D line if it were scaling it by two. But let's see, this is scaling up by, well, not quite two. Let's see. I suppose if you did it perfectly, it's supposed to be an equilateral right triangle. So square root 2. Hm. If it were two dimensional, you'd expect scaling up by squart root two would give you square root 2 squared as much stuff. And square root 2 squared is, of course, two, which is the amount of stuff you got. Odd how dragons turn out to be exactly two dimensional. But in the end, you get a fill-up thing with a fractal edge. And that's a lot like a filled in dragon dungeon. 2D area of pink steel on the inside, infinite fractal patina on the outside. Except the dragon curve still gets weirdness points for getting its 2D-ness from an infinitely squiggled up line rather than triangles that are 2D to begin with. And now you're plagued by another thought. What about an infinitely long line, a true line, rather than the line segments you've been dealing with? In a way, an infinitely long line doesn't have a length. Not a defined one. Like the infinitely short line, there's no real number capable of describing it. And if there's no number for it, how can you multiply that number by 2? If, when you make a line twice as long, you don't get a line with exactly twice the length, is a line really one dimensional? And if the Koch curve is made out of a line squiggled up enough to be more than 1D, what happens if you pull it back apart into a line? You try to imagine the correspondence. If this point ended up here, then this point would end up infinitely that way. And this one, twice as infinitely that way, which makes no sense. And this one also infinitely that way. And this one and this one. But they can't all end up at the same infinity, because they each have to have infinite line between them. So you suppose you're going to need a super long line that's like infinitely many infinitely long lines all put together. And maybe that line will not be one dimensional, but however many dimensions this thing is. You wonder how you would figure out the exact fractional dimension of the Koch curve or Sierpinski triangle. 3 to the dimension gives you four times as much stuff. But how do you find that number? If only there were a way to figure that out. Suddenly the bell rings, so you pack up to leave as quickly as you can, comfortable in the knowledge that you're never going to have to hear about logarithms ever again.