Doodling in math: Dragon dungeons

So you're back in math class, completely missing whatever it is your teacher is saying about logarithms, because you're too busy trying to keep yourself from getting brainwashed into thinking there's some sort of fundamental truth to the words and notations she wants you to memorize rather than them just being one arbitrary choice used to represent concepts that no one in the education system expects you to understand anyway. And to protect yourself from this assault of meaningless syntax, you have been forced to rely on dragons, that is, fractal dragon curves. The thing about dragon curves is that, like hobbits, they are tricksy. You start with a simple L and then replace each line of the L with a new L, alternating whether they go on one side or the other. And you just keep going. You get a feel for it after a while. The dragon and its particular perpendicularities. Then, somehow, all of a sudden you've got Smaug. I mean, other fractals can be cool too. But some of them are less surprising. Not naming any names-- [COUGH, COUGH] Koch curve. I mean, really? Make a spiky thing. And then make all the lines of that into spiky things. And make all the lines of that into spiky things, and surprise-- you get a really spiky thing. Not that the Koch curve is terrible-- but it's no dragon-- well, maybe a Hungarian Horntail. Or you could just take the middle part to get some sort of fractal architect fortress or dungeon. But there are some practical concerns for the fractal dragon dungeon architect, such as the amount of paint needed to paint your dungeon in bright happy colors or blood. Usually, in the 3D world, you paint 2D surfaces. So you could measure the amount that needs to be painted in square meters. But in 2D, you're painting a one dimensional surface-- just a line of paint-- which you could measure in meters. So say this wall needs one meter of paint. But when you iterate the fractal, every line gets this equilateral triangle bump put in the middle third. All the lines are equal and all equally one third the length of the previous thing. So each line needs one third a meter of paint, for a total of one and a third meters of paint. And then each of those sections does the same thing, growing in a 4 to 3 ratio. So you multiply 1 and 1/3 by 1 and 1/3 and-- hey, why do we write 1 and 1/3 like this, when what we really mean is 1 plus 1/3. This looks like 1 times 1/3, which is just 1/3, not 4/3. Wow. Why would anyone think it's a good idea to notate 5 and 1/2 in the same way we would notate 5 times 1/2? And how many mistakes has that caused? Anyway, you continue your paint calculations with a personal vow to never write compound fractions without the and part ever again. Meanwhile, you don't bother to simplify it because-- meh-- distractions. But then you get bored with that and start thinking about how you could grab the ends of each iteration and stretch the whole thing out like a piece of bent string to measure it. Each time it's like you take the whole amount from last time, divide it into thirds, and add one more third. Not only does it increase each time, but the amount that it's increasing keeps increasing. So you know it's going to approach infinity. If you had the full final Koch curve and started pulling the ends apart, you'd just keep pulling and pulling and pulling. And it would keep unfolding and unraveling. But you'd never be able to pull it taught, unless, of course, you have infinitely long arms and can increase your pull speed until it's infinitely fast. But Einstein might have a thing or two to say about that. The point is you realize you're going to have to leave your dungeon fortress unpainted even though that leaves it more exposed to the elements. You wonder if Sauron ever worried about his tower rusting or suffering water damage. Maybe you can make it out of snakes. What? I meant stainless steel, though you could make it out of snakes. No. Inch worms. No. Inch dragons, which are the tiniest dragons. No. A snake that ate a sheep and an elephant. No. A camel and now the elephant. But I didn't mean snakes in the first place. I meant stainless steel, which is not snakes at all. But wait. If you'd have needed infinite snakes-- no. If you'd have needed infinite paint to paint it, would you need infinite steel to build it? It does get bigger with each iteration. And every time you add a new set of towers, you add four times as many towers as you did the last time, approaching an infinite number of towers with infinite paintable surface. Yet somehow it seems like there's some sort of limit it will never go past. So seeing as this dungeon has limited space, it is probably necessary to plan an entire dungeon city as a matter of public safety. At least that's what you'd propose to the dark master of two dimensional dragon dungeon town, along with the budget and construction timeline. Because nothing says evil overlord like paperwork and bureaucracy. So you make the city one big Koch curve and-- hey, why do they call it the Koch curve when it's clearly more like a Koch spiky thing? Anyway, since each iteration scales by a factor of 3, the towers in the second iteration will be 1/3 third the height, which is-- um-- well, it's less than 1/3 the area. And if it were half the size like in Sierpinski's triangle, it would be-- oh wait. The area for an equilateral triangle with height 1 would actually be 1/2. But all that matters is comparing the area here to the area there. So let's just call this 1 steel area amount, which happens to be equal to 1/2 of whatever height unit squared. Yeah, sure. Anyway, a triangle half the height is 1/4 times the area. But actually that's for a solid triangle. But for Sierpinski's triangle, there's a space in the middle. So it's like 1/3 the area. Anyway, the point is for a solid triangle, it looks like 1/3 the height is 1/9 the area, whether the area is in arbitrary units or steel area amounts, 1/9. It's funny because it's the same for squares. 1, 4, 9-- then 4 times would be 16, then 25. They're all, well, square numbers-- 1 squared, 2 squared, 3 squared. And yet, it also seems like that if this is 1 triangled, this is 2 triangled. 3 triangled. You don't get the same area you would if it were a square. But however much area the first one has, it's still-- 4 times, 9 times, 16 times, 25 times-- 36 times as much. But actually, that makes sense because triangles are like half squares. So, of course, this is nine times the-- hey. Why do we write times like x? That's confusing. Because 25x means 25 times x. But that looks like 25x squared, which is 25 times x squared. But you might think that it's 25 times xx squared, which is 25x to the fourth. So we should probably replacing the old ambiguous times symbol with tiny pictures of newspapers. The division symbol will be similar, but make sure the headlines include something about politics. And-- wait, this is ridiculous. No one reads newspapers anymore. The new symbol will be a tiny website-- be sure to include mini columns, tabs, a search bar, and a picture album-- while the division symbol will be a tiny television tuned to whichever channel plays-- wait, no. The time symbol will be an hourglass and the division symbol either a Roman numeral 1, 2, or 3, depending on your university's sports budget. OK. So 2 times this size is 4 times the area. You wonder if it works that way for other shapes. For circles, area is pi r squared. Though, really you should say it pi r squared, or you might confuse it with pi r squared, which is important. Because when you do make it twice as big, it's the r you're multiplying by 2. So you have to make sure you multiply the r by 2 before anything else happens to it-- Like 2r squared. Not 2 r squared. And after you square the 2, you end up with a factor of 4, which is the same sort of 4 you get in squares and triangles, which is fun, because it suggests ways to divide up a circle into 4 equal regions. And then if you had something made out of squares, then of course, every time you scale it up, all the individual squares follow their square rules. And now that you think of it, maybe in theory to make any shape out of squares, if you had, like, infinitely many. And maybe you could do the same thing in 3D with cubes. I mean, if Minecraft has taught you anything, it's that any shape can be approximated by cubes. The more cubes the better. So whenever you make a 3D dragon dungeon and want to make it twice the size, it would take 8 times as much material, weigh 8 times as much, cost 8 times as much. You wonder if the pattern continues. And 4 dimensional dragon dungeon designers have to deal with their dungeons being 16 times as massive every they scale them up by two. And when you scale them up by 3, 3 to the fourth means it would be 81 times as heavy. You think your 4D dragon dungeon designers have to make their dungeons very small so that they don't go over budget or collapse on themselves. And you feel privileged to be a 2D dragon dungeon architect, where things are much more reasonable. Maybe you should switch to 1D, where when things get twice as big, they get twice as big, which all sounds good. But what was that with Sierpinski's triangle? Maybe this rule is wrong. Because you get three times as much stuff when you make it twice as tall. But does 3 equal 2 to the power of-- wait. Why do we write 81 as an 8 next to a 1 when usually putting things next to each other implies multiplication? But it's not 8 times 1, which is 8. And it's not 8 plus 1 either. It's 8 times 10, plus 1. If this really means that, you should just write it like that in the first place to avoid confusing 81 with 8 or 9, or whatever. Oh, except you might confuse 1 0 with 1 times 0. So you should probably write it as 2 times 5, which, of course, can just written as 2 5. So the result is that 81 is 8 times 2 times 5 plus 1. Much better. Because if by 825, you meant 8 times 2 times 5 times 2 times 5 plus, 2 times 2 times 5 plus 5, you should've just written it that way to begin with. But before you can fully implement your grand plans for notational reform, you realize math class is ending, which is super great because you have only one more class on logarithms to endure before the class moves on to something hopefully more interesting.