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  • Maybe you like to draw squiggles when you're bored in class.

  • Somehow the wandering path of the line,

  • that goes the monotone droning of the teacher,

  • perfectly capturing the way it goes on, and on,

  • about the same things, over and over, but without really going

  • anywhere in a deep display of artistic metaphor.

  • But once you're a veteran of bored doodling,

  • you learn that some squiggles are better than others.

  • Good squiggles really fill up the page,

  • squiggling around themselves as densely as possible,

  • in a single line that doesn't cross itself.

  • It's like the ideal would be to sit down

  • at the beginning of your least favorite class,

  • put your pencil on the page, and keep drawing a single line,

  • filling up more and more space until the bell rings, which

  • is basically what your teacher is doing, except with words.

  • You might find yourself developing some strategies.

  • For example, you're careful not to cut off a chunk of space,

  • because you might want to get back in there later.

  • And if you leave only a little room to get to a certain

  • section, then when you go there, you fill up a lot of it before

  • you leave that section again, or else, instead of a doodle,

  • you'll have an unhappy don't-dle.

  • Or maybe you decide to make a meta-squiggle.

  • A squiggle made out of squiggles.

  • This can be done kind of abstractly,

  • or extremely precisely.

  • For example, let's say you're drawing this simple squiggle,

  • then you draw that squiggle, using that squiggle.

  • But to make it fill up space nicely,

  • you make the outside parts bigger.

  • Then to make it precise, you make the number

  • of squiggles always the same.

  • It's easy to keep squiggling this squiggle all the way

  • across the page if you keep the rhythm of it in your head.

  • This one's like, down a squiggle,

  • up a squiggle, down a squiggle, up a squiggle, down a squiggle,

  • up a squiggle, down a squiggle, up a squiggle, down a squiggle,

  • up a squiggle.

  • But after you've done that awhile,

  • you decide to go a level deeper.

  • A squiggle, within a squiggle, within a squiggle.

  • That's right, we're going three levels down.

  • This serious business could go something like this.

  • Right a squiggle, left a squiggle, right a squiggle,

  • left, woop, right a squiggle, left a squiggle,

  • right a squiggle, left, woop, right a squiggle,

  • left a squiggle, right a squiggle, left, woop.

  • And the next one is even crazier.

  • Like, and up a squiggle, down a squiggle,

  • up a squiggle, down, woop, up a squiggle down a squiggle,

  • up a squiggle, down, woop, up a squiggle, down a squiggle,

  • up a squiggle, down, woop, wop, all the way over here.

  • And, down a squiggle, up a squiggle,

  • down a squiggle, up, woop, down a squiggle, up a squiggle,

  • down a squiggle, up, woop, down a squiggle, up a squiggle,

  • down a squiggle, up, wop, all the way over here.

  • OK, but say you're me and you're in math class.

  • This mean that you have graph paper.

  • Opportunity for precision.

  • You could draw that first curve like this.

  • Squig-a, squig-a, squig-a, squig-a, squig-a, squig-a,

  • squig-a, squig-a.

  • The second iteration to fit squiggles going up and down

  • will have a line three boxes across on top and bottom,

  • if you want the squiggles as close on the grid

  • as possible without touching.

  • You might remind yourself by saying, three a-squig, a-squig,

  • a-squiggle, three, a-squig, a-squig, a-squiggle.

  • The next iteration has a woop, and you

  • have to figure out how long that's going to be.

  • Meanwhile, other lengths change to keep everything close.

  • And, two a-squig, a-squig, a-squiggle.

  • Three, a-squig, a-squig, a-squiggle.

  • Three a-squig, a-squig, a-squiggle.

  • Two, nine.

  • Two, a-squig, a-squig, a-squiggle.

  • Three a-squig, a-squig, a-squiggle.

  • Three, a-squig, a-squig, a-squiggle, two, nine.

  • We could write the pattern down like this.

  • So what would the next pattern be?

  • Five.

  • Two a-squig, a-squig, a-squiggle.

  • Three a-squig, a-squig, a-squiggle.

  • Three a-squig, a-squig, a-squiggle.

  • Two, nine.

  • Two a-squig, a-squig, a-squiggle.

  • Three, a-squig, a-squig, a-squiggle.

  • Three a-squig, a-squig, a-squiggle, two.

  • Nine.

  • Two a-squig, a-squig, a-squiggle.

  • Three a-squig, a-squig, a-squiggle.

  • Three a-squig, a-squig, a-squiggle.

  • Two, nine.

  • And 15 all the way over to here.

  • And now, Yeah.

  • I can talk that fast totally.

  • OK, But let's not get too far from your original purpose,

  • which was to nicely fill a page with this squiggle.

  • The nicest page filling squiggles

  • have kind of the same density of squiggle everywhere.

  • You don't want to be clumped up here,

  • but have left over space there, because monsters

  • may start growing in the left over space.

  • On graph paper, you can be kind of precise about it.

  • Say you want a squiggle that goes through every box

  • exactly once, and can be extended infinitely.

  • So you try some of those, and decide

  • that, since the point of them is to fill up all the space,

  • you call them space filling curves.

  • Yeah, that's actually a technical term,

  • but be careful because your curve might actually

  • be a snake, snake, snake, snake, snake, snake, snake, snake,

  • snake, snake, snake, snake, snake, snake, snake, snake,

  • snake, snake--

  • Also, to make it neater, you draw the lines on the lines,

  • and shift the rules so that you go through each intersection

  • on the graph paper exactly once.

  • Which is the same thing, as far as space is concerned.

  • Here's a space filling curve that a guy named

  • Hilbert made up, because Hilbert was awesome, but he's dead now.

  • Here's the first iteration.

  • For the second one, we're going to build

  • it piece-by-piece by connecting four copies of the first.

  • So here's one.

  • Put the second space away next to it, and connect those.

  • Then turn the page to put the third sideways under the first,

  • and connect those.

  • And then the fourth will be the mirror image

  • of that on the other side.

  • Now you've got one nice curve.

  • The third iteration will be made out

  • of four copies of the second iteration.

  • So first build another second iteration curve

  • out of four copies of the first iteration--

  • one, two, three, four-- then put another next to it,

  • then two sideways on the bottom.

  • Connect them all up.

  • There you go.

  • The fourth iteration is made of four copies

  • of the third iteration, the same way.

  • If you learn to do the second iteration in one piece,

  • it'll make this go faster.

  • Then build two third iterations facing up next to each other,

  • and two underneath sideways.

  • You can keep going until you run out of room,

  • or you can make each new version the same size

  • by making each line half the length.

  • Or you can make it out of snakes.

  • Or if you have friends, you can each make an iteration

  • of the same size, and put them together.

  • Or invent your own fractal curve so

  • that you could be cool like Hilbert.

  • Who was like, mathematics?

  • I'm going to invent meta-mathematics like a boss.

Maybe you like to draw squiggles when you're bored in class.

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算数の授業での落書きスクイグル・インセプション (Doodling in Math Class: Squiggle Inception)

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    林宜悉 に公開 2021 年 01 月 14 日
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