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# Doodling in math: Squiggle inception

Video transcript

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.