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# Doodling in math: Dragons

Video transcript

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.