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# Doodling in math: Dragon scales

Video transcript

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.