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# Doodling in math: Dragon dungeons

Video transcript

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.