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# Doodling in math: InfinityÂ elephants

## Video transcript

So you mean you're
in math class, yet again, because they make
you go every single day. And you're learning
about, I don't know, the sums of infinite series. That's a high
school topic, right? Which is odd, because
it's a cool topic. But they somehow manage
to ruin it anyway. So I guess that's why they
allow infinite serieses in the curriculum. So, in a quite understandable
need for distraction, you're doodling
and thinking more about what the plural
of series should be than about the topic at hand. Serieses, serises,
seriesen, seri? Or is it that the singular
should be changed? One serie, or seris, or serum? Just like the singular
of sheep should be shoop. But the whole concept
of things like 1/2 plus 1/4 plus 1/8 plus
1/16, and so on approaches 1 is useful if, say, you want to
draw a line of elephants, each holding the tail
of the next one. Normal elephant, young
elephant, baby elephant, dog-sized elephant,
puppy-sized elephant, all the way down to
Mr. Tusks, and beyond. Which is at least
a tiny bit awesome because you can get an infinite
number of elephants in a line and still have it fit across
a single notebook page. But there's questions,
like what if you started with a camel, which, being
smaller than an elephant, only goes across a
third of the page. How big should the
next camel be in order to properly approach
the end of the page? Certainly you could calculate
an answer to this question, and it's cool that
that's possible. But I'm not really interested
in doing calculations. So we'll come back to camels. Here's a fractal. You start with these
circles in a circle, and then keep drawing
the biggest circle that fits in the spaces between. This is called an
Apollonian Gasket. And you can choose a different
starting set of circles, and it still works nicely. It's well known in
some circles because it has some very interesting
properties involving the relative curvature of
the circles, which is neat, and all. But it also looks cool and
suggests an awesome doodle game. Step 1, draw any shape. Step 2, draw the biggest circle
you can within this shape. Step 3, draw the biggest circle
you can in the space left. Step 4, see step 3. As long as there is space left
over after the first circle, meaning don't start
with a circle, this method turns any
shape into a fractal. You can do this with triangles. You can do this with stars. And don't forget to embellish. You can do this with elephants,
or snakes, or a profile of one of your friends. I choose Abraham Lincoln. Awesome. OK, but what about other
shapes besides circles? For example, equilateral
triangles, say, filling this other triangle,
which works because the filler triangles are the opposite
orientation to the outside triangles, and
orientation matters. This yields our friend,
Sierpinski's triangle, which, by the way, you can also
make out of Abraham Lincoln. But triangles seem to work
beautifully in this case. But that's a special case. And the problem
with triangles is that they don't
always fit snugly. For example, with
this blobby shape, the biggest equilateral triangle
has this lonely hanging corner. And sure, you don't have
to let that stop you, and it's a fun doodle game. But I think it lacks some of
the beauty of the circle game. Or what if you could change
the orientation of the triangle to get the biggest possible one? What if you didn't have
to keep it equilateral? Well, for polygonal
shapes, the game runs out pretty quickly,
so that's no good. But in curvy, complicated
shapes, the process itself becomes difficult. How do you find the
biggest triangle? It's not always obvious
which triangle has more area, especially when you're starting
shape is not very well defined. This is an interesting
sort of question because there is
a correct answer, but if you were going to
write a computer program that filled a given shape
with another shape, following even the simpler
version of the rules, you might need to learn
some computational geometry. And certainly, we can
move beyond triangles to squares, or even elephants. But the circle is
great because it's just so fantastically round. Oh, just a quick little
side doodle challenge. A circle can be defined
by three points. So draw three, arbitrary
points, and then try and find the circle
they belong to. So one of the things that
intrigues me about the circle game is that, whenever you have
one of these sorts of corners, you know there's going to be
an infinite number of circles heading down into it. Thing is, for every one
of those infinite circles, you create a few
more little corners that are going to need an
infinite number of circles. And for every one
of those, and so on. You just get an
incredible number of circles breeding
more circles. And you can see just how
dense infinity can be. Though the astounding thing
is that this kind of infinity is still the smallest,
countable kind of infinity. And there are kinds of infinity
that are just mind bogglingly infiniter. But wait, here's an
interesting thing. If you call this distance 1
arbitrary length unit, then this distance plus
this, dot, dot, dot, is an infinite series
that approaches 1. And this is another, different,
series that still approaches 1. And here's another, and another. And as long as the outside
shape is well defined, so will the series be. But if you want the simple kind
of series, where each circle's diameter is a certain
percentage of the one before it, you get straight lines. Which makes sense
if you know how the slope of a straight
line is defined. This is good because it suggests
a wonderful, mathematical, and doodle-able way to
solve our camel problem, with no calculations necessary. If instead of camels,
we had circles, we could make the
right infinite series just by drawing an angle
that ends where the page does and filling it up. Replace circles with
camels and, voila, infinite Saharan caravan
fading into the distance. No numbers necessary. Well, I have an infinite
amount of information I'd like to share with
you in this last sentence. [VOICE SPEEDS UP]