So let's say that this is
an equilateral triangle.

And what I want to do is
make another shape out

of this equilateral triangle.

And I'm going to
do that by taking

each of the sides
of this triangle,

and divide them into
three equal sections.

So my equilateral triangle
wasn't drawn super ideally.

But I think you'll
get the point.

And in the middle section
I want to construct

another equilateral triangle.

So it's going to look
something like this.

And then right
over here I'm going

to put another
equilateral triangle.

And so now I went from
that equilateral triangle

to something that's looking
like a star, or a Star of David.

And then I'm going
to do it again.

So each of the sides
now I'm going to divide

into three equal sides.

And in that middle
segment I'm going

to put an equilateral triangle.

So in the middle
segment I'm going

to put an equilateral triangle.

So I'm going to do it for
every one of the sides.

So let me do it right there.

And then right there.

I think you get the idea,
but I want to make it clear.

So let me just, so then like
that, and then like that,

like that, and then almost
done for this iteration.

This pass.

And it'll look like that.

Then I could do it again.

Each of the segments I can
divide into three equal sides

and draw another
equilateral triangle.

So I could just there,
there, there, there.

I think you see
where this is going.

And I could keep going
on forever and forever.

So what I want to
do in this video

is think about
what's going on here.

And what I'm actually
drawing, if we just

keep on doing this
forever and forever,

every side, every iteration,
we look at each side,

we divide into
three equal sides.

And then the next iteration,
or three equal segments,

the next iteration,
the middle segment

we turn to another
equilateral triangle.

This shape that we're
describing right here

is called a Koch snowflake.

And I'm sure I'm
mispronouncing the Koch part.

A Koch snowflake,
and it was first

described by this gentleman
right over here, who

is a Swedish mathematician,
Niels Fabian Helge von

Koch, who I'm sure
I'm mispronouncing it.

And this was one of the
earliest described fractals.

So this is a fractal.

And the reason why it
is considered a fractal

is that it looks the same,
or it looks very similar,

on any scale you look at it.

So when you look at it at this
scale, so if you look at this,

it like you see a
bunch of triangles

with some bumps on it.

But then if you were to
zoom in right over there,

then you would still see
that same type of pattern.

And then if you were
to zoom in again,

you would see it
again and again.

So a fractal is anything
that at on any scale,

on any level of zoom, it kind
of looks roughly the same.

So that's why it's
called a fractal.

Now what's particularly
interesting,

and why I'm putting it at this
point in the geometry playlist,

is that this actually has
an infinite perimeter.

If you were to keep doing
it, if you were actually

to make the Koch
snowflake, where

you keep an infinite number
of times on every smaller

little triangle
here, you keep adding

another equilateral
triangle on its side.

And to show that it has
an infinite perimeter,

let's just consider
one side over here.

So let's say that
this side, so let's

say we're starting
right when we started

with that original
triangle, that's that side.

Let's say it has length s.

And then we divide it
into three equal segments.

So those are going to
be s/3, s/3-- actually,

let me write it this way.

s/3, s/3, and s/3.

And in the middle segment, you
make an equilateral triangle.

So each of these sides
are going to be s/3.

s/3, s/3.

And now the length of this new
part-- I can't call it a line

anymore, because it
has this bump in it--

the length of this part right
over here, this side, now

doesn't have just a length
of s, it is now s/3 times 4.

Before it was s/3 times 3.

Now you have 1, 2, 3, 4
segments that are s/3.

So now, after one
time, after one pace,

after one time of doing
this adding triangles,

our new side, after
we add that bump,

is going to be four times s/3.

Or it equals 4/3 s.

So if our original
perimeter when it was just

a triangle is p sub 0.

After one pass, after
we add one set of bumps,

then our perimeter is going to
be 4/3 times the original one.

Because each of the sides are
going to be 4/3 bigger now.

So this was made
up of three sides.

Now each of those sides
are going to be 4/3 bigger.

So the new perimeter's
going to be 4/3 times that.

And then when we take
a second pass on it,

it's going to be 4/3
times this first pass.

So every pass you take,
it's getting 4/3 bigger.

Or it's getting, I guess,
a 1/3 bigger on every,

it's getting 4/3
the previous pass.

And so if you do that an
infinite number of times,

if you multiply any
number by 4/3 an

infinite number of
times, you're going

to get an infinite number
of infinite length.

So P infinity.

The perimeter, if you do this
an infinite number of times,

is infinite.

Now that by itself
is kind of cool,

just to think about something
that has an infinite perimeter.

But what's even neater is that
it actually has a finite area.

And when I say a finite
area, it actually

covers a bounded
amount of space.

And I could actually
draw a shape around this,

and this thing will
never expand beyond that.

And to think about
it, I'm not going

to do a really
formal proof, just

think about it, what happens
on any one of these sides.

So on that first pass we
have that this triangle

gets popped out.

And then, if you think about it,
if you just draw what happens,

the next iteration you draw
these two triangles right

over there.

And these two characters
right over there.

And then you put some
triangles over here,

and here, and here,
and here, and here.

So on and so forth.

But notice, you can keep
adding more and more.

You can add essentially an
infinite number of these bumps,

but you're never going to
go past this original point.

And the same thing is going
to be true on this side

right over here.

It's also going to be true
of this side over here.

Also going to be true
at this side over here.

Also going to be true
this side over there.

And then also going to be
true that side over there.

So even if you do this an
infinite number of times,

this shape, this Koch
snowflake will never

have a larger area than
this bounding hexagon.

Or which will never have a
larger area than a shape that

looks something like that.

I'm just kind of
drawing an arbitrary,

well I want to make it
outside of the hexagon,

I could put a circle
outside of it.

So this thing I drew in blue, or
this hexagon I drew in magenta,

those clearly have a fixed area.

And this Koch snowflake
will always be bounded.

Even though you can add these
bumps an infinite number

of times.

So a bunch of really
cool things here.

One, it's a fractal.

You can keep zooming in
and it'll look the same.

The other thing, infinite,
infinite perimeter,

and finite, finite area.

Now you might say, wait Sal, OK.

This is a very abstract thing.

Things like this don't actually
exist in the real world.

And there's a fun
thought experiment

that people talk about
in the fractal world,

and that's finding the
perimeter of England.

Or you can actually
do it with any island.

And so England looks
something like--

and I'm not an
expert on, let's say

it looks something
like that-- so at first

you might approximate
the perimeter.

And you might measure
this distance,

you might measure this
distance, plus this distance,

plus this distance, plus that
distance, plus that distance,

plus that distance.

And you're like look, it
has a finite perimeter.

It clearly has a finite area.

But you're like, look, that
has a finite perimeter.

But you're like, no,
wait that's not as good.

You have to approximate it a
little bit better than that.

Instead of doing
it that rough, you

need to make a bunch
of smaller lines.

You need to make a
bunch of smaller lines

so you can hug the coast
a little bit better.

And you're like, OK, that's
a much better approximation.

But then, let's say you're
at some piece of coast,

if we zoom in enough,
the actual coast line

is going to look
something like this.

The actual coast
line will have all

of these little divots in it.

And essentially, when you did
that first, when did this pass,

you were just measuring that.

And you're like, that's not
the perimeter of the coastline.

You're going to have to
do many, many more sides.

You're going to do something
like this, to actually get

the perimeter of the coast line.

And you're just like,
hey, now that is

a good approximation
for the perimeter.

But if you were to zoom in on
that part of the coastline even

more, it'll actually turn out
that it won't look exactly

like that.

It'll actually come
in and out like this.

Maybe it'll look
something like that.

So instead of having these
rough lines that just measure it

like that, you're going
to say, oh wait, no, I

need to go a little bit closer
and hug it even tighter.

And you can really keep
on doing that until you

get to the actual atomic level.

So the actual coastline of
an island, or a continent,

or anything, is actually
somewhat kind of fractalish.

And it is, you can
kind of think of it

as having an almost
infinite perimeter.

Obviously at some
point you're getting

to kind of the atomic level,
so it won't quite be the same.

But it's kind of
the same phenomenon.

It's an interesting thing
to actually think about.