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

And what I wanna do is make another shape

out of this equilateral triangle.

And I'm gonna do that by taking each of the sides of this triangle,

and divide them into three equal sections, 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 wanna construct another equilateral triangle.

So the middle section right over here,

I am going to construct another equilateral triangle.

So it's going to look something like this.

And then, right over here,

I'm gonna put another equilateral triangle.

And so now I went from that equilateral triangle

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

And then I'm gonna do it again.

So, each of the sides now, I'm gonna divide into three equal sides.

In that middle segment, I'm gonna put an equilateral triangle.

I am going to put an equilateral triangle.

So in the middle segment, I am going to put an equilateral triangle.

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

So let me do it right there, and right there.

I think you get the idea, but I wanna make it clear, so let me just...

So then, like that, and then, look like that, like that.

And then, almost done for this iteration, this pass.

And then it'll look like that.

Then I can do it again.

Each of the segments I can divide into three equal sides

and draw another equilateral triangles,

like there, there, there, there, there, there.

I think you see where this is going.

And I could keep going on forever and forever.

So what I wanna 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 iteration, we look at each side,

we divide them in three equal side,

and then the next iteration were three equal segments,

and the next iteration,

the middle segment we turn to another equilateral triangle.

The shape that we're describing right here

is called the Koch Snowflake.

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

The Koch Snowflake,

and was first described by this gentleman right over here,

who was a Swedish mathematician Niels Fabian Helge von Koch.

I'm sure I'm mispronouncing it.

And this is 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 looks 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, 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 where we started

with that original triangle, that's that side.

And let's say it has length S.

And then we divide it into three equal segments.

We divide it into three equal segments.

So those are gonna be S/3, S/3, let me write it this way.

S/3, S/3, and S/3.

In the middle segment, you make an equilateral triangle.

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 its bump in it.

The length of this part right over here, this side,

now doesn't have just the length of S.

It is now S/3 * 4.

Before it was S/3 * 3,

now you have one, two, three, four 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 had that bump is going to be 4 * S/3, or equals 4/3 s.

So if our original perimeter when it was just a triangle is P sub 0,

after one pass, after we had one set of bumps,

then our perimeter is going to be,

so it's going to be 4/3 * the original one.

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

So if this was made up of three sides,

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

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

And then we take a second pass on it.

That's gonna be 4/3 times this first pass.

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

it's getting I guess a third bigger on every,

it's getting 4/3 the previous pass.

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

if you multiply any number by 4/3 an infinite number of times,

you are gonna get an infinite number of！an infinite length.

So, P infinity, P infinity,

the perimeter if you do it 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.

That I could actually draw a shape around this

and this thing will never expand beyond that.

And to think about, I'm not gonna do a really formal proof,

just think about what happens on any one of these sides.

So in that first pass, we have 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 could keep adding more and more,

you can add an infinite number of these bumps,

but you're never gonna go past this original point.

And the same thing is gonna be true on this side right over here.

It's also gonna be true on this side right over here.

Also going to be true at this side over here.

Also going to be true at this side over there.

And then also going to be true at 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 it will not have a larger area

than a shape that looks something like that.

And I'm just kind of drawing an arbitrary,

well I wanna 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,

eventhough 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.

Another thing, infinite perimeter, and finite area.

Now you might say, "Wait, uh, okay, this is a very abstract thing.

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

And there's an 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,

you know, I'm not an expert on the, you know,

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 + this distance

+this distance + that distance + that distance + that distance.

You know, look.

It has a finite perimeter.

Clearly, it has a finite area, but you know,

look that has a finite perimeter.

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

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

Instead of doing it that rough,

you need to make a bunch of smaller lines.

You got to make a bunch of smaller lines

so you can hug the coast a little bit better.

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

But then, let's say at some piece of coast, if we zoom in,

if we zoom in enough, if we zoom in enough,

the actual coastline's gonna look something like this.

The actual coastline will have all these little tidbits in it.

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

you were just measuring, you were just measuring that.

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

You're gonna have to do many many more sides.

You're gonna have to do something like this

to actually get the perimeter of the coastline.

And you say, "Hey, that is a good approximation of the perimeter."

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

it will actually turn out that it won't look exactly like that.

It will actually come in and out, like this.

Maybe look something like that.

So instead of having these rough lines, that just measure it like that.

You're gonna say, "Oh wait,

now 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'll get into the kind of 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.