Hello and welcome to Unit 3 of the course.

The topic for this unit is the box counting dimension.

It’s a different notion of the dimension,

slightly different than the self-similarity

dimension that we covered in Unit 1.

As we’ll see the box counting dimension is

little bit more concrete.

It’s easier to specify an algorithm for its calculation.

And it’s applicable to irregular fractals.

Fractals that aren’t exactly self similar.

The sort of fractals that

we encounter in nature all the time.

Another thing about the box counting dimension is

this is gonna serve as

a sort of a bridge or starting point

as we start to think about scaling more generally

so scaling in fractal behavior

not only in concrete geometric object,

but in process is unfold in time

and in statistical distributions.

So that will be coming up later in the course.

and this will be kinda transition to

key mathematical background

that will help us transition to that.

Lastly in this introductory video,

I wanna issue little warning that

calculating the box counting dimension

and kinda getting feel for it involves a little bit of tedium.

I usually try to avoid this.

But sometimes a little bit of tedium,

a little bit of work I think it’s essential to

like really understanding of mathematical idea

and box counting dimension is one of those times.

So it’s not gonna be terrible.

But will be just a little bit of tedium,

little bit of work some exercises that you’ll have to do.

But when you get through it

you will have a really good understanding

of the box counting dimension.

So to get started before

we dive into the box counting dimension.

I wanna think a little bit more about

why we even need dimensions.

What’s wrong with just length and area?

So what’s wrong with the idea of length and area?

Why can’t we apply these notions

meaningfully to most fractals?

Here is a way to think about this.

So here is the Koch curve again.

And we could ask what’s the length of this.

How long is the Koch curve?

Well we saw in Unit 1

we did a little calculation

that as we go through the steps of constructing this

the length just gets longer and longer and longer.

And so I would say in a certain sense,

the length is infinity or may be undefinied.

It just keeps growing.

And that seems a little weird.

I mean would like this to have a length may be

but it’s so jaggedly so wiggly that

the length turns out to be infinite.

So that’s cool and interesting

but may be not so helpful

if you wanna describe the shape qualified

do some math with it.

Similarly we could ask what about the area.

What’s the area of this?

Well, the area is going to be zero.

Because this is made up of lines

and lines are one-dimensional

and the lines

they just, you know

no matter a line

so wouldn't get have an area

so the area for this will be zero.

So on the one hand, we’ve got, yes it’s a fractal.

But it’s a relatively straight forward geometric object.

This got some complications.

But it seems like we are able to do

something mathematically with this.

It’s not infinite I mean yes it’s infinite length

but it fits between my hands, has 0 area.

But it does kinda seems to take up some space.

Because it's bumpy.

And so this situation is pretty unsatisfactory.

We wanna be able to do more than just say

Well, if you think about it as a line it’s infinite.

If you think about as an area it’s zero.

And so the answer is

length that means we're assuming it's one-dimensional.

Area means we're assuming it's two-dimensional.

And the answer is well it’s not either those.

It’s a different sort of dimension.

And so we need to think about get the dimension right

in order for us be able to talk about

the size length or area or something else.

And so that’s something that

the box counting dimension will do.

It will tell us what the dimension is.

It will also let us think about how to talk about

the size in certain sense of these types of sets

in a way that’s gonna be

a lot more satisfactory than this.

So in the next video we’ll get started

with the box counting dimension.

And we’ll start slow and as you probably guessed

we’re gonna counting up a bunch of boxes.

It’s tiny bit tedious.0

But it’s the best way I think to get

a really good understanding of these important ideas.