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.