In the previous subunit I talked about

the fractal dimension of various objects

such as coastlines.

But I haven't yet told you how these

real world fractal dimensions were computed.

It was possible for us to compute

fractal dimension from the Koch curve

and the Sierpinski triangle because

these are perfect mathematical fractals,

not real world objects.

But there's a lot of interest in computing

approximate fractal dimension in the real world

because it can often reveal insights about

natural or human created systems.

There are a lot of different methods for analyzing

fractals and whole books devoted to this subject.

Here I'm going to show you one commonly used

method for estimating fractal dimension,

the box counting method.

The box counting method is

closely related to this idea that

as you change the size of the ruler

that you measure a fractal by,

you get a different length as you go

further and further into smaller and smaller length scales.

So here's what the box counting method consists of.

You take a particular object.

Here I have a picture of the British coastline.

So what we do is overlay this figure

by a grid of boxes.

Each box has a certain length of its side,

which is the scale at which we're measuring this figure.

And what we do is count

the number of boxes in which

part of the black outline of the coast appears.

For example, it does not appear in this box,

even though this is in the middle of Great Britain,

so we don't count it.

So if we follow that procedure

and count the number of boxes

containing part of this black outline,

I got 36.

The length of the side was 10 units for each box.

Now I go to the next step

and I increase the size of the boxes.

So I'm now calculating the number of boxes,

but at a different scale.

Here because the length of the side

of the box was larger, I got fewer boxes

that contained part of this figure.

Then I would go up again.

Here the size of the box is larger again, 12.

And I got 27 boxes that contained part of the figure.

So you keep doing this,

accumulating this list of numbers.

Let's look at the relationship

between Hausdorff dimension,

which we already learned about,

and box-counting dimension.

If you recall, for the Hausdorff dimension

we had a relationship that is

the number of copies of a figure

from a previous level.

If we take the log of that,

that was equal to the dimension

times the log of the reduction factor

from the previous level.

It can be shown that if you do this

box-counting method, this can be approximated

by looking at the log of the number of boxes

and that's equal to the dimension

times the log of 1 over the length of the side.

D is called the box-counting dimension

and if you want to see the derivation of this

and other details about the relationship

between these dimensions,

take a look at Chapter 4 of the Fractal Explorer

which is a website about fractals.

And there's a link from our

Course Materials page on this.

Now the question is,

how do we actually get this D from our values

from numbers of boxes and

lengths of sides.

Well if you're up on your algebra

you might have noticed that this equation

is actually the equation of a straight line.

If we plot it on a graph

where the axes are here,

the log of one over the length of the side,

this x value,

and the y axis is log of the number of boxes.

And D would be the slope of that straight line.

So what we can do is we can

take the measurements that we made

at each level for the box counting

and we can plot it, each measurement, on this graph.

So here's some hypothetical measurements

that we might have gotten,

where the number of boxes goes down

as the length of the side goes up.

Notice this is 1 over the length of the side,

so as length of the side goes up, this goes down.

You can see that if this is actually true

these should form a straight line

whose slope is the dimension.

So we can estimate the dimension

by plotting these points,

doing our measurements for the boxes

and then plotting these points.

Drawing a straight line through them,

figuring out what the slope of that line is,

and that's our measured dimension.

And that's roughly what people did

to calculate things like the

fractal dimension of coastlines.