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.