We can now experiment with box-counting dimension
using the NetLogo model called
boxcountingdimension.nlogo.
Here you can see that this allows us to
iterate examples of fractals just like we did
in the previous model.
So let's go for 4 iterations of this.
But we can do at this point now is
compare the Hausdorff dimension,
1.262 that we calculated,
with a box-counting approximation.
So I'm going to do Box-Counting Setup here
and you can see there's an initial box length
set to 10, which you can change.
So here's the initial box right down here
and the increment is going to be 1.0.
So we're going to increase the box size
by 1 unit at each iteration.
Ok, so here this tells us
how many boxes there are and so on.
And watch over here as we do box counting
where the model is going to plot
the log of the number of boxes
versus the log of 1 over the box length
for each iteration. So let's just go ahead
and do that with Box-Counting Go.
Now this is just like we saw where we're
putting a grid of boxes over the figure.
You don't see the whole grid.
You're only seeing the boxes that contain pieces of the figure.
And at each time step, see iteration,
we see what the box length is
and the number of boxes that is being counted.
And here those values are being plotted.
And you see they're sort of beginning to
approximate a straight line.
So if we keep going, the boxes get bigger and bigger.
And then we can stop it by clicking again
on Box-Counting Go at any time.
I haven't actually run it for very long,
but I have some points and what I can do is
say Find Best Fit Line.
That does a linear regression
and computes a box-counting dimension here of 1.122,
which is a little bit different than the
Hausdorff dimension of 1.262.
Now that's because, remember
box-counting is just an approximation.
We can get a better approximation
if we start with a smaller initial box length,
or if we start with a smaller increment.
But that, of course, is going to take longer.
So let's start over with our Koch curve.
Iterate, iterate, iterate...ok.
And our approximation would also be improved
if we iterated more.
Box-counting Setup, and Go.
I can speed this up, but it's still kind of a slow calculation.
Net Logo is not known for its extreme speed of computing.
It's kind of a trade-off.
It's easy to program in, but not super fast.
But anyway, now you can run this, go away.
Go get a cup of coffee, like computer scientists like to do
while waiting for their program to finish.
And let it run for many iterations
and then see how well the
box-counting dimension approximates
the Hausdorff dimension.
And you'll see that the next exercise
is to test that out.