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We can now experiment with boxcounting 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 boxcounting approximation.

So I'm going to do BoxCounting 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 BoxCounting 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 BoxCounting 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 boxcounting dimension here of 1.122,

which is a little bit different than the

Hausdorff dimension of 1.262.

Now that's because, remember

boxcounting 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.

Boxcounting 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 tradeoff.

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

boxcounting dimension approximates

the Hausdorff dimension.

And you'll see that the next exercise

is to test that out.