This is a NetLogo model that shows
several examples of well known mathematical fractals.
It's called "ExamplesoFractals.nlogo",
you can download it from the link below or from the course materials page,
and here's how it works. You have a set of possible examples.
So for instance I will set up the model by clicking on Koch Curve.
And you can see the initial line of the Koch curve here.
And if I do iterate, it replaces the line by our familiar shape at level 1.
And it also over here on the right side, it shows me what N is, the number of copies of the line.
1, 2, 3, 4. And the reduction factor, 3.
And it also shows me the length of the fractal at this level.
And it's gonna show me a plot of fractal length over time.
So if I continue to iterate, we get our familiar series of iterations of the Koch curve.
Another very well-known fractal is the so called Cantor Set.
The Cantor Set works by starting with a line just like the Koch curve.
And at each iteration the middle 1/3 of the line is removed. And that's it.
At the next iteration each middle 1/3 of each line is removed.
And so on...
Ok, so here we can see that N, the number of copies, is 2.
And M is the factor by which each line segment shrinks.
That is 3. This is 1/3, a hole of 1/3, and 1/3.
And so on, and so the dimension of the Cantor Set
here is log 2 divided by log 3, which is .6.
So you can see that as we get closer and closer to infinity here by iterating,
the fractal shrinks in length. And because of the number of holes in it,
it's in between 1 dimensional and 0 dimensional,
where a 0 dimensional object is just a point.
So this is an interesting fractal in which the length shrinks
and the dimension is less than 1.
Our Sierpinski Triangle we also can iterate.
Here in the picture we showed before, these 3 triangles were actually filled in.
So just imagine that they are here. And you can create our Sierpinski Triangle.
You can try out these other fractals on your own
and see what the fractal dimension and length of them are.
So now you can do the exercise that's described in the next segment.