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