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The question we were left with is
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"What is the real length of an object, such as a coast line,
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which is a fractal, and which has self-similar ruggedness, or roughness, at many different scales?"
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As we'll do for all the complex concepts for this course, we'll address this question by looking at a simplified, idealized model.
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First, I need to point out that fractals are deeply connected to dynamical systems via the concept of iteration.
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That is, fractals are created via an iterative process reminiscent of the iteration we did on the Logistic Map example.
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To illustrate this, I'm going to focus on a particular, simple, yet beautiful fractal called the Koch Curve.
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It was invented in the early 1900s by a Swedish mathematician named Helga von Koch.
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We construct this fractal by starting with a simple curve.
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Here, a straight line segment, and iterating a simple rule on that curve.
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The simple rule says, at each successive time step, take each segment
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--here we only have one segment--
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delete its middle third,
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and replace the middle third by an angle
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where each side is one third the length of the original segment.
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Now we have a new figure with four segments,
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each segment of length one third the length of the original segment.
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So now, to go on to the next step, to iterate this, we apply our same rule to each of these four segments.
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That is, for each segment,
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we erase the middle third and
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we replace the middle third with an angle
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consisting of sides that are one third the length of the original segment.
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Of course, I am doing all this by hand, so the proportions are only approximate, but you get the idea.
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Now we have a new figure that has sixteen segments and
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we can iterate one more time by applying our rule.
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Delete the middle third of each segment and
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replace the middle third by an angle
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whose sides are one third the length of the original segment.
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Well, hopefully you get the idea, but let's actually try to do this a little bit more neatly.
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I've adapted one of the NetLogo library models
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called KochCurve.nlogo
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and I've put it on our course materials website.
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Let's look at it here.
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If we do "setup" we get our line segment--
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that's this red line segment
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--and every time I do "step" it's going to apply our rule.
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There we go.
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Apply it once.
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Now it applies the same rule to every segment, and so on.
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After we iterate it several times we can see that, in some sense,
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this could be considered to be an idealized coastline.
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Now we could ask, as we iterate it an increasing number of times,
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"What's the length of this coastline?"
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We can first measure the length of the curve at Level 0
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which we get to by clicking on "setup."
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That's just that original segment.
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Then we can measure the length at Level 1, Level 2, Level 3, 4, and so on.
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Measuring the length at these successive levels is basically
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the process of shrinking our ruler at each step.
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Let's go back to Level 0, where we have the original segment.
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So now, let's make a table by creating a note here.
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I'm making the font small so I can have it all on one line.
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And I'm going to measure for each level
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I'm going to measure the segment length--call that SegLength--
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the number of segments--call that NumSegs--
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and the curve length.
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For initial segment--that's at Level 0--the segment length, that's the length of our line segment.
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Let's just call that L.
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The number of segments is one, so the curve length is just L.
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OK, let's pretty this up a bit.
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I'm just going to stretch it out,
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so this will be our reminder of what we're measuring at each level.
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Now we go to Level 1 and apply our rule.
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Each new segment is one-third the length of our original segment,
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and there are a total of four segments
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so the length of the whole figure is 4/3L.
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What about Level 2?
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We apply our rule to each of the segments from Level 1.
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That is we apply it four times.
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The new figure looks like this.
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Now, here's a quick quiz.
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What is the length of the figure at this level?