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