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In the previous subunit I talked about
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the fractal dimension of various objects
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such as coastlines.
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But I haven't yet told you how these
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real world fractal dimensions were computed.
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It was possible for us to compute
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fractal dimension from the Koch curve
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and the Sierpinski triangle because
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these are perfect mathematical fractals,
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not real world objects.
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But there's a lot of interest in computing
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approximate fractal dimension in the real world
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because it can often reveal insights about
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natural or human created systems.
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There are a lot of different methods for analyzing
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fractals and whole books devoted to this subject.
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Here I'm going to show you one commonly used
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method for estimating fractal dimension,
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the box counting method.
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The box counting method is
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closely related to this idea that
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as you change the size of the ruler
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that you measure a fractal by,
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you get a different length as you go
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further and further into smaller and smaller length scales.
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So here's what the box counting method consists of.
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You take a particular object.
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Here I have a picture of the British coastline.
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So what we do is overlay this figure
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by a grid of boxes.
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Each box has a certain length of its side,
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which is the scale at which we're measuring this figure.
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And what we do is count
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the number of boxes in which
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part of the black outline of the coast appears.
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For example, it does not appear in this box,
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even though this is in the middle of Great Britain,
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so we don't count it.
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So if we follow that procedure
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and count the number of boxes
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containing part of this black outline,
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I got 36.
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The length of the side was 10 units for each box.
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Now I go to the next step
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and I increase the size of the boxes.
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So I'm now calculating the number of boxes,
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but at a different scale.
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Here because the length of the side
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of the box was larger, I got fewer boxes
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that contained part of this figure.
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Then I would go up again.
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Here the size of the box is larger again, 12.
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And I got 27 boxes that contained part of the figure.
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So you keep doing this,
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accumulating this list of numbers.
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Let's look at the relationship
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between Hausdorff dimension,
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which we already learned about,
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and box-counting dimension.
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If you recall, for the Hausdorff dimension
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we had a relationship that is
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the number of copies of a figure
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from a previous level.
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If we take the log of that,
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that was equal to the dimension
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times the log of the reduction factor
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from the previous level.
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It can be shown that if you do this
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box-counting method, this can be approximated
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by looking at the log of the number of boxes
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and that's equal to the dimension
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times the log of 1 over the length of the side.
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D is called the box-counting dimension
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and if you want to see the derivation of this
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and other details about the relationship
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between these dimensions,
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take a look at Chapter 4 of the Fractal Explorer
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which is a website about fractals.
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And there's a link from our
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Course Materials page on this.
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Now the question is,
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how do we actually get this D from our values
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from numbers of boxes and
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lengths of sides.
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Well if you're up on your algebra
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you might have noticed that this equation
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is actually the equation of a straight line.
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If we plot it on a graph
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where the axes are here,
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the log of one over the length of the side,
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this x value,
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and the y axis is log of the number of boxes.
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And D would be the slope of that straight line.
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So what we can do is we can
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take the measurements that we made
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at each level for the box counting
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and we can plot it, each measurement, on this graph.
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So here's some hypothetical measurements
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that we might have gotten,
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where the number of boxes goes down
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as the length of the side goes up.
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Notice this is 1 over the length of the side,
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so as length of the side goes up, this goes down.
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You can see that if this is actually true
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these should form a straight line
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whose slope is the dimension.
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So we can estimate the dimension
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by plotting these points,
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doing our measurements for the boxes
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and then plotting these points.
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Drawing a straight line through them,
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figuring out what the slope of that line is,
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and that's our measured dimension.
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And that's roughly what people did
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to calculate things like the
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fractal dimension of coastlines.