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In this subunit I give some examples of the use of fractal dimension in both abstract and real world fractals.
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In the previous subunit we derived a generalized definition for dimension, which could be applied to fractals.
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At each level we look at the logarithm of the number of copies there are of the object at the previous level...
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...and the reduction factor in the size of a side or a segment from the previous level.
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Using this definition, we calculated that the dimension of the Koch curve was approximately 1.26.
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Now, if you didn't understand the derivation of this, don't worry, you can still use the formula.
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And I should note that this is one of several methods used to calculate the fractal dimension of an object.
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It's called the "Hausdorff Dimension", after the German mathematician Felix Hausdorff.
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Let's look at another famous fractal, called the "Sierpinski Triangle", which was proposed by the Polish mathematician, Waclaw Sierpinski, in 1916.
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For this fractal, we start with a triangle. Our rule for iteration is to remove the triangle formed by connecting the midpoints of the three sides.
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So we take the midoint of each of the three sides of the triangle...
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...and we connect them together and remove the triangle that results.
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We're now left with three smaller triangles, each of whose sides are exactly one half the length of the original triangle side.
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Let's iterate through a few more levels...so we iterate once more...
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... we do the same rule to each triangle - each of these three triangles...
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So now we have nine smaller triangles, each of whose sides is one half the length of the side of the previous level.
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And we can do that again, and again...
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...and we start to get a really nice, interesting looking figure.
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Now, considering our definition of fractal dimension, here's a simple quiz question for you:
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...what is the specific formula for the fractal dimension of this figure?
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Now, this is a bit tricky,...
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because the term in the denominator is the reduction factor of the side...
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...not of the whole triangle,
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So it's the reduction factor in the length of the side of the triangle,..
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so remember that.