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