Now let's look at some actual applications of fractal dimension in the real world. I'm sure you've always wanted to know the dimension of cauliflower. Fortunately, it's been calculated at approximately 2.8. That is, if you look at a cross-section of cauliflower, this flat thing, you find that it's actually a little more than 2 dimensional. It's between 2 and 3 dimensional, due to the dense fractal self-similarity of the cauliflower branches. You may have noticed that the logistic map bifurcation diagram has a tree like structure which indeed is fractal. So for instance, if we blow up a little part of it here. And some parts of it are quite self similar. We blow up this part here, we see that this looks very much like the whole thing. And people have calculated the fractal dimension to be about .5. It has so many holes in it that it's not even quite 1 dimensional. The fractal dimension of coastlines has been calculated. Notice that the west coast of Great Britain has a higher dimension than the smoother coast of Australia. or the even smoother coast of South Africa. And all of these are a little bit more than 1 dimensional. If you look at them as going along a curve, sort of like the Koch curve. People have looked at fractal dimension of more abstract kinds of phenomena, like stock prices. This is from a paper that was published in 2000 looking at the Oslo Stock Exchange and looking at an index which listed a hundred day daily price records, a hundred week weekly price records, and a hundred month monthly price records. And you can see that all of them have very similar kinds of ups and downs, even though they're at very different time scales. The person who wrote this paper asked the question, "Are stock prices following a random walk?" And the project here was to compare the fractal dimension of these curves with the fractal dimension of a random walk to see if the dimensions were the same. If you look at random walks, they also have a lot of detailed ups and downs and self similar structure. But after some complicated mathematics this paper was able to answer "No." The fractal dimensions are not the same. Therefore, it's not likely the stock prices are following a random walk. Of course, in these real world examples there's no exact proportion of size reduction or clear cut number of copies. It's very different from the exact mathematical fractals like the Koch curve and the Sierpinski triangle. So there's a lot of caveats in actually applying fractal analysis to time series. A group of scientists looked at fractal dimension as applied to Jackson Pollock's drip paintings. They looked at the fractal dimension in these paintings and found that if you plot the year of the work versus their measurement of its fractal dimension you get this kind of increase, and this is claimed to be the evolution of complexity as measured by fractal dimension of Pollock's paintings over time. Using measures like fractal dimension for quantifying aspects of art has been a controversial subject for a long time. Later on we'll talk to John Rundell, a geophysicist who's been interested in using ideas from fractal geometry for a long time in the natural sciences.