Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
In our analysis of the Koch curve, we were fortunate that it behaves so nicely—that is, it lends itself to being measured. Many objects in nature are not so "nice." They may exhibit properties of self-similarity either only at limited scales (e.g., a fern leaf)—or only in a rough, approximate manner—or both.
Nevertheless, the concept of fractal dimension can generally be used to help describe and analyze naturally occurring phenomena and objects. In order to use this tool, however, we must replace our requirement of strict self-similarity with a notion of approximate, or statistical, self-similarity. Let's look at an example.
A famous application of fractals was posed as the question: "How Long is the Coastline of Britain?". This question embodies the fact that the value obtained when measuring the length of a complicated shoreline, such as that of Britain, depends on the length of the "ruler" that is used. Indeed, as with the Koch curve, we can convince ourselves that the length can be as long as we choose.
Benoit Mandlebrot saw that, if we view the coastline as a fractal, we can start to make some sense of its measurement. The problem is that the curve does not repeat its exact shape at different scales, as the Koch curve does. Rather, statistical features repeat at different-length scales. This might include the number of bays or peninsulas of a certain scale that one finds when measuring with a specific ruler.
One might find that one quadrant of the entire curve contains three bays and four peninsulas of length one unit (here we're letting a unit equal the length of one quadrant). If we then look at one-eighth of the curve, our unit becomes smaller, and the larger bays and peninsulas that showed up in the first view become more-or-less flat. New bays and peninsulas become evident, however, now that we have a more detailed view. We might find that the number of smaller bays and peninsulas (of length ) is similar to before—say, three bays and five peninsulas. So, although the exact shape is not the same at both scales, the number of significant features is about the same. This gives us the idea that the coastline is approximately self-similar.
We can use these properties to find the dimension of our coastline, but we need a new technique. The strategy we used previously to find the dimension of the Koch curve won't work in this case, because we do not have exact self-similarity, but, rather, only statistical self-similarity. To find out more about a method that might work, let's look again at the Koch curve and use rulers of different sizes to measure its length.
Recall that the first time we tried this, we found that the length of the curve approaches infinity as we take closer and closer looks. This time, however, instead of being concerned with the absolute length, we'll focus on how the length changes with the size of the ruler with which we choose to measure.
We start with a ruler of length one and find that the length of the curve is 4 units. Now, if we measure with a ruler as long (what might be considered a "more sensitive" ruler), we find that the length is 16 × units. As we use smaller and smaller rulers, the following table begins to take shape:
Notice that nowhere so far are we concerned with finding copies that look exactly like the entire curve—we care only about how the measured length of the curve changes with the ruler size. Hopefully, it is becoming apparent that this technique will work on curves that are not as uniform as the Koch curve. To find the relationship between these quantities, we can plot them on a graph.
Notice that the scales with which we are dealing suggest that we should look at a logarithm graph (log-log) of these data. This kind of plot is often useful when dealing with quantities (like these) that change exponentially. To make the log-log graph, we simply take the logarithm of all the quantities and re-plot the data.
Now, to find out how these two values are correlated, we can look at the slope of the best-fitting line. For simplicity's sake, we'll just choose the start and end points:
Subtracting this from 1 yields log , which is the same expression for dimension that we obtained earlier by looking at self-similar copies.
So, to find the dimension of our original coastline, which will allow us to come up with some sort of meaningful measurement, we can take a set of data that includes both the length of the ruler we use and the total length that we find. If we then plot the data on a log-log graph, we can find the relationship between the choice of ruler and the total length. This will generate a line (or we can choose a line of best fit), the slope of which is related to the dimension of the coastline.
Note that the slope of this line is equivalent to 1 minus the dimension of the coastline—or, alternatively, the dimension of the curve is equal to 1 minus the slope of our line. With this knowledge of the approximate dimension, we can select a unit of an appropriate size with which to make our measurements. This unit is not a length and not an area, but something in between—call it "larea" for now. Furthermore, it is specific to the coastline with which we are concerned, so it doesn't provide a means of determining whether a certain coastline is "longer" than another. However, it does enable us to talk about the relative curviness of shorelines. For instance, we would expect a coastline with a fractal dimension close to 1 to be much more featureless than a coastline whose dimension is closer to 2.
Statistical self-similarity abounds in nature. The surface of a dry landscape has the same features at many different scales. The branching of trees follows similar rules. One of Mandelbrot's great contributions was seeing how fractals relate to the natural phenomena and rhythms of our world.
Another person who was fascinated by natural rhythms was the American painter Jackson Pollock. Pollock was born in 1912 in Wyoming, and he relocated to New York at the age of 18. Through developing his craft as a painter, he changed his technique dramatically in 1947. The drip paintings he began to create, eschewing all traditionally accepted concepts of form and rigidity in favor of pure emotion and crazily strewn lines, brought him fame.
On the surface, it seems that his technique could be easily replicated by anyone with a bucket of paint, a canvas, a garage, and a penchant for extreme moods. Recent mathematical analysis of his paintings has shown, however, that copying a Pollock is not as easy as it may at first appear.
Richard Taylor, a physicist who pursued his analytical interest of Pollock's work while earning a masters degree in art theory from the University of New South Wales, studied the statistical self-similarity of Pollock's paintings. His method was to take a digital scan of a Pollock painting and section it into squares of different sizes for analysis, much as we sectioned off the coastline of Britain previously. For each square size, computers are used to identify certain physical traits of the paintings, somewhat analogous to the bays and peninsulas from the coastline example. Researchers found that Pollock's paintings exhibit statistical self-similarity, and are, therefore, fractals.
Fractals were not widely known until the ‘60s, and Pollock died in 1956, so it is highly unlikely that he was intentionally trying to paint mathematical objects.
Nevertheless, the fractal nature of his art is striking—and unique. In fact, it is used, in conjunction with other methods, to authenticate paintings purported to be Pollock originals. This "fractal fingerprint" method involves computing the fractal dimension of such a work and comparing it to the range of dimensions known to be exhibited in Pollock's paintings.
Taylor claims that his technique "shouldn't be regarded as a final word on Pollock authenticity, [although] it's a pretty nifty use of fractal math."
It is clear that fractals, and fractal dimensions, initially discovered as abstract mathematical objects, have a fascinating connection to the natural world. Indeed, many of the objects that we encounter on a daily basis cannot be measured within the traditional confines of one, two, and three dimensions as independent parameters. Rather, they must be evaluated on the basis of their scaling and self-similarity to be truly understood.