In 1967, legendary mathematician Benoit Mandelbrot posed a subtly deviant question: How long is the coastline of Great Britain? I would have been the first one to pull out an atlas and a piece of string, measure around the outline of the country, and give the simplest possible answer. But what if this did not satisfy Mandelbrot’s inquiry? Well, I could then perhaps declare a spontaneous leisure tour of the countryside and ride around the coastline on my bicycle to measure the distance. This would generate an interesting result, for my new distance would far exceed my string-and-atlas measurement. Mandelbrot is not known for taking the easy way out; what if he still demanded a more rigorous answer? With the madness of mathematics in my heart, I could travel around the coastline with a yard (or meter) stick carefully measuring each beach and every crag in the dangerously rocky coastlines. Again, I would obtain an even longer measurement for the perimeter. Perhaps I could even measure the coastline one centimeter at a time or break out my trusty electron microscope and measure the distance around each cell and particle on the coastline! This would give an astronomical distance measurement. Is there any end to this absurdity? No-the coastline of Great Britain is a fractal.A fractal is a system that resembles itself at different levels of magnification. Erosion is a fractal process-a small eroded section often resembles the larger eroded surface it is a part of. Despite the mathematical nature of erosion, useful applications of fractal mathematics in the field have remained evasive until recently. Scientists at the University of Rochester in New York have used fractals not only to model but to predict erosion as well. It is time for this groundbreaking research to graduate from the laboratory to the field. The recent findings open a warehouse of mathematical tools to the erosion community. Fractal mathematics fills volumes of dusty books, and the critical discovery by Yonathan Shapir, Ph.D., and his colleagues pierces the membrane between physical erosion and years of complicated mathematical theory. The future of erosion research promises to be an exciting one as the Rochester laboratories pass the baton to geologists, engineers, and landscape architects. Fractals, Fractals EverywhereA fractal is essentially anything that shows self-similarity at different levels of magnification. That is, if a small section resembles the whole object, that object is a fractal. A small distance of rocky coastline is similar in shape to the entire coastline seen from the air, so the coastline is a fractal. A weathered and lonely eroded rock resembles its parent mountain range, so mountains are fractals. Anything that branches out and repeats itself is also a fractal. Notice tiny streams resemble the rivers they flow into. A tree branch resembles the whole tree, and the same concept applies within an individual leaf. Even the human circulatory system is a fractal. In fact, fractals are abundant in living systems-just look closely at a piece of broccoli the next time you sit down for a salad.Mandelbrot introduced the concept of fractals in the mid-1970s. A few years later, he published his groundbreaking book, The Fractal Geometry of Nature. The equations governing fractals are numerous and nonlinear, but the solutions, when visualized graphically, produce breathtaking images. Still, mathematical fractals remain isolated from the physical world. They are trapped on chalkboards and textbook pages and have surprisingly few applications, despite their uncanny similarity to natural processes.Erosion modeling and prediction is the ideal field for fractals to be of use. Coastlines and riverbanks are fractal, mountains are fractal, and even root structures that prevent erosion are fractal. Furthermore, weather patterns are intricately related to fractals by chaos theory. Scientists at the University of Rochester have recently disturbed fractals’ theoretical slumber. The research team used fractals to predict when an extreme point would erode through a given surface. Fractal mathematics has stepped off the dusty page and into the laboratory, and now fractals are ready for practical development within the erosion community.
Mathematical WondersIt is important to understand the basics of fractal mathematics before delving into how fractals are used to model landscapes. We’ll start at the beginning and work our way up to nonlinear fractals, fractal landscapes, and erosion prediction.One could spend an entire summer contemplating the wonder of one of the first fractals, the von Koch snowflake. Discovered by mathematician Helge von Koch in 1904, this snowflake is a prime example of the fractal geometry related to the problem of measuring Great Britain’s coastline.
Figure 1. Four iterations of the von Koch snowflake. To display the final image, we would need to iterate it for infinity.Often in geometry, we start with a simple shape and branch out to increasingly complicated figures. The same holds true for the von Koch snowflake: We start with an ordinary triangle. Then, innocently enough, we add on another triangle, producing a star shape (Figure 1). Notice that there are actually six new triangles around the perimeter. Next, let’s add another triangle to each one of those, producing six new stars and 18 new triangles to further toy with. Remember the fractal nature of Great Britain’s coastline? The snowflake is beginning to resemble the complexity of an eroded coastline. But we are not finished yet. To generate the von Koch snowflake in its entirety, we need to go through an infinite number of triangle-adding iterations. Simple geometry of triangles has produced something so strange and beautiful that we need a whole new field of mathematics to cope with it. This new mathematics is, of course, fractals.Don’t pass off the von Koch snowflake as an aesthetic curiosity just yet. There is something almost disturbing about the finished flake. Each time a triangle is added to a side of the shape, it increases the distance of that side by 4/3. Therefore, if our original triangle had sides with distance 1, the sides after one iteration have distance 4/3. This means that the distance around the shape’s perimeter has increased from 1 + 1 + 1 = 3 for the original triangle to 4/3 + 4/3 + 4/3 = 4 for the new shape. With each iteration, the distance around the shape increases. After an infinite number of iterations, the von Koch snowflake is finished. If we were to set out to measure its perimeter, we would be baffled. Even though the flake fits neatly onto the page, the length around its perimeter is infinite!Another common fractal is known as the Sierpinski triangle, discovered in 1915 by mathematician Waclaw Sierpinski. This fractal is generated by drawing triangles within triangles over and over again, ad infinitum. Once we create an infinite number of triangles, we can zoom in on any spot and reproduce the entire Sierpinski triangle again! No matter how many times we magnify the shape, we always get a perfect Sierpinski triangle. The Sierpinski triangle comes with its own paradox. Suppose instead of drawing triangles in triangles, you were cutting them out. Start by cutting out the large triangle in the center, and then repeat by cutting the center triangle out of each smaller section (Figure 2). Could you ever, even after cutting out an infinity of triangles, carve out the entire area of the original triangle?
Figure 2. Five iterations of a Sierpinski triangle. How many iterations would it take to turn the entire Sierpinski triangle white?The von Koch snowflake and Sierpinski triangle are certainly bizarre shapes, but it is easier to harness these figures with mathematics than it is with the random coastline of Great Britain. Mathematicians prefer to work out complicated theories with simple shapes before adding in the randomness and nonlinearities found in the real world. After we begin to understand linear fractals, we can attempt to apply them to nonlinear situations.Closer to Nature: Nonlinear Fractals
Figure 3. There are limitless possibilities for fractal art. The software is surprisingly easy to master, and amazing images can be completed in minutes.Nature is nonlinear. It is chaotic and defiant of even our most cherished laws of physics. A weather prediction program went awry after running for an hour and gave birth to chaos theory. Weather is inherently chaotic, and forecasters face downright daunting mathematics every day attempting to tell us if it will rain or not. Now, combine the near impossibility of accurately predicting rain with the tedium of predicting wind, snow, ice, and other erosion-causing events. It becomes frighteningly evident that no matter how many super computers we have grinding away, it will take more than chaos theory, classical dynamics, and geological models to predict erosion.For fractals to be of any practical use in predicting erosion, they would need to be examined in a nonlinear context. Mathematicians took such shapes as the von Koch snowflake and first developed a sound mathematical structure. They then applied the math to nonlinear equations in the complex (“imaginary”) plane. From this numerical experimentation came the psychedelic fractal art that is common on posters and T-shirts today (Figure 3).
Figure 4. The famous Mandelbrot set takes seconds to generate and a lifetime to master. Here we have (a) the original set, (b) 4x magnification, and (c) 8x magnification. No matter how many times we zoom in, we will always see this level complexity.The most celebrated nonlinear fractals are the Julia and Mandelbrot sets. As with the Sierpinski triangle, we can arbitrarily zoom in on these fantastic fractals and see amazing structure at any level (Figure 4). But unlike Sierpinski triangles with their identical self-similarity, nonlinear fractals do not reproduce themselves exactly upon magnification. Each closer look provides an image that is totally new. Adjusting a few initial parameters leads to drastic changes in the final fractal, which is why fractals are so much fun to toy with on idle afternoons. Nonlinear fractals are far more reasonable for modeling geological and living systems. When was the last time you saw a perfect triangle in nature?On the other hand, when was the last time you saw something as bizarre as the Mandelbrot set in nature? By increasing the complexity of fractals, we make a big step in modeling erosion. Nonlinear fractals are self-similar but not exactly the same under magnification. A single beach might resemble the entire coastline, but we cannot expect it to look exactly the same. Mandelbrot and others soon discovered how to use the new nonlinear mathematics to generate fractal surfaces.Fractal SurfacesWith the help of raw computing power, we can manipulate fractals to form impressive landscape images. Fractal landscapes are used in a variety of computer artwork, including movies. Unfortunately, the well-trained eye will quickly raise suspicion about the accurate simulation of real erosion by these fractals. The computer-rendered mountains rarely take erosive processes into account. Even so, fractal landscapes are an important step in unleashing fractal mathematics to the erosion community.The science of generating fractal landscapes is not as complicated as one would think, but the one major requirement is randomness. Fractal terrains start with-you guessed it-a triangle. Connecting the midpoints of a triangle forms four new triangles. Then the important step is to shift each midpoint out of the page a random distance. Now our shape no longer resembles anything like a triangle; it is three-dimensional and rough. Repeat this process in a fractal-like nature with the smaller and smaller morphed triangles, and the surface gets a rougher, rocky appearance. Add a computer-generated light source, and you end up with fractal mountains that look strikingly real (Figure 5).
The random distance the midpoints are pulled out of the page is based on Brownian motion. By adjusting this parameter, you can change the roughness of your modeled surface. Incidentally, Brownian motion is the random movement of a tiny particle suspended in a liquid. I wonder if Robert Brown suspected in 1828 that the random motion he observed would later be used to create amazing fractal landscapes.
Figure 5. Fractal landscapes are an important step in modeling erosion. There is software available to generate not only mountains but entire fractal planets.The mathematics involved in fractal surfaces can be mind-boggling, but there are user-friendly software packages available, such as MojoWorld (www.pandromeda.com), that allow a user to create unique fractal mountains, clouds, and entire planets without first completing a Ph.D. in computer science.I spoke with Ken Musgrave, Ph.D., who has set the standard in fractal landscape research for years and is the creator of MojoWorld. Musgrave has been working with fractals since the mid-1980s when he worked with Benoit Mandelbrot himself at Yale University. Musgrave is honest about the limitations of fractal landscapes. While they might look realistic at first glance, a trained eye will notice that erosion is generally missing from the gorgeous fractal mountains. People often get excited about fractal mountains, but the landscapes, while fascinating, are not the missing link between fractal mathematics and erosion. Expecting to use fractal-landscape software to scientifically model erosion is similar to attempting to use a von Koch snowflake to better understand a real snowflake. At first glance it seems like an appropriate model, but closer investigation reveals that the two are only similar to a first approximation. Musgrave explains, “People are profoundly insensitive to the actual appearance of terrains in nature.” Because of this, we pass off fractal landscapes as mountains when, upon closer examination, they do not look scientifically like real mountains at all.
Figure 6. Ken Musgrave has added erosion to fractal surfaces in these before (a) and after (b) images. To incorporate erosion into the initial fractal rendering is essentially impossible.Musgrave and his colleagues have used fractals to model erosion (Figure 6), but combining erosion with fractal landscapes is virtually impossible. This is because drainage networks and erosion are context sensitive-that is, incomputable. Take, for instance, a mountain stream. A boulder upstream drastically changes the geography downstream. Each fractal change in the landscape produces a change in the erosion process, but each erosion change in turn alters the drainage network and thus the landscape. You can’t change one without changing the other. The problem is not with the speed of computers desperately crunching numbers; instead it’s with the fundamental annals of computer science. Even with the striking resemblance of fractal landscapes to real mountains, fractal-modeling software is of little practical use to the erosion control community except for as a fun way to goof around creatively after a hard day’s work.So here we are with 30 grueling years of fractal mathematics, nonlinear fractals, and computer programs that can create entire fractal planets, yet we have essentially nothing for practical use in the field of erosion control. It’s frustrating when a whole field of mathematics just cannot find its real-world application. The University of Rochester scientists must have found this situation unbearable just before they discovered how to model and even predict erosion using fractals.Fractals, Erosion, and Shapir
www.fractalarts.comYonathan Shapir, Ph.D., approached the problem of fractals from a physical instead of mathematical perspective. He and his colleagues have successfully linked fractals to the physical erosion process. They’ve also verified their mathematical results experimentally. The group continued by using fractals to predict when a single critical point will erode through a surface. This groundbreaking research finally opens up fractals to practical use in erosion. Since the Rochester results are still in laboratory phases, the possibilities are wide open for the erosion and geology community to use new data on larger scales for the first time ever.The Rochester group began by examining the relationships between cyclical processes and erosion/deposition. Cyclical processes are, of course, found frequently in nature; just consider rainfall, ocean waves, spring runoff, and so on. The scientists found that cyclical erosion or deposition processes create fractal surfaces. After working out fractal equations analytically (equations and proofs), Shapir and his colleagues verified their results using numerical calculations. With a solid mathematical foundation in place, they turned to experimental methods. They began with computer simulations. Each data point consisted of 50-5,000 runs, and each run consisted of 500-10,000 cycle simulations to ensure statistical accuracy. The results of the computer simulations were exactly what the theory predicted: Cycles of erosion and deposition create a self-similar fractal surface. The final test was to see if physical surfaces display the fractal properties predicted by the theory. The research group used a silver solution deposited and eroded over several cycles. They examined the surface roughness with an atomic-force microscope at different phases, and sure enough, it had the predicted fractal properties. Observing fractal behavior within the laboratory deposition chamber is one thing, but what about on beaches? We know that the coastline is a fractal and that the high and low tides are cyclical processes. I asked Shapir about applying his research to beaches. “The formation of beaches under the cyclical low and high tides is a typical example of cyclical fractal growth,” he says. “Our theory could, in principle, turn into a predictive model. For that, we need as an input how the coast forms under high tide alone-in particular, the fractal properties-and under low-tide conditions alone. We will then be able to calculate how these two processes combine into a cyclical process and predict the fractal properties of the coast formed in this cyclical process.”Once Shapir’s group demonstrated experimental reliability of their mathematics, they combined fractals with a branch of math known as “extreme-value statistics” and went on to predict when a single point will erode through. Again, this research is still in its infancy; there is a plethora of research open to the erosion field based on these preliminary results. The Rochester group discovered that the extreme point of a surface is related to the surface roughness. This result is significant; it means that by examining solely surface roughness, you can predict when a point will erode to a given depth. Using similar methods, the scientists modeled erosion backward to determine what a surface looked like before it eroded. The next step in this novel research is to test the predictions experimentally.There are immediate applications for Shapir’s research. For instance, the fractal models can be used to figure out at what point the surface inside a battery will erode through. Other immediate applications include determining how long a steel pillar can support a bridge before it rusts and how long a container can hold an acid before it springs a leak. But even more important is the fact that a physical tie has been formed between fractal mathematics and the erosion process. What we have now is a gold mine of mathematics waiting to be extracted by geology and landscape architecture. The world of erosion is a strange one; chaotic forces carve the landscape with fractal patterns. Shapir himself notes, “Often things are not formed by a single process, but by a combination of growth and recession. What’s amazing is that so many growth and recession cycles can be described by just a few fractal solutions.” Not long ago, one could look at a horribly eroded surface and feel assured that math could never explain such a muddy mess. That is no longer the case; the math that stemmed out of infinity, chaos, and randomness has finally found its way to an appropriate application. Just as we borrow the mathematics of ellipses to predict and understand elliptic orbits of the planets, we will soon be able to borrow fractal mathematics to predict and understand the fractal nature of erosion.
