To test the relationship between the fractal dimension of computer generated landscapes and their appearance, I wrote a landscape modelling program in Matlab. The program can model landscapes using a subdivision method, as a sort of control, as well as by the use of a cellular automaton method, a less common method. The goal was twofold: To see if landscapes created using a cellular automaton method can be made to exhibit fractal characteristics; and to see whether dimension corresponds to landscape appearance, and if so, how.
A number of years ago, several scientists and mathematicians attacked classical Euclidean geometry as being too limited with respect to the world we live in. Among this group was Benoit Mandelbrot, who has been credited with developing what he calls a "geometry of nature." What has since come to be known as fractal geometry offers a new way of looking at our world. This geometry breaks the mold of integral dimensions, suggesting that nature consists of fractional dimensions. For example, the dimension of a coastline is somewhere between one and two, while a mountain range has dimension between two and three. The idea behind fractal geometry is self-similarity, which is inherent in nature, and is required for determining fractal dimensions. The landscape modeler, written in Matlab, is designed to create realistic looking three dimensional landscapes. In addition to modeling, the program calculates either the fractal dimensions of the landscapes, or an estimate of the dimension when the landscape isn't self-similar, and thus it can be used as a tool to analyze this aspect of the landscapes.
The program uses two different methods to model landscapes: A subdivision method and a cellular automaton method. The first of the two is a common method for modeling landscapes, making use of self-similarity by recursive subdivision. The second is a much less common method, and is the focus of this project. Cellular automata are often used for biological purposes, and although they have been used for landscape modeling [4], little research has been done in this area.
To verify the correctness of the landscape modeler, I tested landscapes created by subdivision, and found their fractal dimensions to average 2.4 as constructed. Then I went on to analyze the statistical roughness, or estimated dimension, of the cellular automaton landscapes in an attempt to create realistic looking landscapes employing this less used method. An object is fractal only when it exhibits self-similarity, and this is not necessarily the case with the landscapes produced by the cellular automaton method. For this reason, I will use the term statistical roughness rather than fractal dimension when I don't know the landscape to be self-similar.
The cellular automaton method used in this program is based on a square grid, and is run for a given number of iterations. Changing the size of the grid or the number of iterations changes the appearance of the landscape created, as well as the statistical roughness. And so I was curious as to how exactly changing the grid size and iterations would affect the appearance and roughness of the landscape. I also wanted to find out whether the generated landscapes would look more realistic as the statistical roughness approached 2.4, the same as the dimension of the landscapes created by subdivision. I quickly found out that running the cellular automaton algorithm on a grid of size 15 x 15 for 15 iterations results in a roughness around 2.4, and I used these initial findings for further analysis.
With the number of iterations fixed at 15, I found the statistical roughness to increase with grid size. For small grid sizes the roughness grows as size increases, and then the growth levels off, as shown below.
This increase in statistical roughness is due to the lack of scaling exhibited by the cellular automaton algorithm. Regardless of the size of the grid, the features of the cellular automaton remain the same, due to the behavior of the individual cells. Thus the resulting landscape on a large grid has more peaks and valleys, resulting in a larger statistical roughness.
Fixing the grid size at 15 and varying the number of iterations affects statistical roughness in a different manner. As the iterations increase from one to around five, the roughness drops rapidly, and then levels off abruptly.
The cellular automaton is randomly initialized, and thus a small number of iterations results in a very rough landscape and a high statistical roughness. The first few iterations smooth the cellular automaton out significantly, but after about five iterations the automaton is nearly as smooth as it will get, and so further iterations make virtually no difference.
Another question to consider is whether the landscapes created using the cellular automaton method can be made to exhibit fractal characteristics, allowing us to measure the actual fractal dimension. That is, do they exhibit self-similarity in that their statistical roughness is the same at different scales. And it turns out that the answer to this depends on the two characteristics described, the size of the grid and the number of iterations. When the grid is very large or when very few iterations of the cellular automaton algorithm are run, the resulting landscape is not fractal in that the small details have a higher statistical roughness than does the main shape of the landscape, and thus there is a lack of self-similarity. In the case of few iterations, this occurs because the landscape is quite random with many small bumps, and no large mountains have formed. Similarly, when the grid size is large the landscape is spread out, and all of the features tend to be small. When a small grid or a large number of iterations is used, the landscape lacks self-similarity in that the details are of low statistical roughness as the resulting landscapes tend to be smooth. So it would seem that somewhere in the middle, the right balance could be found to create a self-similar, fractal landscape, and indeed I found this to be the case.
Combining the two variables, grid size and iterations, the statistical roughness comes closest to that of the landscapes created by subdivision when the cellular automaton algorithm runs 16 iterations on a grid of size 15. And these two values also give the best results in terms of self-similarity. That is, running 16 iterations on a grid of size 15 produces fractal landscapes that exhibit a dimension of 2.4 at all scales. Not surprisingly, these values also result in landscapes which look quite similar to the landscapes created using the subdivision method. These results verify the assumption that the appearance of computer generated landscapes, regardless of method used, varies with statistical roughness. The higher the roughness the more rough and jagged the resulting landscape is, while a low roughness results in smooth landscapes. More specifically, a statistical roughness somewhere between 2.4 and 2.5, the same as the dimension of actual mountain ranges, produces the most self-similar, realistic looking landscapes.
1. Barnsley, Michael. Fractals Everywhere; Harcourt, Brace and Jovanovich: Boston, 1988.
2. Mandelbrot, Benoit B. The Fractal Geometry of Nature; Freeman and Co.: New York, 1983.
3. Peitgen, Heinz-Otto and Dietmar Saupe. The Science of Fractal Images; Springer-Verlag: New York, 1988.
4. Turcotte, Donald L. "Scaling in Geology." Proceedings of the National Academy of Sciences of the United States of America; Vol. 92: pp. 6697-6704, July 18 1995.