Fractal geometry is the study of irregular shapes. Nature is full of these irregular shapes from trees and leaves, through mountains and rocks, to weather patterns and clouds. However these irregular shapes display a self-similarity across multiple scales that can be identified and used to inform our understanding. Nature’s self-similarity has a randomness to it. As an example, oak trees branch like oak trees and maple trees branch like maple trees from first branches to twigs, but all the branchings are not identical. Mathematics calls this self-affinity. The study of nature’s self-affinity was first presented by Benoit Mandelbrot in The Fractal Geometry of Nature. On the first page of the first chapter Mandelbrot asks the question,
Why is geometry often described as cold and dry? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, and coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
(Mandelbrot 1983, 1)
The fractal dimension is a measure of the irregularity of a shape. A prime mathematical example of an irregular self-similar entity is the Koch curve (Figure 8.1). The Koch curve is constructed through an iterative process. First a line segment is divided into thirds. Then the middle third of the curve is taken out and replaced with a triangular blip with sides equal in length to the thirds of the original line; thus the line is now four segments long. This procedure is then repeated on each of the now four line segments. The iterative process goes on to infinity in mathematical concept. In real world drawing of the Koch curve there is a limit to the iterations due to line width used to draw the curve. The fractal dimension as defined by Mandelbrot can be calculated as:
where a is the number of pieces and s is the reduction factor.
4 = (1/(1/3))D
4 = 3D
log (4) = D log (3)
D = log (4)/log (3)
D = 1.26
The dimension of the Koch curve tells us that it is more than a one-dimensional line and less than a two-dimensional plane. Other fractal curves have different fractal dimensions. The Cantor set, where a line is divided into thirds and then the middle third is taken out, has a fractal dimension of (D = log (2)/log (3) = 0.63). The Cantor set after many iterations reduces to a cluster of clusters of points. The Minkowski curve divides the original line into fourths and then constructs a line with eight segments. The fractal dimension of the Minkowski curve is (D = log (8)/log (4) = 1.5). The Peano curve divides the line into thirds and then creates a curve with nine segments creating a curve with the dimension (D = log (9)/log (3) = 2.0); thus the Peano curve is a line that, when iterated to infinity, will fill the two dimensional plane. As a comparison, a straight line divided into three segments that maintains the three segments has a dimension of (D = log (3)/log (3) = 1) (Bovill 1996, 23–27).
FIGURE 8.1 The Koch curve illustrates the construction of a fractal.
Source: Bovill 1996.
Mandelbrot asks the question, “How long is the coast of Britain?” (Mandelbrot 1983, 25). He comes to the conclusion that the length is undefinable because, as one uses a smaller and smaller measuring unit, the length gets longer and longer as more inlets and points are included in the measurement (Figure 8.2). However, a fractal dimension can be calculated for the coastline by measuring its length with shorter and shorter measuring units. Graphing the Log(coastline length) versus the log(1/length of the measuring unit) produces a straight line. The slope of the line is the measured dimension (d = 0.26). The fractal dimension is (D = 1 + d). Thus the fractal dimension of the coast of Britain is (D = 1.26). This is the same dimension as the Koch curve. Mandelbrot thus shows that nature is fractal (Bovill 1996, 27–38).
FIGURE 8.2 Measuring the length of the coast of England with smaller and smaller surveying distances produces longer and longer coastline lengths.
Source: Bovill 1996.
FIGURE 8.3 Measuring the fractal dimension of the coast line at Shell Beach, Sea Ranch, California with the box counting method.
Source: Bovill 1996.