Fractal Geometry

A Simple Explanation

Where classical geometry deals with objects of integer dimensions, fractal geometry describes non-integer dimensions. Zero dimensional points, one dimensional lines and curves, two dimensional plane figures like squares and circles, and three dimensional solids such as cubes and spheres make up the world as we have previously understood it.

However, many natural phenomena are better described with a dimension part way between two whole numbers. So while a straight line has a dimension of one, a fractal curve will have a dimension between one and two depending on how much space it takes up as it twists and curves (Peterson, 1984). The more that flat fractal fills a plane, the closer it approaches two dimensions. Likewise, a "hilly fractal scene" will reach a dimension somewhere between two and three. So a fractal landscape made up of a large hill covered with tiny bumps would be close to the second dimension, while a rough surface composed of many medium-sized hills would be close to the third dimension (Peterson, 1984).

A More Complete Explanation

We can demonstrate this by first defining a fractal set according to Nn = C/rnD where Nn is the number of fragments with a linear dimension rn, C is a constant, and D is the fractal dimension (Turcotte, 1992). By rearranging this equation we get D = [ln(Nn+1/Nn)] / [ln(rn/rn+1)]. Now we can take a line segment of unit length (figure 1) and divide it into parts. In figure 1a the segment is divided into two parts, so r1 = 1/2, and one of the parts is kept (the other one is thrown out), so N1 = 1. If we divide the remaining segment into two parts and again keep only one fragment, then r2 = 1/4 and N2 = 1. In this case, D turns out to be zero which is the Euclidean equivalent to a point. No matter how many iterations are performed, at order n, Nn = 1 so D will always be zero. This makes sense when we consider that as we continue to divide and keep part of the segment, the line length approaches zero as the order approaches infinity.

Figure 1. Fractal dimensions with line segments of unit length.

We can demonstrate a Euclidean line just as easily (figure 1b). Here the line segment is divided into two parts, but we keep all the fragments, so r1 = 1/2 and N1 = 2. From the next iteration we get r2 = 1/4 and N2 = 4. Therefore, D = ln(2) / ln(2) = 1. This too makes sense because the line segment will always be of unit length.

However, it is just as easy to create a line segment with a fractal dimension between zero and one. In figure 1c we divide the line segment into three parts and retain the two end pieces, so r1 = 1/3 and N1 = 2. When we repeat this process, r2 = 1/9 and N2 = 4. Therefore, D = ln(2) / ln(3) = 0.6309. If we start with a line segment of unit length and divide it into five parts (figure 1d), then r1 = 1/5. By keeping the two end pieces and the center piece, N1 = 3. After one iteration, r2 = 1/25 and N2 = 9. Now D = ln(3) / ln(5) = 0.6826. Of course, for both figures 1c and 1d, approaching an infinite number of iterations will make the remaining line fragments shorter and shorter until we have many points. This infinite set of clustered points is called "dust," but this is unimportant for the purposes of this paper (Turcotte, 1992). What is important is that any fractal dimension between zero and one can be created using this method.

Just as it was possible to find a fractal dimension between zero and one, we can apply the same methods to a square and find dimensions between zero and two. The examples in figure 2 help demonstrate this (Turcotte, 1992). In all of these examples, the unit square is divided into nine equal squares with r1 = 1/3. For the next iteration, each of the smaller squares are themselves divided into nine fragments each with r2 = 1/9. As with the line segments, these iterations continue n times. In figure 2a only one square is kept each time, so N1 = Nn = 1. Therefore, D = 0 and we have another example of the Euclidean dimension of a point. Two squares are retained in figure 2b during the first iteration, four squares are kept after the second iteration, and so forth so that N1 = 2, N2 = 4, etc. Therefore, at the second order D = ln(2) / ln(3) = 0.6309. Of course, this is the same dimension as the line segment example from figure 1c. We discover a Euclidean line of one dimension when we keep three squares in figure 2c since N1 = 3 and N2 = 9. When we only remove the center square at each iteration in figure 2d so that N1 = 8 and N2 = 64, the fractal dimension is 1.8928. And, of course, if we keep all the squares (figure 2e), the fractal dimension is two, or a Euclidean plane. If we wanted to find dimensions between two and three, we would just continue this method with a unit cube.


Figure 2. Fractal dimensions with plane figures of unit area.

Another way to look at fractal development is to systematically add to a line of unit length, rather than removing parts. The Koch curve, named after Helge von Koch who introduced this geometric figure in 1904, is now used as a common description of fractals. The construction is simple (Jürgens et al., 1992). We begin with a straight line of unit length in figure 3a and proceed to divide it into three equal parts. Then we replace the middle third with an equilateral triangle and take away its base (figure 3b). So at this stage we have increased the length by four-thirds. We can continue this procedure indefinitely, although figure 3c only advances it one more iteration.


Figure 3. An example of the Koch Curve iterated twice. (a) A line of unit length. (b) The line increases in length by 4/3. (c) The length is again increased by 4/3, so now it is 16/9 as opposed to the initial unit length.

It is easy to see the self similarity inherent in this and the previous constructions, but the complexity greatly increases once we move beyond linear fractals into the world of nonlinear fractals such as the Mandelbrot set.

Copyright 1993, University of Illinois Board of Trustees

National Center for Supercomputing Applications, Education Group