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).

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.

Copyright 1993, University of Illinois Board of Trustees

National Center for Supercomputing Applications, Education Group

jgasaway@ncsa.uiuc.edu