The concept of "fractal dimension" is attributed to a 20th Century mathematician, Benoit Mandelbrot. His fractal theory was developed in order to try to more precisely quantify the immense complexity of nature in relatively simple equations. His favourite example of such complexity was the craggy coast of Britain, which, seen from far above, looks somewhat wrinkled and convoluted. Yet, as an observer gets closer and closer to the shore, the complexity of the coastline increases -- smooth lines become jagged and more complex, until the observer is so close that she is observing the minute variation in the positions of each individual grain of sand along the shore.
Moreover, we can imagine this observer measuring the length of the coastline with smaller and smaller rulers. As she measures with increasingly precise resolution, her approximation of the length of the coast will keep increasing. In fact, she might well conclude that the length she is looking for diverges to infinity!
And yet, it is obvious that this "infinitely long" coast of Britain encapsulates only a finite area, just as a circle drawn on the globe can contain all of Britain. In some way, we believe that the coast of Britain is more "substantial" than a simple circle, and perhaps more interesting than a 1-dimensional line which defines a circle's circumference. Fractal dimension was developed as a way to quantify this contradictory complexity.
Notice that a line segment is self-similar . It can be separated into 4 smaller replicas of itself, each 1/4th the size of the original. Each looks exactly like the original figure when magnified (scaled) by a factor of 4.
|4 = 41 pieces|
A square can similarly be separated into miniature squares. If the smaller square is scaled 4 times, then it is identical to the larger square; however, observe that here we need 42 = 16 copies of the smaller figure to fill up the original square figure.
|16 = 42 pieces|
Following along in this pattern, a cube can be broken down into smaller cubes, each of which can be scaled by a factor of 4 to obtain the larger cube -- but now it takes 43 = 64 smaller cubes to fill up the larger cube.
|64 = 43 pieces|
From this, we begin to see a pattern emerge: N, the number of smaller pieces required to "fill up" the original figure is equal to S, the scaling factor, raised to the D power, where D is the dimension of the figure. In the above examples, it is easy to find the dimension simply by reading the exponent:
N = SD
This simple concept can be generalized to measure non-integral dimensions as well. Figures with such fractional dimensions are called "fractals", and if you have completed Problem Set 1, you have already seen an example of such a figure, the von Koch Snowflake. Another common fractal is the Sierpinsky Triangle, which is created by successively removing the middle section out of an equilateral triangle.
This is a picture of a Sierpinsky Triangle.
To generate it, we begin with an equilateral triangle. Draw the lines
connecting the midpoints of the three sides, then remove the center
triangle. Note that our new triangle contains three smaller
triangles, each with sides one-half the length of the original. Each
of these smaller triangles looks exactly like the original, when
scaled by a factor of 2.
Now, you repeat this process (iterate) on each of the smaller triangles, as shown here:
Continuing this pattern results in the figure above. The fractal is the limiting case that results when the iteration of this process is carried out infinitely many times. Notice that the lower-left portion of the triangle is exactly the same as the entire trangle, if it were doubled in size, and the lower-left portion of that triangle is the same as its containing triangle, and so on. In other words, the Sierpinsky Triangle is self-similar.
But what is the dimension of the Sierpinsky Triangle? Notice the second triangle is composed of 3 miniature triangles exactly like the original. The smaller triangles could be scaled by 2 to produce the entire triangle (S = 2). The resulting figure consists of 3 separate identical miniature pieces (N = 3).
So, what is the dimension, D? Recalling that N = SD, we simply take the logarithm of this equation to obtain:
|D * log(2)||=||log(3)|
|D||=||log(3) / log(2)|
|D||=||1.585 (not an integer!)|
In fact, a trivial computation from the above equation yields:
D = log(N)/log(S).
This is the formula to use for computing the fractal dimension of any strictly self-similar fractals. The dimension is a measure of how completely these fractals embed themselves into normal Euclidean space.
For the following figures, we have given N, S, and D.
N = 2, S = 3, D = log(2) / log(3) = 0.6309
N = 5, S = 3, D = log(5) / log(3) = 1.4649
N = 4, S = 3, D = log(4) / log(3) = 1.2618
N = 8, S = 4, D = log(8) / log(4) = 1.5
Just for fun, and for a self-test of your understanding of what we've talked about here, find the dimension of the fractals generated in each of the following ways:
If you are interested in a more rigorous development of fractal theory, you should consult The Fractal Geometry of Nature by Benoit Mandelbrot. It is a seminal text in the field, and it is written at an "educated layman's" level. For an even more rigorous discussion of more recent results, including computer applications, there are a few good texs by Michael F. Barnsley, one of which is entitled Fractals Everywhere, but this is pretty dense reading. Both of these books will have good bibliographies, as well.
There is also a Fractal FAQ, if you're looking for more resources on the web.
This document is based upon a similar introduction in the PWS OnLine Series, Copyright © 1995 by PWS Publishing Company. A couple of the graphic images displayed here are also from that document.