Dimension of a Fractal
In the previous cases it is easy to find the dimension by simply reading the
exponent. However it's not always so easy. Consider the Sierpinski Triangle -
an example of a fractal.
Let's look at how it is generated: Begin with a triangle.
Draw the lines connecting the midpoints of the sides and cut out the center
Note that we have in our new triangle 3 "miniature" triangles. Each side = 1/2 the length of a side of the original triangle. Each "miniature" triangle looks exactly like the original triangle when magnified by a factor of 2 (magnification or scaling factor).
Take the result and repeat (iterate).
And again ...
Iterate this forever
Notice that the lower left portion of the triangle is exactly the same as the entire
triangle when magnified by a factor of two. It is self-similar.
Now we compute the dimension of the Sierpinski Triangle: Notice the second
triangle is composed of 3 miniature triangles exactly like the original.
The length of any side of one of the miniature triangles could be multiplied
by 2 to produce the entire triangle (S = 2). The resulting figures consists
of 3 separate identical miniature pieces. (N = 3).
What is D?
- (not an integer!)
- In general,
This method of finding fractal dimension can be used for only strictly self-similar
fractals. Other ways of computing fractal dimension include: mass, box, compass, etc.
Fractal dimension has turned out to be a powerful tool. Now mathematicians are
able to measure forms which were previously immeasurable such as mountains,
clouds, trees and flowers. Fractal dimension indicates the
degree of detail or crinkliness in the object
and how much space it occupies
between the Euclidean dimensions.
1994,1995,1996,and 1997 by Mary Ann Connors. All rights reserved.
If you wish to use any of the text or images in Exploring
please contact its author Mary Ann Connors at the following address.
Dr. Mary Ann Connors
Department of Mathematics & Statistics
Lederle Graduate Research Tower
University of Massachusetts
Amherst, MA 01003