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 triangle.

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).

Repeat again.

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?

or

(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.


(c) Copyrighted 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 Fractals please contact its author Mary Ann Connors at the following address. Thank you.

Dr. Mary Ann Connors
Department of Mathematics & Statistics
Lederle Graduate Research Tower
University of Massachusetts
Amherst, MA 01003
Email:
mconnors@math.umass.edu
mconnors@math.umass.edu