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 selfsimilar.
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
selfsimilar
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.
©
Copyrighted 1994,1995,1996,1997,
1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005,
2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017
and 2018 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
email:
mconnors@math.umass.edu
or
email: mconnors@westfield.ma.edu