4a: What is fractal dimension? How is it calculated?
Description
This article is from the Fractal FAQ, by Ermel Stepp stepp@muvms6.mu.wvnet.edu with numerous contributions by others.
4a: What is fractal dimension? How is it calculated?
A common type of fractal dimension is the Hausdorff-Besicovich
Dimension, but there are several different ways of computing fractal
dimension.
Roughly, fractal dimension can be calculated by taking the limit of the quo-
tient of the log change in object size and the log change in measurement
scale, as the measurement scale approaches zero. The differences come in
what is exactly meant by "object size" and what is meant by "measurement
scale" and how to get an average number out of many different parts of a
geometrical object. Fractal dimensions quantify the static *geometry* of an
object.
For example, consider a straight line. Now blow up the line by a factor of
two. The line is now twice as long as before. Log 2 / Log 2 = 1,
corresponding to dimension 1. Consider a square. Now blow up the square
by a factor of two. The square is now 4 times as large as before (i.e. 4
original squares can be placed on the original square). Log 4 / log 2 = 2,
corresponding to dimension 2 for the square. Consider a snowflake curve
formed by repeatedly replacing ___ with _/\_, where each of the 4 new lines
is 1/3 the length of the old line. Blowing up the snowflake curve by a factor
of 3 results in a snowflake curve 4 times as large (one of the old snowflake
curves can be placed on each of the 4 segments _/\_).
Log 4 / log 3 = 1.261... Since the dimension 1.261 is larger than the
dimension 1 of the lines making up the curve, the snowflake curve is a
fractal.
For more information on fractal dimension and scale, access via the WWW
http://life.anu.edu.au/complex_systems/tutorial3.html .
Fractal dimension references:
[1] J. P. Eckmann and D. Ruelle, _Reviews of Modern Physics_ 57, 3
(1985), pp. 617-656.
[2] K. J. Falconer, _The Geometry of Fractal Sets_, Cambridge Univ.
Press, 1985.
[3] T. S. Parker and L. O. Chua, _Practical Numerical Algorithms for
Chaotic Systems_, Springer Verlag, 1989.
[4] H. Peitgen and D. Saupe, eds., _The Science of Fractal Images_,
Springer-Verlag Inc., New York, 1988. ISBN 0-387-96608-0. This book
contains many color and black and white photographs, high level math, and
several pseudocoded algorithms.
[5] G. Procaccia, _Physica D_ 9 (1983), pp. 189-208.
[6] J. Theiler, _Physical Review A_ 41 (1990), pp. 3038-3051.
References on how to estimate fractal dimension:
1. S. Jaggi, D. A. Quattrochi and N. S. Lam, Implementation and
operation of three fractal measurement algorithms for analysis of remote-
sensing data., _Computers & Geosciences_ 19, 6 (July 1993), pp. 745-767.
2. E. Peters, _Chaos and Order in the Capital Markets_, New York, 1991.
ISBN 0-471-53372-6 Discusses methods of computing fractal dimension.
Includes several short programs for nonlinear analysis.
3. J. Theiler, Estimating Fractal Dimension, _Journal of the Optical Society
of America A-Optics and Image Science_ 7, 6 (June 1990), pp. 1055-1073.
There are some programs available to compute fractal dimension. They are
listed in a section below (see "Fractal software").
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