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07 What is fractal dimension? How is it calculated?




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This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.

07 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 quotient 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, via the WWW

Fractals and Scale (by David G. Green)
http://life.csu.edu.au/complex/tutorials/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 Q22 "Fractal software").

Reference on the Hausdorff-Besicovitch dimension

A clear and concise (2 page) write-up of the definition of the
Hausdorff-Besicovitch dimension in MS-Word 6.0 format is available in
zip format.

hausdorff.zip (~26KB)
http://www.newciv.org/jhs/hausdorff.zip

 

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