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

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

Continue to: