Fractal dimension

Q4a: What is fractal dimension? How is it calculated?
A4a: 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 ob-
ject.  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, correspond-
ing 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).

Q4b: What is topological dimension?
A4b: Topological dimension is the "normal" idea of dimension; a point has
topological dimension 0, a line has topological dimension 1, a surface has
topological dimension 2, etc.

For a rigorous definition:

A set has topological dimension 0 if every point has arbitrarily small
neighborhoods whose boundaries do not intersect the set.

A set S has topological dimension k if each point in S has arbitrarily small
neighborhoods whose boundaries meet S in a set of dimension k-1, and k is the
least nonnegative integer for which this holds.