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| | 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 |
| 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 |
| |
| |
| | 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. |
| |