fractal dimension

fractal dimension

   <mathematics> A common type of fractal dimension is the
   Hausdorff-Besicovich Dimension, but there are several
   different ways of computing fractal dimension.  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.  [sci.fractals FAQ].