fractal dimension
<mathematics> A common type of fractal dimension is the
HausdorffBesicovich 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].
Nearby terms:
fr « fractal « fractal compression « fractal
dimension
» FRAD » fragile » fragment
