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].