The question is: "What is the length of a coastline, or the border of this fern?"
A line is a figure in one dimensional, 1D, space. If we divide the line into N equal elements (N is the number of steps) of length s/N (the scale of measurement) then the length, or extent, of the line segment is
E = Ns^D
E = N(s/N)^1 = s for one dimension. We can make the subdivision as small as we want and the length of the line obviously remains unchanged at s units.
The circumference of a square is also a 1D object. If the sides of the square were s long then the extent, E, of the square would be
E = 4s^1 having units of area, s^1. N, the number of steps, is 4 for a square.
Any regular figure, a hexagon for example, will have a circumference whose extent is in one dimension, E(hexagon) = 6s^1.
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 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, access via the WWW http://life.anu.edu.au/complex_systems/tutorial3.html.
Fractals and Scale by David G. Green
C. J. Pennycuick, Newton Rules Biology, Oxford University Press, 1992