The
fractal dimension of a geometric object is a measure of how much area it fills. (For a curve embedded in the plane. For a two-dimensional set of points, it is a measure of how much volume, etc.)
There are several methods of calculating fractal dimension. The simplest is the box-counting dimension, which coincides with the more general Hausdorff dimension for most interesting cases. Suppose you have a one-dimensional fractal embedded in the plane, like the Koch snowflake. Denote by N(\epsilon) the number of squares of side length \epsilon needed to cover the fractal. Then its fractal dimension is
ln N(\epsilon)
lim ----------------
\epsilon->0 ln (\epsilon^-1)
The box-counting dimension of the middle-third Cantor set is (ln 2)/(ln 3). The box-counting dimension of the Sierpinski gasket is (ln 3)/(ln 2). The box-counting dimension of the Henon attractor is approximately 1.27.