Random Fractals

Natural objects do not contain identical scaled down copies within themselves and so are not exact fractals as described above. However, natural objects can often be classified as random fractals, meaning that each smaller part of it is statistically similar to the whole. Random fractals can be generated by modifying the iteration process of the last section to include a probabilistic element. Consider the generation of a random Koch curve. The initiator and generator are as before, but in the following steps ($k=2$ onwards), the prefractals are obtained by replacing each line segment with the generator in such a way that the triangle of the generator points randomly (for example, determined by a coin toss) to either side of the original line. The figure for the final fractal shape looks very irregular compared to the exact Koch curve but is closer to the shape of natural objects such as coastlines. Just as for exact fractals, one can introduce a dimension to characterize random fractals. One example is the box counting dimension. In this method, space is divided into equal sized cubes (or squares if the figure lies on a plane) of linear dimension $r$. One then counts the number of cells, $N(r)$ that are needed to cover the given shape. If

$$N(r) \propto r^{-D_b}$$ (2.4)

as the length $r$ is changed, one says that the distribution of points is $D_b$-dimensional. This definition obviously agrees with the Euclidean dimension for straight lines and planes but gives fractional values for more complicated shapes such as coastlines. Note that the equation above is of the same form as that which comes from the definition of the self-similarity dimension mentioned above.