Classical Landscapes

Visually convincing fractal landscape forgeries were pioneered by Richard Voss, using variations of Brownian motion and fractional Brownian motion. Perhaps his most familiar image is Fractal Planetrise, widely distributed as the back cover illustration of The Fractal Geometry of Nature. Click the image to enlarge.

Voss generated this fractal planet by adapting to the suface of a sphere the construction of Brownian motion, a sum of displacements at uniformly distributed locations and of normally distributed amplitudes. Voss' program picked great circles with uniformly randomly distributed poles. The great circles are thought of as geological faults, and one hemisphere of the great circle is displaced in height from the other. The height differences are normally distributed, and color is assigned according to height. Adding oceans and polar caps, a plausible planet emerges after about 10,000 great circle displacements.

The moon surface behind which the planet is rising has a 1/f distribution of craters. That is, for each number D, the number of craters N(diam > D) of diameter greater than D satisfiesN(diam > D) = 1/D. This sort of power-law distribution often is associated with fractals, so we speak of this as a fractal distribution of craters.

In addition, Voss applied fractional Brownian motion to construct terrain syntheses. For regular Brownian motion, travelling a distance r will result in a change in height of sqrt(r), on average. Fractional Brownian motion has an associated roughness exponent H, 0 < H < 1, and travelling a distance r in a fractional Brownian motion landscape will result in a change in height of r^H, on average. High H gives smooth, weathered mountains; low H gives rough, tectonically active mountains. For surfaces in 3-dimensional space, the dimension D is related to the roughness exponent H by D = 3 - H.

Here are three examples of fractional Brownian mountains, with H = 0.85, H = 0.5, and H = 0.2, so d = 2.15, 2.5, and 2.8. respectively. This clearly illustrates a connecton between roughness and dimension. Click each image to enlarge.