Why are Seacoasts Fractal?

In a famous paper written decades ago, Benoit Mandelbrot asked how long the coastline of Britain really was. The answer depends on what kind of meter stick you use. The closer one looks at any scale of a rocky coast map, from well above the 100 kilometer level to the kilometer level, and so on to the meter level, the more indented and lengthy the "coastline" becomes. Not only that, but the coast's underlying geometry seems be fractal, meaning that it is extremely fractured and also self-similar: the shape looks, in a statistical sense, the same at all levels of magnification.

Now, scientists in France have inquired into the physical processes that actually could carve out a fractal coast. Their simulation of a rocky coast evolution depends on an iteration of erosion action. First, waves are allowed to erode the weak points in a smooth shoreline. This makes the shore irregularly indented and longer. This erosion exposes new weak points, but at the same time mitigates the force of the sea by increasing the wave damping. These steps are then repeated over and over. The resultant coast is fractal, with an effective dimension of 4/3.

According to Bernard Sapoval and A. Baldassarri of the Ecole Polytechnique (Palaiseau, France) and their colleague A. Gabrielli of the "Enrico Fermi" Center (Rome), this new study provides the first suggestion of how a fractal shoreline comes about. (Sapoval et al., Physical Review Letters, upcoming; bernard.sapoval@polytechnique.fr, 33-169334172.)