More Examples of Natural Power Laws and Fractal Sets
Coastlines and landscapes
Mandelbrot points out that the length L of a coastline depends on the length d of the particular measuring rod (unit distance in a map) with which the measurement is made through the power law L=a d ^1-D, where a is a proportionality constant and the fractal dimension D ranges from 1.1 to 1.5, independent of tectonic setting and geologic age. Examples are the coast of Britain, D=1.25; D=1.13 for the Australian coast, D between 1.12-1.39 for the Japanese coast. A higher fractal dimension means a greater degree of roughness and complexity, so the smooth eastern coast of Florida has a fractal dimension very close to unity, while the very rugged Norwegian coast has D=1.52.
The natural forms created by erosion are particularly easy to imitate with fractal surfaces. It is in fact a common place to see forgeries of landscapes created by computer artists that are very realistic. For instance, the fractal dimension of the surface used to construct the mountain forgery shown here is D=2.2. What is most interesting, and as yet unexplained, is that deviations of more than 10% from this value would not reproduce the natural mountain shapes as well, making them look outlandish.
Topography is produced by the interaction of the two promordial forces of the Earth, tectonics and erosion; so it is quite remakable that thier combined effect appears to produce such a narrow fluctuation in fractal dimension. Of course, there are mountains such as well developed conic volcanoes that cannot be regarded as fractal, at least until the forces of erosion modify its original Euclidean shape.
Fault and fracture networks
In terms of their frequency N versus their linear length R, crack networks obey power laws of the form N=R^D, where D varies between 1.3 and 1.6. Distributions of exposed joints and fractures in well studied regions are commonly found to have fractal dimension around 1.6.
Clouds and rain
The turbulent process involved in the formation of clouds seems not to have any characteristic length scale in the range 1 to 10,000 km. All clouds in this group have an area A to perimeter P relationship given by the power law A ~ P^1.33
The number N or rock fragments created by weathering, high explosives, mining, impact of projectiles, etc. is related by a power law to the linear size r of the fragment, such that N = C(1/r)^D, where D, the fractal dimension, varies between 2 and 3, depe nding on the type of rock and the fragmentation process. For instance the fractal dimension D is 2.42 for granite fragmented by a nuclear explosion, D=2.16 for broken sea ice, and D=3.05 for asteroids.
Frequency-size of impact cratersAn excellent place to look at craters is on the Moon's surface. The self-similar property of the craters enable one to know how big or small a geven crater is by looking at just one picture without scale reference. One cannot tell whether the crater is 50 cm, 50 km or 500 km long. This is the characteristic symptom of scale invariance.
The number N(>d) of impact craters having a diameter larger than d satisfy the power law relationship N(>d) ~ d^-D, where D is the fractal dimension, which has been found to have a value very close to 2.0 for the Moon, Mars and Venus. The size of asteroids is also known to follow a fractal distribution with dimension D around 2.1.
There exist time-dependent events that happen in clusters, in a self similar manner, that is, consisting of bursts of activity followed by long pauses, in a pattern that repeats itself at all time scales. Fractals can exist in both space and time.
There are manu other structures that display fractal shapes or fractal statistical distributions, in all branches of science. Some remarkable ones include fractal form of the viscous fingering of low viscosity liquids injected under pressure into other fluids, the structure of the rings of Saturn, the shpae of lightening discharges, the body's network of blood vessels, oak trees, cauliflower, broccoli, etc.