How Long is the Coast of Great Britain?

 

Figure 1: The coastline of Great Britain

In 1967, Benoit Mandelbrot published [7] ``How Long is the Coastline of Great Britain'' in Nature. In it, he posed the simple question of how one measures the length of a coastline. As with any curve, the obvious answer for the mathematician is to approximate the curve with a polygonal path, each side of which is of length . (See Figure 2.)
 

Figure 2: Approximating the coastline of Great Britain

Then by evaluating the length of these polygonal paths as , we expect to see the length estimate approach a limit. Unfortunately, it appears that for coastlines, as , the approximated length as well.

In a later book, [10, pp 28-33,] Mandelbrot discusses the extensive experimental work on this problem which was done by   Lewis Fry Richardson. Richardson discovered that for any given coastline, there were constants F and D such that to approximate the coastline with a polygonal path, one requires roughly intervals of length . Thus, the length estimate can be given as

The reason has to do with the inherent ``roughness'' of a coastline. In general, a coastline is not the type of curve we are usually used to seeing in mathematics. Although it is a continuous curve, it is not smooth at any point. In fact, at any resolution, more inlets and peninsulas are visible that were not visible before. (See Figure 3.) Thus as we look at finer and finer resolutions, we reveal more and more lengths to be approximated, and our total estimate of length appears to increase without bound.

 
Figure 3: Each increase in scale reveals new degrees of roughness

Mandelbrot made two proposals based on these observations. First, that when measuring an irregular object like a coastline, it is important to include at what scale the measurement is being made, as obviously by using a different scale a different ``length'' can be arrived at. (Lest we think this is a purely academic pursuit, Mandelbrot noted that the reported lengths of shared boundaries between a large and a small nation, such as Spain and Portugal, could vary by as much as 20%. Empirically, it turns out that this difference can be accounted for by a factor of two in the step size used to estimate the length. [10, p. 27,])

Second, Mandelbrot proposed to think of the exponent D as a dimension, and to think of this dimension as the ``natural'' dimension in which to measure the coastline. Just as attempting to measure the ``length'' of a square gives an infinite result, attempting to measure a coastline in the wrong dimension gives a useless answer.

Mandelbrot noted that many other objects in nature tended to be both rough and show the same roughness on all scales, and so might lend themselves to being described by a fractal dimension. Thus, the beginnings of a theory of fractal dimension appeared. Now we require an actual mathematical definition of fractal dimension.