The fractal dimension of Mellon Udrigle


Series 4, Episode 6, “Inner Hebrides to Faroe Islands”, of the BBC television series Coast, bravely tackled the fractal nature of coastlines. (The episode is available on BBC iPlayer until . The segment I’m discussing here starts at 35:30.)

Visiting a stretch of the coast of Scotland near Mellon Udrigle, presenter Nicholas Crane and mathematician Tony Mulholland from the University of Strathclyde measured a short length of rocky coast using rulers of four different lengths, resulting in the increasing series of distances shown at the right.

They namechecked Benoît Mandelbrot, whose 1967 paper “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension” connected the problem of measuring the length of a coastline or frontier with the mathematical notion of fractal dimension. But I think they missed an even more interesting story, because (as Mandelbrot noted) the coastline problem had been empirically investigated a little earlier by Lewis Fry Richardson.

(I learned about Richardson’s story from Tom Körner’s book The Pleasures of Counting and what follows is a paraphrase of Körner’s account.)

Richardson was a Quaker and a pacifist. In World War I he served with the Friends’ Ambulance Unit attached to the French 16th Division, and transported wounded soldiers during the Third Battle of Champagne (1917). After the war he worked for the Meteorological Office on a system for forecasting weather by numerical computation, but when in 1920 the Met was placed under the control of the Air Ministry, Richardson resigned on principle.

While teaching physics at Westminster Training College, he carried out a mathematical study of the nature of warfare. His 1950 book Statistics of Deadly Quarrels systematically collects and analyzes data on wars and other conflicts between 1820 and 1945 to attempt to get quantitative answers to questions like “are some countries or groups inherently more belligerent than others?” “are wars getting more frequent or deadly over time?” “does a common language increase or reduce the chance that two countries or groups will fight?” and so on. He observed, for example, that occurrences of outbreaks of war between pairs of countries appear to obey Poisson statistics, suggesting that many events take place (provocations, accidents, disputes, assassinations etc) each of which has a very small chance of leading to a war.

The problem of the lengths of coastlines arose in the consideration of whether countries with longer borders are more likely to go to war. In order to normalise his data by the lengths of borders, Richardson needed to determine these lengths. However,

An embarrassing doubt arose as to whether actual frontiers were so intricate as to invalidate that otherwise promising theory. A special investigation was made to settle this question. […] At first I tried to measure frontiers by rolling a wheel of 1.8 centimetres diameter on maps; but there is often fine detail, which the wheel cannot follow; some convention would be needed as to what detail should be ignored and what retained: considerable skill would be needed to guide the wheel in accordance with any such decision; and in practice the results were erratic.

Much more definite measurements have been made by walking a pair of dividers along a map of the frontier so as to count the number of equal sides of a polygon, the corners of which lie on the frontier. […] Its total length, \(Σl\), has been studied as a function of the length, \(l\), of its side. This process comes down to us from Archimedes, and is standard in pure mathematics. […]

The west coast of Britain from Land’s End to Duncansby Head was chosen as an example of a coast that looks more irregular than most other coasts in an atlas of the world. […] As to how the total length \(Σl\) may be expected to vary with the length \(l\) of the side, I have no theory. Quite empirically the logarithms of these variables were plotted against one another; and a straight line was drawn through the points. More evidence would be needed before one could say whether the deviations from the straight lines are of any interest. I am inclined to regard them as random. The important feature for present purposes is that the slope of the graph is only moderate even for such a ragged line as the western shore of Great Britain. On the straight line in [the plot of \(\log Σl\) against \(\log l\)] the total length [\(Σl\)] varies inversely, as the fourth root of the side [\(l\)], that is $$ Σl ∝ {1 \over \root 4 \of l}. $$

Richardson found that similar lines of fit could be drawn for other coastlines, with the situation summed up by “the useful empirical formula \(Σl ∝ l^{−α}\) where \(Σl\) is the total polygonal length, \(l\) the length of the side of the polygon, and \(α\) is a positive constant, characteristic of the frontier.”

Mandelbrot analyzed Richardson’s exponent \(−α\) as \(1 − D\), where \(D\) is the fractal dimension of the frontier. On Coast this concept was mentioned but the programme segment wasn’t long enough to explain it, even at the empirical level of Richardson’s analysis.

So let’s follow Richardson’s procedure ourselves for the Coast measurements, using gnuplot to do the fitting. Here’s the file

0.5 61.5
1.0 51.0
2.0 30.0
14.0 14.0

In the anomalous case of the 0.5 m ruler, I’ve taken the number of steps, 123, as being correct, based on Nicholas Crane being filmed clearly counting to 123. This gives a total length of 61.5 m, not the 64 m reported by the programme. Here’s the gnuplot program:

set logscale x
set logscale y
set xlabel "Length of ruler (m)"
set ylabel "Measured length of coast (m)"
f(x) = b * x**(-a)
fit f(x) '' via a, b
plot '' title "Measurements", f(x) title "Best fit"

The result is shown below, and the best-fit procedure finds the value \(a ≅ 0.46\), giving this very wiggly piece of coast the fractal dimension of 1.46.

When the length of ruler in metres is plotted against the total length of coast in metres, using logarithmic scales on both axes, the four measurements made on the Coast programme line very close to the line of best fit, with a slope of about −0.46.