The Coastline Paradox
On the face of it, it is a simple enough: how long is the coastline of the United Kingdom? The answer, though, is far from straightforward. Two of the most respected and renowned data sources come up with wildly different answers, the CIA Factbook putting it at 12,429 kilometres , while the World Resources Institute stating that it is 19,717 km. How can two respected authorities come up with an answer that has a difference of over 7,000 kilometres?
There are two factors at play. The first is the fractal dimension or “wiggliness” of the British coastline. Part of the charm of this island’s coastline is its proliferation of coves, inlets, peninsulas, which is self-evident when you study a map. It has a fractal dimension of around 1.25, not as high as Norway’s 1.52, but considerably greater than that of, say, Australia at 1.13 and South Africa at 1.05. What this means in layman’s terms is that the higher the fractal dimension the more irregular and jagged it is, making it more difficult to measure with any degree of accuracy.
The second factor is what is known as the coastline paradox. In essence, while it is relatively easy to measure with a high degree of accuracy a surface that is straight, the degree of accuracy being increased with the higher precision of the measuring device deployed, a jagged surface like a coastline poses an altogether different set of problems. Measuring in finer detail does not improve the accuracy; it merely adds to the total.
To illustrate the point, if we measure the coastline of Great Britain, the main island in which most of England, Wales and Scotland is situated, using units of 100 kilometres, the result is around 2,800 km. Using units of 50 kilometres, the result increases by approximately 600km to 3,400km. Even if digital technology is deployed to create a digitized representation of the coastline of Great Britain and its surrounding islands, one using 2.282 million individual vertices produces an answer of 11,023 miles while one just using 2,282 reveals the answer as 3,876 miles, shrinking the coastline by over 7,000 miles.
The problem of varying coastline lengths was first spotted by Lewis Richardson, who while researching the possible effects of border lengths on the probability of war in the early 1950s, noticed that the Portuguese reported the length of their border with Spain as 987 km while the Spanish recorded it as 1,214 km.
The commonly used method of estimating the length of a border was to lay out straight-line segments of an equal length (l) on a map or aerial photograph with each end on the boundary. Richardson discovered that the sum of the segments increases as the common length of the segments decreases or, to put it plainly, the shorter the ruler, the longer the border. In certain circumstances, Richardson found, if l approaches zero, the length of the coastline approaches infinity.
The Richardson effect spawned a seminal paper, printed in Science in 1967, by Benoit Mandelbrot, How long is the coast of Britain? His conclusion was that there is no definitive answer and it all depends on how you measure it. The obvious takeaway from that is that any statistic about the length of a country’s coastline or border is meaningless without knowing the unit of measure.
