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## Why Knowing the True Length of a Coastline or Border is Impossible

by | Sep 3, 2020

Lewis Fry Richardson was an English mathematician, and being a Quaker, was also a pacifist. One area he studied was human warfare with the hopes of lessening the incidence and severity of wars. He developed a hypothesis that the likelihood two neighboring countries would go to war was proportional to the the length of their border. In collecting data about the length of the border between Spain and Portugal he noticed that a Spanish encyclopedia listed the border as 987 km while a Portuguese encyclopedia claimed it was 1,214 km. That’s quite a difference! He found similar discrepancies when looking at the reported borders between other countries; different sources often gave different lengths for borders between countries.

What was the source of the disparate border lengths? Then answer lay in the scale of the resolution of the instrument doing the measuring. Because a border is often defined by a natural geographic feature like a river or a coastline, it is often irregular. Thus, the exact length measured will depend on how detailed the we get with respect to measuring every turn, squiggle and jut of the natural geography defining the border.

For example, consider the coastline of Great Britain: if it “is measured using units 100 km long, the total length of the coastline is approximately 2,800 km long whereas if units of 50 km are used, the total length is approximately 3,400 km, a difference of 600 km.” Source.

Thus, as the size of one’s ruler gets smaller, the length of the coastline or border gets exponentially longer. This leads to the flummoxing conclusion that the length of a border or coastline is not an objective, fixed distance but rather are of indeterminate length not capable of being known except with reference to the scale of the resolution of the measuring.

A good example of how widely borders and coastlines can differ due to scale is found in a report by the Congressional Research Service in 2006 and noted: “The ‘general coastline’ data [for the U.S] are based on large scale nautical charts, resulting in a coastline measure for the 50 states totaling 12,383 miles. Another measure using smaller scale nautical charts more than doubles this measurement to 29,093 miles, while the figures used by the NOAA in administering the Coastal Zone Management program (16 U.S.C. §1451) come to 88,612 miles (not including the Great Lakes).” So various measurements of the length of the U.S. coastline resulted in measurements of 12,383, 29,093 and 88,612 miles!

Based on this astounding observation about borders and coastlines, mathematician Benoit Mandelbrot created a branch of mathematics now known as fractal geometry. A “fractal” is  “a curve or geometric figure, each part of which has the same statistical character as the whole.” Source. Fractals are characterized by self-similarity of shape regardless of the scale at which you zoom in. Fractal geometry will be the topic of a future IFOD.

1. I have always loved this story because it is a great example of why “measurement” is so hard. To me this is the kind of knowledge school children should be learning — not facts like “how long is X” — but “how would you measure X” and what does it mean to even ask that question? When “facts” are at your fingertips it is more important to understand how things are arrived at. Coastlines are also a perfect tool for introducing the ideas of scale and the complexities of fractals. (BTW — I know you are interested in the brain — the brain’s vasculature is also a beautiful example of a fractal)
Thanks John!

2. I wrote my thesis in college on Mandelbrot, fractal equations & their potential uses. It’s fascinating and utterly limitless. The best part though are the visual representations. His epilogue goes something like this, “ Imagine music is outlawed but writing music scores and sheets still occurs & they are appreciated for their elegance. Until one day, a hundred years later the music is heard.” That’s what a fractal is like.

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