From The Coastline Paradox To Fractals

Published by Elias Wirth on

In this article, we describe how a natural phenomenon, the coastline paradox, played an important role in creating the field of fractal geometry. Furthermore, we explain the coastline paradox and how fractal geometry can be used to measure the length of a coastline.

Let us begin our journey with a short explanation of the coastline paradox.

Explanation of the coastline paradox

In the year 1950, the English mathematician Lewis Fry Richardson was researching the correlation between shared border length and the probability of war among two adjacent countries. [1] Therefore, he wanted to know the length of the border shared between the countries of Portugal and Spain. When he looked up the official length, he noted something strange. The two countries reported two different values for their shared border length. That would not be unusual, measurement errors are pretty common. The problem was that these values differed by more than 200  kilometers. Such a discrepancy could not have been a mere rounding error. How could two measurements for one and the same object be completely different?

This was the birth of the coastline paradox. It was named this way because the same effect can be noted when comparing different measurements of coastlines.

Measurements of coastlines

Naturally, we seek to explain the paradox, just like Richardson did decades before us. The coastline paradox is best explained visually. Let us consider an island that is completely surrounded by water.

This figure depicts an island and its coastline.
Figure 1. This figure shows an island with a black border.

At the time, coastlines were measured by laying a ruler with length \(l\) over a map repeatedly. The ruler’s ends had to touch the coastline. The number of times the ruler had to be laid out, \(n\), multiplied with the length of the ruler and taking into account the scale of the map would then result in the measured length of the coast. [2]

We now measure the length of the coastline of the island in Figure 1 with a ruler.

This figure shows one possible measurement of the island's coastline. The larger ruler is used 5 times.
Figure 2. This figure shows one possible measurement of the island’s coastline. The larger ruler is used 5 times.

If someone were to ask us right now, we would say that the length of the coast on the map is 5 times the length of the ruler.

But as we can see, the ruler is rather large for the coastline with many different edges, nooks, and crannies. Let us measure the coastline with a smaller ruler one more time, just to make sure that we obtain the right value.

This figure shows one possible measurement of the island's coastline. The smaller ruler is used 11 times.
Figure 3. This figure shows one possible measurement of the island’s coastline. The smaller ruler is used 11 times.

Now that we have two measurements for the same object, the coastline of the island, we are able to compare them. The length of the smaller ruler is clearly longer than half of the larger ruler’s length and thus it is obvious that the two measurements are not the same.


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Measurement tools and the problems they create

When considering Figure 2 and Figure 3 we realize that the measured length of the coastline increases as we decrease the length of the measuring tool. We have just described the coastline paradox that Richardson stumbled upon with his research:

The length of a coastline is not well defined.

The length of a coastline depends on the length of the measurement tool. Richardson also described that the length of any coastline tends to infinity as the length of the measuring tool decreases. For quite some time, no more developments were made in this area of mathematics.

Coastlines and fractals

More than ten years later, however, Benoit Mandelbrot published a paper in spiritual succession to Richardson’s research. [3] In this paper, Mandelbrot invented a new field of mathematics, fractal geometry, that studies the fractal dimension of non-rectifiable curves.

He moved on from the coastlines in specific and coined the term fractal, an object that is self-similar, that means similar to itself, no matter the level of detail at which it is perceived. [4] Consider Figure 4 for an example of a fractal and a self-similar object.

The graphic shows the Mandelbrot set, an example of a fractal.
Figure 4. The graphic shows the Mandelbrot set, an example of a fractal. [5]

Measuring the immeasurable coastlines with fractal geometry

How is it possible to talk about the length of the border of a fractal if it the measured length tends to infinity as the measurement tool size is decreased? Mandelbrot thought about this a lot and came up with a way to assign a special number between 1 and 2 to fractals that made it possible to describe them, the fractal dimension.

We now introduce the concept of the fractal dimension for the example of the coastline. Since all coastlines are basically infinitely long, we can not describe them using their length. We can, however, describe how fast the measured length of the coastline grows when decreasing the measurement tool length. And this ratio is basically the fractal dimension.

Mandelbrot applied the same concept to all fractals and not just coastlines or other phenomena in nature. He created plots of measurement tool length versus the measured length of the fractal in a log-log plot. (This means that both the x- and y-axis are scaled logarithmically.) He then defined the slope of the plotted curve to be the fractal dimension of the curve. [6] He thereby totally circumvented the problem of having to deal with infinitely long curves by measuring the rate at which the measurements tend to infinity instead.

Note that this is just a very basic description of a small part of fractal geometry.

Conclusion

To summarize, fractal geometry is one of the mathematical fields whose origins can be found in a natural phenomenon, the coastline paradox. So mathematics can be applied to nature, but nature can also be used to create new mathematical areas. The two share a very special bond in that way.

Fractals are everywhere, from lightning bolts over broccoli to the Math Section’s title page. Before we end the article, let us take a look at a couple more of them in this slideshow.


References

[1] Hide, R. (1994), Collected papers of Lewis Fry Richardson. Vol. 1, Meteorology and numerical analysis, General editor, P.G. Drazin, 1993, Pp. xiv + 1016. Vol. 2, Quantitative psychology and studies of conflict, General editor, Ian Sutherland, 1993, Pp. xv + 762. Cambridge University Press. (Editorial boards for both volumes: Oliver M. Ashford, H. Charnock, P.G. Drazin, J.C.R. Hunt, P. Smoker and Ian Sutherland.) Price £95.00 per vol. ISBNs 0 521 38297 1, 0 521 38298 X. Q.J.R. Meteorol. Soc., 120: 1425-1426. doi:10.1002/qj.49712051916

[2] Wikipedia contributors. (2018, August 29). Coast. In Wikipedia, The Free Encyclopedia. Retrieved 20:18, September 8, 2018, from https://en.wikipedia.org/w/index.php?title=Coast&oldid=857080130 

[3] Mandelbrot, Benoit B. “II.5 How long is the coast of Britain?”. The Fractal Geometry of Nature. Macmillan. pp. 25–33. ISBN 978-0-7167-1186-5.

[4]Weisstein, Eric W. Fractal. From MathWorld–A Wolfram Web Resource. Retrieved 20:16, September 8, 2018, http://mathworld.wolfram.com/Fractal.htm

[5] The graphic is made using the octave GNU software, which can be downloaded for free from the official website.

[6] Weisstein, Eric W. Coastline Paradox. From MathWorld–A Wolfram Web Resource. Retrieved 20:00, September 8, 2018, http://mathworld.wolfram.com/CoastlineParadox.html


Elias Wirth

The Math Section is my personal website dedicated to applications of mathematics in everyday life. The intention behind this project is to make mathematics more approachable to the public while staying mathematically rigorous. I have a Bachelor's degree in Mathematics from the University of Berne and am currently working on my Master's degree in Mathematics at the ETH Zurich.

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