Is the coastline of Britain infinite?

The answer is yes. In a sense, Mandelbrot would say.

Benoît B. Mandelbrot, who sadly past away last week, was a novel and revolutionary mathematician. He was the father of “fractal geometry”, a field of mathematics concerned with describing complex but real objects such as “the shape of a cloud, a mountain, a coastline or a tree”, rather than the idealized spheres, ellipses or lines, features of the more familiar Eucledian geometry. “Fractal” was the term Mandelbrot coined himself to describe the mathematical shapes that could reproduce some of nature’s most intricate objects.

The actual mathematical definition of fractals is less straightforward than stated, and it involves a concept dubbed self-similarity. The cauliflower, for example, is a self-similar object: each of its branches is a scaled-down version of the cabbage as a whole, and the branches themselves are composed of smaller branches which still resemble the entire cauliflower (and so on as we zoom in). That is, a self-similar object is one that is composed of smaller copies of itself. A mathematical example is the Koch curve or Koch snowflake, a fractal that precedes Mandelbrot himself.

Zooming in on the Koch curve (via Wikimedia Commons)

Kochsim.gif

Interestingly enough, the Koch curve, just as the coast of Britain, is infinite. This is a conclusion that you can get to yourself by building your own Koch curve. Start with a triangle of equal sides. Then follow a set of simple rules:

  1. divide each side of the triangle into three segments, all of equal length;
  2. draw a triangle of equal sides with base in the middle segment and pointing outwards;
  3. remove the segment that is the base of the smaller triangle you just built.

If you follow these steps indefinitely (for each segment of the new curve), you will get a snowflake-looking shape.

Building the Koch curve: the first few repetitions (via Wikimedia Commons)

500px-KochFlake.svg.png

While the triangle you started with has a finite length, the perimeter of the shape increases every time you repeat steps 1 to 3. In the limit of infinite repetitions, where your curve is made up of an infinite number of small segments, you are faced with a shape with infinite perimeter. The outline of that shape, which has infinite length, is what is called the Koch curve.

It may not bother you that the Koch snowflake has infinite perimeter; after all it is just a mathematical shape which is built by repeating a set of rules ad infinitum, a rather unrealistic process in itself. So how can something as real and tangible as the length of the coast of Britain be infinite, that inconceivable and abstract mathematical entity?

It all comes down to the fact that intricate coastlines are like fractals, self-similar. Zooming in on a part of Britain’s seacoast will reveal a complexity similar to that of the unmagnified region. The coastline has features at all scales — from cliffs to rocks to grains of sand — so if you want to measure its length, you need to decide how small your ruler is going to be. The length of Koch’s curve becomes larger as the segments that it is made of become smaller. Similarly, the length of the coast of Britain increases as the size of the unit of measure decreases. In the limit (admittedly mathematical) where the base ruler has zero length, the coastline will be infinite.

In a sense, Mandelbrot would say, the coastline of Britain is infinite.

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2 thoughts on “Is the coastline of Britain infinite?

  1. Bart Penders

    The same would go for any other county, of course. I wonder… Given that the North Sea (and its oild and gas riches) are divided amongst the neighbouring countries based upon the length of their coast line. Should we start redrawing some lines here….

    Like

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