Wednesday, March 08, 2006

(39) The precise measurement of the British coastline

Around noon on December 26th 1988, I continued to read from Chapter Four of James Gleick’s book Chaos:Making a New Science.

Chaos is an interdisciplinary idea which has thrown up unanticipated difficulties in making long term predictions in situations which were previously thought to be quite simple to predict with considerable, if not absolute, deterministic precision.
This was the first book on the subject that I had read.

I had flipped through it before I came to read Chapter Four that afternoon, but I certainly had not read through it in detail before then.

Sub-titled A Geometry of Nature, the chapter is mainly concerned with the ideas of Benoit Mandelbrot.
The shapes of classical geometry, lines and planes, circles and spheres, triangles and cones, are a powerful representation of reality, but for understanding complexity they are the wrong kind of abstraction. Mandelbrot observes that clouds are not spheres, mountains are not cones, and lightning does not travel in a straight line.
The new geometry mirrors the rough, twisted, tangled, pitted reality of the real universe. His work makes a claim about nature’s complexity, which is that such odd shapes are more than just distortions of the classic forms of Euclidian geometry. They are often meaningful and may reveal the very essence of a thing.

What is the essence of a coastline, for example?

Mandelbrot asked this question in a key paper of his in 1967: How Long is the Coast of Britain?
He argued that a coastline is really infinitely long, for it depends on the length of your ruler.

Open a set of dividers to a length of one yard, and walk them along the coastline. The resulting number of yards is just an approximation of the true length, because the dividers skip over twists and turns smaller than one yard, but if you note the number anyway, then set the dividers to a smaller length — say, one foot — and repeat the process, you arrive at a somewhat greater length, because more of the detail will be captured.

Set the dividers at four inches, and start again, and you will get a still greater length.

This mental experiment is a way of quantifying the effect of observing an object from different distances, at different scales.

The length of England’s coastline estimated from a satellite would be a smaller guess than that of an observer trying to walk its coves and beaches, who will in turn make a smaller guess than a snail negotiating every pebble.

You might think that these estimates will continue to get larger until they reach some final value, the true length of the coastline.

But Mandelbrot found that as the scale of measurement becomes smaller, the measured length of a coastline rises without limit, bays and peninsulas revealing ever-smaller subbays and subpeninsulas — at least down to atomic scales, where the process does finally come to an end.


I had not seen much listed in the TV guide in the papers for that Boxing Day that appealed to me.
I had noted the following programme on BBC 2: 4. 00:

The Shape of the Nation. A gull’s eye view of Britain’s coastline starting at Land’s End travelling anti-clockwise accompanied by the music of Britain.

It did not sound exceptionally interesting. All the same, at about 4.45 p.m. I tuned into it.

I expected that there would be a few places of interest around Britain’s coastal regions.

But to my surprise I discovered that it was a continuous film, shot from a jet aircraft, of the entire British coastline.

It was, of course, a conscious decision on my part to watch this programme, but even so I find the very fact that it was broadcast on the same day as I happened to read that part of Gleick’s book, plus my uncertainty about its precise content, to make the coincidence noteworthy enough to warrant inclusion here.

And there is also the consideration that, although for the point he was making about the rough and uneven and, perhaps, ultimately unmeasurable qualities of a coastline Mandelbrot might have taken any, he had actually selected the very coastline that the jet was negotiating; the British.

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