The thing about fractals

The perpetually unanswerable question ‘How long is a piece of string?’ now has some serious competition in the shape of ‘How many miles of coast does the National Trust look after?’

In 2004, this seemed a straightforward question with a straightforward answer – 709 miles. However, a recent calculation has disputed this figure and attests that the Trust now owns 742 miles of coast. So why the difference?

Of course, since 2004, the Trust has acquired an additional 14.5 miles of coastline, including a stretch of the iconic White Cliffs of Dover. But the answer also lies with the way that the coast is measured.

The length of the coast is fractal, which means that scale becomes a real issue when trying to measure it. The length of the coast will change according to what scale you use to measure it. As it is fractal, the closer you look at it the more detail you are able to see, and therefore the line of measurement gets longer.

If you look at the coast around Falmouth (A) on Google Maps Satellite from a height where you can see most of the UK, the coastline appears fairly smooth. There are perfectly curved bays and ruler straight edges, however, as you zoom in the coast detail becomes clearer until the jagged edges of the coast are all too evident, as are the small headlands and coves which you weren’t able to see from far away. In essence this demonstrates fractals and explains why scale affects distance – the larger the scale, the greater the detail and the longer the measurement.

This is further evident on the scaled maps of Northey Island below.

Northey Island, Essex, on three different scales

Northey Island, Essex, on three different scales

The largest scale green map clearly shows a higher level of detail and so plots a longer line than the zoomed out small scale orange – and the almost ten times difference between the two is fairly dramatic. Although all of the maps are of the same island they share few similarities and demonstrate the wide margin of difference in measurement.

It is this concept of scale which partially causes the discrepancies between the 2004 and 2013 measurements. The new calculation exclusively uses the 1:50,000 Ordnance Survey mapped coastline, which wasn’t available for Northern Ireland in 2004. A uniform approach means that this most recent measurement is as accurate as possible.

But when measuring the coast, the Trust sometimes has to rely on a subjective judgement call. Should estuaries be counted as coast, and if so, where do the estuaries and coast end and rivers begin? The good folks at Ham House on the River Thames (72km or 44 miles from the open coast) or Leigh Woods on the River Severn (116 km or 72 miles), probably don’t consider themselves as coastal – but as they experience the tide they are measured as part of the coastline!

So after that rather technical explanation are there any significant conclusions to draw from this? In short – no. But sometimes it is nice just to know.

  • If you found this post interesting, you might like this post from the Welsh coast too.
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