Monday, 1 August 2016

How long is the coastline of Great Britain?

This is a bit of a long read, so if you really want to know the answer to the question in the title of this post, it's very simple: it depends upon how you measure it. Or, you could say that the coastline of the island of Great Britain is infinitely long. But this doesn't really help anyone who wants to walk or kayak or swim round this island, so I'll attempt to answer the question here. Take a look at the image below and you'll see that I've calculated the distance of the coastline round the island of Great Britain as 11,023 miles. 

Quite a lot of coastline for a small island

But hold on a minute, I also calculated it again and got an answer of 3,876 miles, as you can see below. What's going on here? Well, the first image is an extremely detailed digitised representation of the coastline of Great Britain and surrounding islands (bearing in mind 'detailed' is a relative concept). This first map is represented by 2,282,000 individual vertices which create the polygons you see in the image above. 

In the second map, only 0.1% of these vertices are retained, so the geographical features you see below are represnted by 2,282 individual vertices. You can't see much different between the two at the scale you view them at here but if you were trying to navigate your way into a harbour or sea loch on the west coast of Scotland, for example, it would make a big difference. Click the first image to enlarge it and then compare it to the next one and you will see some differences, but nothing too drastic.

The coastline length is a function of how you measure it

At this point, you might be thinking 'hasn't this got something to do with fractals and Benoit Mandelbrot?' - and you'd be right. He wrote a very famous paper in Science in 1967 on exactly this topic, entitled 'How long is the coast of Britain'. The answer is that there really is no definitive answer - it's all about how you measure it. But let's say you want to swim or kayak around the coastline of Great Britain and nearby islands. How far would you have to travel? I tried to calculate this based on a 1km distance from the shoreline and concluded that it could be done by covering fewer than 2,000 miles - even though the coastline seems to be a lot longer. After all, you wouldn't want to go in and out of every little cove and estuary.

Be my guest

I created a little gif based on different ways of measuring the British coastline, starting off with a file that included 100% of the vertices from my original Ordnance Survey map layer (see notes below for more on this). I then created files with fewer and fewer vertices retained, all the way down to a non-sensical shape which retained hardly any of the original points. This is what I got - at 2 seconds per frame (note '% of vertices retained' figure in each image):

Coastline length at different measurement scales

It's a bit difficult to see the difference between some of these images at this scale, so I also zoomed in to the west coast of Scotland to produce another little animation. This time, you can really see more of the difference between the layers I produced. The figures on the graphics indicate what percentage of the original vertices were retained in each case. Below, this, I have also provided a still image with different versions of the coast overlaid on top of each other, just to demonstrate the impact of reducing the number of vertices on the representation of the coastline, and hence its length.

This shows Morar, Mallaig and Loch Nevis 

Each line represents a different level of generalisation

I then decided to take a smaller island and extract the individual vertices (also known as nodes) that make up the shapes you see in the maps above. For this, I chose the Isle of Skye because it's one of the biggest British islands and the coast is highly irregular and indented. Using the version of the original shapefile where I retained 1% of the original vertices, Skye is represented by 772 individual nodes joined together to make a single polygon, as you can see below.

This produces a pretty good approximation of the coastline of Skye for most purposes. At this resolution, the coastline of Skye comes in at 330 miles (530km), compared to 456 miles (733km) at the original resolution. But of course we need to remember that if we had digitised around every single rock around the coastline the length would be nearly infinite. If you measured the coastline with a matchstick, for example, you'll get an extremely high value (and a sore back).

Skye represented with a polygon comprised of 772 vertices

Here's what this looks like when you show them one by one, in an animated gif - just to give you an idea of how it is plotted spatially. This is shown at 15ms per frame, so the dot fairly zooms around the coastline. All of this also gives you a little insight into how a GIS deals with geometry and what goes into the shapes that you see on your screen. It also helps explain why the very detailed, highly accurate spatial data files we can download from Ordnance Survey aren't always the most appropriate ones to use in small scale mapping. Or, maybe I just wanted to make another geogif, but either way I think I learned something.

A dot going round the Isle of Skye at 99,000 mph (forever)

So, how long is the coastline of Great Britain? Well, if you want to swim or kayak around all islands then you should think about training for a distance of around 2,000 miles and if you want to walk the coastline of Great Britain only then it's most likely going to be a bit more, or maybe a bit less - but that depends upon how you plan your route. Despite all the uncertainty, however, I think we can all agree that you'll need to go more than 1,024 miles.

Yes, this is Britain (kind of)

Last of all, I also did a little gif showing the 174 vertices of Great Britain when the file is massively reduced - so I'll end with this.

Another one, just for fun

Notes: I used the OS OpenData Boundary Line product for the coastline. This was a polyline file so I converted it to a polygon and then generalised it several times using the Visvalingam algorithm in mapshaper. Contains OS data © Crown copyright and database right 2015. You'll see if you search online that my measurements are close to those of others - so I'm at least as right or wrong as some people. If you're interested, you might want to look up the coastline paradox as well and, of course, Lewis Fry Richardson. Other big British islands? After the island of Great Britain, it's Lewis and Harris at 741 miles of coastline (1,193km), the mainland of Shetland at 692 miles (1,113km), Skye at 456 miles (733km) and North Uist at 334 miles (537km). Remember that this refers to coastline length and not land area.