How long is the coast of Canada?

If I asked you what country had the longest coastline you might be able to tell me pretty quickly that the answer is Canada. And even if you didn’t know off the top of your head, after a bit of a look at a map you would probably be able to see that the answer is Canada, in part due to its massive size, but also because it has a huge number of islands. But you would find it infinitely harder to tell me exactly how long that coastline is, and that brings us to the coastline paradox.

At the most basic level the coastline paradox says that the length of coastlines is not fixed and depends on the scale you measure it at. Or to put it another way, the closer you look, the longer it gets.

We know that the length of coastlines has baffled cartographers for centuries, but the concept of the coastline paradox was first brought to light quite recently. In 1951 Lewis Fry Richardson, a mathematician and pacifist, decided to try and figure out if the length of the border between two countries had any effect on how likely they were to go to war. To help answer this question Lewis decided to look into the border between Spain and Portugal. He started by trying to find out how long it was. However, he was surprised to find that Spain had plotted the border as 987km, whereas Portugal said it was 1214km. Lewis discovered that reason for the 227km difference in length was that the countries had measured the exact same border using different scales. This led Lewis to conclude that the lengths of coastlines and some borders are not just extremely variable; they are in fact infinite.

This inspired another mathematician, Benoit Mandlebrot, to write the paper How long is the coast of Britain? and to conclude that the accurate measurement of coastlines is nearly impossible. To help explain this he later invented fractals, which are curves that get more and more complicated the more closely you look at them. These curves can be used to explain a lot of things in nature, for example snowflakes and rivers.

To illustrate the point, imagine two friends, Bridget and Jean, decide to map the coastline of a country, say Canada, as a fun lockdown project. But when they arrive for their first day of measuring, they realise Bridget brought a metre rule, whereas Jean brought a 6meter tape measure. They decide to do a practice run. Jean goes first and measures a bit with her tape measure. Then Bridget checks Jean’s work with her shorter ruler and gets a much bigger length. Then along comes Margaret with a 30cm ruler, and she finds the coast to be even longer again. At this point the three friends would realise they were victims of the coastline paradox and that they would need a lot more than a 4week circuit breaker to figure this one out.

So, the next time someone asks you ‘how long is the coast of Canada?’, you can pretty confidently tell them that the most accurate answer is ‘It depends’. Then you can show off with your now expert level knowledge of the coastline paradox. And then, if you’re feeling really bold, you can follow up by dazzling them with chat about fractals.