How Long is the Coastline of Britain?

I’m reading a charming “slow travel” memoir by Dan Kieran, and came across this passage (reflections during a period that he and two friends took a milk float across southern England) about the paradox of measuring the coast of the island, and pointing the way (although Kieran doesn’t mention it) to fractals.

At the time I was going slightly mad, but when I got home a friend pointed me towards the paper called ‘How long is the coast of Britain?’, published in 1967 by a mathematician named Benoît Mandelbrot. It suggests that the depth of your journey–in terms of how closely you perceive it–really does increase its length in mathematical terms. The answer to the question posed by the paper is that there is no answer. It’s a paradox because the length of a coastline depends entirely on the way you measure it. It comes down to context, and the context in which you perceive something is a function of the brain.

To explain the coastline paradox you first have to accept that there is a definitive geological coastline of Britain, which would depend on high or low tide and all manner of other variables. Assuming you’re prepared to accept this, you then have to imagine measuring this coastline with a three-foot ruler. You would, eventually, come up with the coastline’s length. But what if you then repeated the experiment with a one-foot ruler? The smaller ruler would give you a greater distance because you would be able to get into lots of nooks and crannies that the three-foot one would have to stretch across.

Now you’re probably thinking, ‘Well, fine, but if you went down to a one-inch ruler you get an even greater distance. At least that would be accurate though, because you can’t get smaller than a one-inch ruler.’ The problem is, of course, that you can. You can shorten the ruler over and over again, going deeper and deeper, getting smaller and smaller, and the length of the coastline will increase each time. So there really is no definitive answer to the question. It’s a rather disconcerting thought, isn’t it? We all work on the assumption that the real world is defined according to measurable things, but once you begin to focus on it, the act of measurement itself becomes fraught with uncertainty.