What Is The Coastline Paradox?

I've been driving along Australia's famous Great Ocean Road. And I'm stopped here near the Twelve Apostles, which are these big sandstone bluffs. Actually, there's only eight of them left because the others have eroded over time. And erosion is really what's given us this coastline the way it looks now.

So that brings to mind a question for me. Which is, "How long is the Australian coastline?" Well, if you were to measure it out in lengths of 500 kilometers, you would find that it's about 12 and a half thousand kilometers long. But the CIA World Factbook puts the figure at more than double that: over 25,700 kilometers.

But how can it be that we have two different estimates for the length of the same coastline? Well, this is called "The Coastline Paradox." The answer is, it depends on the length of measuring stick that you use. So, if you connect up the dots from cliff to cliff to cliff, you get a shorter length of coastline than if you measure with a smaller measuring stick and measure into every inlet.

So what length of measuring stick should we use? Well, in theory, you can go all the way down to the size of a water molecule. And if you do that, then the length of Australia's coast is virtually infinite. Do you believe me that you could have a finite area object like Australia bounded by an infinite perimeter? It doesn't seem to make sense.

But I can give you another example of this: it's called the Koch snowflake. So what you do is you take a triangle with sides of length 1 and then on each side add another triangle with sides of length a third. Continue doing that again and again forever. What you end up with is a shape which is a finite area but an infinite perimeter.

Shapes like these are called fractals, and many coastlines have the same fractal structure, which means they have some sort of self-similarity on many different scales. So you can zoom in and zoom in, and the coastline looks roughly the same.

So if you want to know the length of a coastline, you need to first specify the length of your measuring stick because that's what the answer depends on.