the coastline paradox.

In: Earth Science

It’s a weird quirk of measuring the edge of something that has *fractal-curve properties*. It starts with the question of “How long is the coastline of England?” Seems straightforward enough, just use your map, measure the coastline, and scale it properly. Except, you missed some detail, the coast is actually bumpier than your map, so you get a better map that has those missing bumps. Except, again, this map isn’t bumpy enough. You can get more detailed in your measurements, but then your coastline will always have more detail.

And, as you add that detail, the coastline can get arbitrarily large.

Consider a 1 meter x 1 meter blanket, what’s the surface area? 1 meter^2 right?

Well now what if its a quilt? It has a foot print of 1 m^2 but the lumps and bumps mean the actual quilt has more than 1 m^2 of surface area to it

What if you zoom in even further? The threads don’t form a perfectly smooth plate so there’s even more area from going up/down over each thread, and the bodies of those threads aren’t perfectly smooth so there’s even more area there!

The end result is you starting with a blanket that clearly had a surface area of 1 m^2 and ended up with an answer far higher than 1 by the time you were done

Similarly coastlines look smooth from a distance but get really jagged as you start getting closer. If you’re just counting how many 1 km long lines you can put in a row against the coastline you’ll get a different number than if you use 100 km long lines or 10 meter long lines, and if you start looking at millimeter long lines you can fit far far far more than the 1 km long line estimate would have led you to believe because they fit into all the nooks and crannies that the longer lines passed right over.

The paradox is that you can get wildly different measurements while measuring the same thing just depending on the size of the unit you are using. Increasing the precision of your measurement doesn’t improve accuracy, it fundamentally changes the value being measured

Suppose you want to measure the circumference of a circle. Obviously, the edge is round. But your ruler is straight. So, you measure as best as you can with your straight rules. Now, if the ruler is just the right length, you can measure 4 times round the edge, and you’d think the circle was actually square. So, you switch to a smaller ruler. This time, you get a pentagon. The length is closer to being correct, but still wrong. So, you switch to a shorter ruler. Now, your measured ‘circumference’ is longer still, but still not right. As you measure with smaller and smaller rulers, you get better and better estimates which keep increasing, but the straight edge can never match a circle.

Coastlines aren’t straight. They’re full of all sorts of wiggles, bays, coves etc. If a straight edge can’t measure a circle properly, how can it measure a coastline? Depending on the scale of the map (the length of the ruler), you can get vastly different values.

A straight line is the shortest distance between two points.

If you add a bump to a line, the line gets longer.

If you measure a coastline, for example, there’s a lot of detail to consider. Each detail is a bump in an otherwise straight line.

If you’d measure all levels of detail, the line will have a heck load of bumps.

Those bumps make the line really long.

The “paradox” term is loosely applied. It refers to how you can seemingly always go finer in detail, hence no measurement is universally “correct” in a strict sense.

Though, practically, even if you assume discrete space and, say, the Planck length or something as a smallest unit, the “correct” measurement would be useless. Taking all that detail into account (assuming for the sake of argument that atoms and all were completely static while measuring) would leave you with a ridiculously bumpy line, and thus a ridiculously huge length.

It might turn out (and I’ll just pick a random number for simplicity’s sake) that the coastline of Great Britain is 500 trillion kilometres long. Even if that were the “correct” measurement, it would be useless. It’s a ridiculously high number, way out of scale for any application of it we’d have.

A coastline isn’t a simple shape. It’s very very complex. If you measure a coastline with, say, miles, then you’ll end up rounding off these complexities which means you’re getting a length that is actually *less* than the true length. As you measure with smaller and smaller units, you round off less and less complexities, and the length gets larger and larger. That’s my understanding at least