Eli5 why the coastline paradox is a paradox?
In: 1301
Because you have a finite area that a seemingly infinite coastline encompasses. It’s also weird how your measurement precision *increases* the actual measurement, potentially infinitely.
The term paradox does not necessarily mean something that is impossible, but can also be applied to some things that are just so counter intuitive that you would not remotely think of them
The counterintuitive thing here is that you would think that the measurement of a coastline is consistent. Most people find it incredibly counter intuitive when they are told that the answer varies from a fixed number to infinity depending on how you measure it.
When you measure something, you’d think that measuring with a smaller granularity would get you a more precise reading. With a coastline, the smaller the unit you use the larger the answer becomes, going all the way to infinity. So what’s the true length of any coastline? No one really knows. That’s the paradox.
The measured length of a coastline depends on the size of the smallest feature that you take into account. That is, the closer you look, the longer the coastline becomes. And there’s no upper boundary on it, so the measured length of the coastline can be made larger than any finite number by taking a close enough look at it. Basically, the true length of a coastline can be thought of as infinite. But at the same time we can clearly see that the coastline is a finite object which is clearly bounded. So we have a finite object of infinite length. That is the paradox.
3Blue1Brown mentions England’s coast line is about 1.21 dimension (fractal): https://www.youtube.com/watch?v=gB9n2gHsHN4
One way to think about it is to look at pictures of the Earth from space. It looks like a perfect sphere. Spheres are smooth round objects, right?
Except we know it’s not a perfect sphere, because it’s a bit wider at the Equator when we measure more closely.
More than that, if we zoom in on the Earth a bit more, we might start to see the mountain tops. The mountains aren’t a smooth sphere, they’re jagged points.
And then we go a bit closer, and we see buildings and hills sticking out. It’s even less of a sphere. It’s all over the place.
Zoom in even more, you’ll see the potholes in the road. Even further, every rock, every imperfection in the dirt.
So each time you look more closely at the Earth it gets even less like what looked like a perfect sphere at the start. Let’s say we calculate the surface area of the Earth. If we calculate it as a sphere, we’ll get an estimate, but we’re missing out the mountains. They’ll add a bit. Maybe we can include the mountains, but then what about the hills? They’ll add a bit too.
The more accurate you get, the more detail you have to add into the measurement.
The coastline works the same way. The more closely you look at it the more imperfections there are to take into account that increase the total length.
It isn’t a paradox in the formal sense. People are a little fast and loose with the word paradox. Like the Fermi paradox
vasuace2 on different types of paradoxes
[https://www.youtube.com/watch?v=kJzSzGbfc0k](https://www.youtube.com/watch?v=kJzSzGbfc0k)
Pretty sure people are using the term infinity incorrectly. Using Planck length as the measurement unit, you could measure around an island and indeed get a very large number. You think that is an infinite value? One could then use Planck length/2 and take another measurement and indeed get a larger and more accurate measurement of the island. Do you really think any of these values are infinite? Wowsers.
My interpretation is that focusing on the math only is the wrong way to look at it.
For example, a well defined and very curvy shape will have an increased edge area as one uses smaller and smaller straight units of measure. However, because it’s well defined, this value will converge on a true value.
In the case of a real beach though, as you zoom in there are more and more boundaries to measure and the actual definition of what is coast and what is ocean becomes much less distinct. What grain of sand is the edge?
It’s possible that if we froze time and inspected the beach molecule by molecule, this particular paradox might no longer be defined as one. At least I believe at that scale eventually the measurement would converge to a true value.
I don’t believe this should count at all as a paradox though. It seems to be a description of a easily observable fact that boundaries are relative. Scale is a huge factor in any kind of measurement. However, I understand that Webster isn’t backing me up on my definition of paradox and that many such exceptions apply.
Latest Answers