The closer you look, the longer it gets: The Coastline Paradox

Rob Bell:
Hello, and welcome to Sketchplanations The Podcast, discussion on facts and ideas to help fuel your own interesting conversations.

I'm engineer and broadcaster Rob Bell, and with me is the creator and talent behind Sketchplanations, Jono Hey, and entrepreneur and past winner of The Apprentice, Tom Pellereau.

Hello, Chaps.

Jono Hey:
Hello.

Tom Pellereau:
Hi.

Rob Bell:
Now then, have you ever heard of the coastline paradox, listeners?

I hadn't, but once I'd seen Jono's sketch that explains it, it really got me thinking.

Now, I'm fascinated by it, and I often wonder if it applies to many other things than just coastlines.

You should be able to see Jono's sketch that explains this concept as the artwork for this episode.

But if you fancy a closer look, then go and find it at sketchplanations.com.

Right then, Jono, let's get into it.

The coastline paradox, over to you.

Jono Hey:
Yeah, the coastline paradox, broadly speaking, is actually very simple.

If you said, what is the length of the coastline of Great Britain, for example, there isn't actually a fixed answer.

Rob Bell:
What?

Jono Hey:
I know.

So there isn't a fixed answer.

We're used to things where if you measure them, it's the length that it is, and you can measure it more and more accurately, and you'll get more and more accurately that length.

It turns out with a coastline or something like that, it depends on the size of the scale that you're measuring it with.

And so, this is the coastline paradox, which is actually the smaller the unit of measure you use, the longer the coastline gets.

And it's a really interesting story for that that we can get into.

But in brief, that is the coastline paradox.

Rob Bell:
I know you're a keen hiker, and you are often at the coast in your leisure time.

Is that how you discovered this phenomenon?

Jono Hey:
No, but you know, I often think about, as I say, we often go down to southwest of England, and I put myself in that position.

And I think if I was to be measuring the coast, like looking at a map of the southwest of England versus standing on the coast path, versus like when the tide's out, going in and out of all the things, I do think about that a lot.

But no, I heard about this, including the story that sort of led up to it from a really fascinating book of which multiple Sketchplanations exist called Scale by Geoffrey West.

And I think I have mentioned it once or twice before.

And he has a really interesting chapter on this, which is basically that measuring, it's so mad, measuring length isn't as simple as it might appear.

And in some ways, it was like so obvious and so assumed that nobody questioned it for what might be even thousands of years until this came up.

So yeah, I don't know, I think of it when I'm down the coast, but I found out about it from this fascinating book called Scale.

Rob Bell:
Can I give a half-baked fact about what I know bits about?

Jono Hey:
As long as it's not an alternative one.

Rob Bell:
It's not an alternative one.

It's about the meter and the standardisation of the meter.

And for years, it was this, I'm going to say, it's not steel, it's not iron.

It might have been iron, a piece of iron that was deemed to be one meter long, and it was kept in France.

And that's as much as I know about it.

Since we have more complex measuring, this is the half-baked bit, I don't know if you can realise, now there is a more accurate way of defining it, which I think is to do with the speed of light or some other constants, some other kind of global physical constants that help you define a meter other than something that's just a physical piece of metal.

Jono Hey:
I have heard that and I can't fully bake it, but I know what you mean, that it completely changed from, here's this one long meter to something to do with distances and speeds, a bit like how long a second is or whatever is.

Yes.

We live quite near the National Physical Laboratory where I think they actually have the atomic clock, which is oscillations of the cesium atom or something like that, which is really fascinating.

Rob Bell:
Yes, because before that, it was just a guy with a stopwatch who was deemed to be the best guy at stopping it at exactly one second.

Sorry, I digress.

Jono, tell us about where this idea of the Coastline Paradox came from.

Jono Hey:
Yes, I can tell you.

I'm going to lean very heavily on this chapter from this book because he tells it so well and it's such an interesting story.

There's various bits about the Coastline Paradox on the web, but nothing is good as what I found in this book.

The story starts quite oddly with this guy called Lewis Fry Richardson, who was an Englishman, he was a polymath more or less.

He was like a mathematician, a physicist, a meteorologist.

He got a psychology degree later in his life.

Rob Bell:
Love those guys.

Jono Hey:
One of the things he did was actually pioneer computational modeling for the weather using hydrodynamics equations much before.

This is around 1910, that sort of time, way before like a high-speed computers and actually some of his techniques still form the basis of much like modern weather forecasting.

Rob Bell:
Brilliant.

Jono Hey:
Decades later, which is quite amazing.

But he has this curious history.

He was a Quaker and a conscientious objector.

He lived through World War I, so he conscientiously objected so he wasn't part of World War I directly.

He was just fascinated with war and conflict and how to prevent it.

The bizarre way that led up to the Coastline Paradox was he was looking for, from what I've read, a quantitative theory for the origins of war and international conflict.

So kind of like a science of war.

He took some of the modeling that they were doing for chemical reactions and the spreads of diseases and things and used them to try and model the accumulation of weapons in different countries.

Like this country increases their military, so this country increases their military, and you get this sort of spreading effect.

He had this great term for it.

He called it a deadly quarrel.

A deadly quarrel is a violent conflict that results in a death.

One of the interesting things about it, which is where it led to something interesting, because instead of looking at it from a sociological angle, or psychological angle, or even historical, he looked at lots of data on historic wars.

He was just there focusing on data and looking back at past wars.

His idea of a deadly quarrel was essentially, he said, well, is there a continuum?

On the one hand, you've got a guy's having a fight and somebody died.

It was an individual murder.

Then the other hand, you've got this top level, you've got World War II, where 50 million people died.

It's a scale in orders of magnitude.

It's a bit like the Richter scale.

You've got on the one hand, the Richardson scale would be one is a murder and eight is World War II.

If 10 people die, your magnitude two is like a riot.

If 100 people or so are dying, it's a skirmish, it's magnitude three, and you go up to level A.

Anyway, he was really interested in war and what caused it and how to prevent it.

He had this hypothesis that the probability of a war between two countries was proportional to the length of the common border, which I think is quite interesting, so you go, Spain and France just match up at the Pyrenees, but other places, US and Mexico or so, have a much longer border that goes on for a long time.

Some places have got big borders, some don't.

Because he was looking at data for things, he started looking at the border length.

And so, for example, he found this weird thing that the border between Spain and Portugal was sometimes listed as 987 km and sometimes listed as 1,214 km.

Rob Bell:
Right, okay.

Jono Hey:
Netherlands and Belgium was sometimes 380, and other times it was 449 km.

Rob Bell:
Okay.

Jono Hey:
And this was a time when, you know, we've had a lot of science with the Industrial Revolution and all that.

We knew how to measure stuff really well.

Apparently, we knew that the height of Everest within a few feet.

So, you know, what could be going on?

How could we be messing up the measurement of this border?

Isn't that just really weird to be out by hundreds of kilometers?

And so, this is where it kind of started, was looking at the length of borders.

And so, to give a counter example, and I think which is why this is so weird, if you were to like try and measure the length of the room you're in, or your living room, and I gave you like a meter stick, imagine you put this meter stick one after each other, and you might get, you know, six and then it doesn't fit in.

So, you get like, okay, it's around six meters.

But then if I gave you a 10-centimeter ruler, and you did the same and laid it all out, you might get six meters 30.

And then if I gave you a one-centimeter ruler, you go, okay, actually, you know, it's six meters 28 or six meters 32.

And the smaller your ruler, the more it converges to the precise length of your living room, right?

Rob Bell:
Yes.

Jono Hey:
That's what you expect.

When you measure something, the more precise you measure it, the more accurately you get that number.

So your measurement converges on the true answer of the length of your living room.

Rob Bell:
Yes.

And this is where I find it becomes fascinating and where you start to get a bit lost because I automatically want to take that to its extreme.

Jono Hey:
Okay.

All right.

Rob Bell:
Just do what I mean.

Jono Hey:
Yeah.

Yeah.

But you can do that with the length of your living room and it will get closer and closer like adding digits to pi, right?

Rob Bell:
Yes.

Jono Hey:
Instead of 3.142, you'll be 3.14159 and so on.

Yeah.

It will get more and more precise.

So that's how we're used to in like Euclidean geometry.

Now, if you do that to a border or a coastline, instead of converging on the true length of the coastline, the smaller your ruler you use, it keeps getting longer.

So instead of it converging on the one value that your coastline is, this is the true length of your coastline.

So you start with 100k ruler for the Spain-Portugal border and you get 900.

Rob Bell:
Because you're missing out any of the little bumps and whatever along the line.

Jono Hey:
Exactly.

And if you keep on going down smaller and smaller, it just keeps getting longer.

And so in some ways this like violates these laws of measurement that we've been using for a few thousand years, which is basically this fundamental point that length depends on the scale that you use to measure it.

And it isn't an objective property of what you're measuring.

Rob Bell:
Yeah.

Yeah.

Jono Hey:
I don't know about you, but that kind of blows my mind.

And I think what he tries to get across in his book, which is like, it's incredible.

We've had all these mathematicians and smart people from the time of the Greeks and the Egyptians working on this stuff.

And nobody really noticed that everybody sort of, you know, once we had the Greeks with the geometry and squares and triangles, and we were trying to build stuff and you go, it's three bricks, four bricks, whatever.

We sort of impose our human measurement on this.

And when you look at, actually, if you look at the natural world, that's not how it behaves.

Measurement works very differently.

Rob Bell:
So if we take that to its extreme and you keep getting smaller and smaller and smaller and the measurement keeps getting smaller.

Tom Pellereau:
When you start measuring around every single little stone.

Rob Bell:
Well, so this is what I'm talking about.

Tom Pellereau:
Not only do you measure the fact that it goes in and out, you kind of go, well, it's got stones here, so we've got to measure around the outside of that, so and in the inside of this one.

Rob Bell:
Then you go even smaller and you go around every bump of every stone and then you go even smaller and every atom of every bump and then you go even smaller.

So are we saying then that the coastline or the border of any nation is infinite?

Jono Hey:
I mean, I think if you had a small enough ruler, it broadly gets there, which is, I don't know, it's just a sort of pretty mad.

Tom Pellereau:
Unless it's completely straight, there are some borders that are completely straight, which is I don't think it matters.

That's mental, isn't it?

Rob Bell:
This is what I mean.

I was so intrigued by it.

The first time I saw this in your sketch, Jono, is that's what?

Jono Hey:
Yeah, that example of some things being straighter than others.

So Richardson apparently published his work in a somewhat obscure journal.

And of course, it was in a paper that was investigating the origins of war.

So it was kind of missed for a long time.

Yeah, I think he had this funny, interesting name for me.

It was called The Problem of Contiguity and Appendix to Statistics of Deadly Quarrels.

So this fundamental insight and measurement was in that.

So it took this guy, Benoit Mandelbrot, to sort of rediscover this work and actually-

Rob Bell:
What, years later?

Jono Hey:
Yeah, years later, I think 1960s or so, to sort of expand it and generalise the findings and figure out the maths to deal with them.

So he specialised in what he called the art of roughness and fractals and self-similarity.

And to your point about how wiggly things are.

So for coastlines, for example, if you look at southwest of England, it has what's called a fractal dimension of 1.25.

Rob Bell:
And fractal dimension, you're talking about how wiggly it is?

Jono Hey:
A measure of how wiggly it is, yeah.

So it's South Africa, apparently, like the coast of South Africa, it's very non-wiggly and it was a dimension of 1.02.

But Norway, if you can imagine, with Norway with this big long fjord, you try to walk a few miles in one direction in Norway, and it takes you a really long time because it's really wiggly and crinkly and that's much larger at 1.52.

So one way to think about that is if you increase the resolution of your measurement by a factor of two, so you go from a meter to scale to measure it at half a meter, the Britain's coastline goes up 25%, but Norway's goes up 50%, which is a huge amount, right?

You start measuring it at a scale of half the size and Norway's coastline goes up like 50%.

Rob Bell:
And if you do that again from half a meter to 25cm, it will go up 50% again.

Jono Hey:
It will keep going up again, assuming that it sort of stays as wiggly as you go in.

Rob Bell:
This is amazing.

Yeah.

Jono Hey:
So we've talked about coastlines and borders, but just to broaden it out a little bit, and you're probably thinking this already, basically, enormous numbers of things in the natural world that we're all familiar with are actually sort of fractal in nature and have these self-similarity, similar properties.

And it can be like really small if you think of like leaves and soil, or a great example is this, I don't know if you've seen it, there's a beautiful cauliflower called the Romanesco cauliflower.

Rob Bell:
Oh, yeah, I think so.

Jono Hey:
Yeah.

Which is basically sort of a pyramid shape.

And as you look at the surface, it's pyramids.

It's made up of pyramids.

And each of those pyramids is made up of tiny pyramids.

And it's just this beautiful, like self-similar thing all the way down.

Or like the branching of a tree.

Or you might have seen like, you know, sand dunes, but then you look at the sand on the sand dunes and they're doing the same pattern that the sand dunes are making on the sand dunes.

Clouds also, like if you were to say, what's the surface area of a cloud?

You'd struggle because you go down smaller and it kind of still looks like a cloud at macro level.

And then other things like, I think, you know, the surface of the ocean, the surface area of the ocean.

Stock markets, I think, also sort of do this.

And then you go bigger, so you got like mountains.

So actually, if you think of what a lot of Mandelbrot's work and fractals, the only way you could start to simulate in graphics realistic looking mountains was by being able to do things which look on the big scale the same they do on the small scale.

Mountains kind of look like small mountains that build up into something that looks like a mountain on a big scale as well.

And if you take an aerial photo of the Grand Canyon, it could just as well be like a rivulet of water running through some gravel in your back garden.

It just looks like a giant version of that.

And then just finally, to connect it to some other things, which are also super relevant, where this math comes in is things in the body.

Like we're so familiar with, you think of the surface area of the brain with all its crinkles and wiggles.

Rob Bell:
The brains are wiggly.

Jono Hey:
Yeah, and you think of the lungs where you got two lungs and then each one is broken into branches and then each one of those is broken into branches and then you have all these tiny little…

Sorry?

Rob Bell:
Bronchioles.

Jono Hey:
Yeah, and then you get down to the alveoli and blood vessels or you think of like, what's the surface area of the inside of a vein or you've got cilia on it.

These things are everywhere.

And yet, we were kind of ignoring it for centuries when we tried to measure stuff, which is just fascinating.

And it's sort of illustrated, I think, so clearly by the coastline paradox.

So I'll shut up now because I just think it's just amazing.

Rob Bell:
Oh, it's very, very cool.

And you're talking about other areas where it applies as well.

Your results are only as good as your method.

Or not as good, but as…

Yeah, your results are only as good.

Is that a thing?

Is that a saying?

Jono Hey:
It's garbage in, garbage out.

I mean, this is definitely the case that if you get…

If you get a smaller ruler and you try and measure the details of your coastline, you will get a much larger number.

So if you're more accurate, it will change the results that you get.

Rob Bell:
But in a bit more effort, you'll get a more accurate output.

Tom Pellereau:
I have a similar question.

I think it comes down to the same thing, about time resolution.

So I go out for a 5K run, right?

And my phone, my watch is monitoring how long, how far I'm going.

But I sometimes hold my phone in my hand, and I'm like, my step is a certain length, and it's not dissimilar to the distance that my arm is going up and down as I'm running.

And I'm like, I presume the resolution of my kind of tracking is only doing it every few seconds, as it were.

So it doesn't tell the difference in my hand going up and down and up and down and up and down.

So I'm like, if this was tracking at a higher resolution, it would think that I'm climbing them, like it would say that I've gone up every single 5,000 steps that I've done with that.

So actually, the higher the resolution of my GPS, maybe, the longer it would seem that I've run.

So I might have actually only traveled a distance on the floor of 5K, but because it's in my hand, it's going up and down.

Occasionally, I do that or I'm avoiding someone, I could very easily quadruple, double, etc.

The distance that-

Rob Bell:
Is that how you're doing it?

Tom Pellereau:
The phone has traveled.

Jono Hey:
It turns out you're only running one case.

Tom Pellereau:
Because I genuinely at one time was like, maybe I should be holding my phone in my hand, because I get almost triple the distance in 3D.

Jono Hey:
I know what you mean there.

The timing of your measurements, if you just do every second, you missed all the movement that you had in between.

I think that's the thing with the stock market having these patterns as well.

It has it on the macro scale, but also if you zoom in on a time scale of what's happening-

Rob Bell:
Closer, closer, closer, yeah.

Jono Hey:
Yeah, you're getting that same kind of effect.

But yeah, great example.

When I don't, I don't know, it's cold or dark out, and I just need to go for a run, sometimes I just go around the houses.

You can go a long way if you just go in and out of every single road and dead end that you come to, and you don't have to go very far from your house.

Rob Bell:
Up and down everyone's driveway along that road as well.

Yeah.

Jono Hey:
Once in Bushy Park, there's this big avenue of trees, there's like two or three trees in a row for like a good quarter, half mile or so.

I did one run which was like, instead of running down the trees, I ran to the right, around the one on the right, and then back to the one on the left, and then around to the one on the right.

I slalomed all the way down and it was interesting.

I made this, it looked great actually.

I thought it should be a real event.

Yeah.

Because it was quite good slaloming these trees, and I ran them really quite a long way, but in a very short space.

I was like the coastline of Norway, trying to get down this Bushy Park line of trees.

Rob Bell:
What was your wiggly?

Jono Hey:
My fractal dimension was huge in that one.

Rob Bell:
Yeah.

Jono Hey:
Huge.

Rob Bell:
I was thinking about this in a very different non-physical sense.

Now that I work in TV and essentially storytelling, I saw a little parallel there.

The more detail about the events in history or a particular character's past or cultural nuances, whatever it may be, the more details you go into when telling that story, the more complex that story can become.

And so when you're making documentaries about something, it's this kind of continuous judgment on how much detail do you add because you want people to come away with something interesting and maybe useful, but at the same time, you need to be wary of putting people off by making whatever it is your documentary too long, but it's just, I can't watch this.

I'm completely lost in this now.

So is this this kind of wiggle and how much detail are you going to go into each of the wiggles within that story?

I thought that was quite a nice kind of parallel to me.

Jono Hey:
That's really good.

It reminds me of there's a Simpsons episode where just shown like old people going on a lot and there's Grandpa Simpson in a supermarket and he's trying to pay and he gets out a coin from his wallet and he's like, this reminds me, it was a Tuesday.

I got up in the morning and I went downstairs.

I set the toaster to three, medium brown, he's chatting to this cashier and there's a massive line of people.

What depths do you go into for your stories?

Tom Pellereau:
That's brilliant.

Jono Hey:
I love that.

Tom Pellereau:
Is that the secret of storytelling, knowing how deep to go?

Because I presume the depth creates memory or stirs memory, so it makes it a memorable story.

Rob Bell:
Details can be really interesting, can't they?

Details can really draw you in to that story, but too many details, you're like, oh, come on, get to the end.

Tom Pellereau:
We're all different in terms of our desire for certain information.

Rob Bell:
Yeah.

But like Jono says, you could always be infinite about how detailed you go into that story.

Tom Pellereau:
The definition of filibustering, isn't it?

Rob Bell:
I'm not, I'll probably leave this bit out, but I quite enjoyed researching it.

Now, I'm not a computer expert at all, which you can probably tell by the fact that I've used the term computer expert.

I heard something on the radio that day about how much smaller and more powerful microchips are becoming.

It's insane now compared to how it was when microchips were first invented.

On a microchip, you have a number of switches or transistors that deal with and process information through electrical signals.

On early microchips, you'd maybe have just a few transistors, but it was still a microchip, so quite small.

Nowadays, you can have billions on a microchip of the same size or even smaller because of how small those transistors are becoming.

And we're talking like two or three nanometers or billions of a meter for each of these transistors, right?

They're right so packed in so close next to each other.

But the difference here to the coastline paradox is that, well, with this technology or with current technology, at least, there's a physical limitation to the size of those transistors, like down to, when you go beyond the size of an atom, you can't get it to do what you need it to do.

Whereas with the coastline paradox, you can keep going smaller and smaller and smaller and smaller, right?

And getting longer and longer and longer and longer totals.

Tom Pellereau:
Although you might be constrained by the same when you start getting down to the atomic layer.

Rob Bell:
On a coastline?

Tom Pellereau:
Yeah.

Rob Bell:
But then you go subatomic, because all you're doing is measuring it.

You don't need to do it.

It doesn't need to do anything.

Tom Pellereau:
Yes.

Yes, you are correct.

Rob Bell:
And then you go half of that and then go half of that.

Tom Pellereau:
There's a game here, half of that, half of that.

Infinity plus one.

Rob Bell:
I don't know how you guys feel about infinity.

I remember learning it about the concept of infinity at school.

It would have been maths or physics probably.

And it was enjoyable.

I enjoyed learning about it.

But I've always felt a little uneasy, you know, with this kind of sense of utter chaos.

Things being infinite because, you know, there's this complete lack of control in my mind.

When are things possible then?

Jono Hey:
That's mad.

I've never had that reaction, but I can see it.

I think like the number zero was around as early as the other ones as well, which is curious in a similar kind of way.

Rob Bell:
Yes.

Zero.

Absolutely nothing.

Jono Hey:
There's nothing.

Why do we need a number for that?

Rob Bell:
Good.

Amazing.

Is there anything else anyone wants to add on the coastline paradox before we round this topic off?

Jono Hey:
The summary is, it's like meaningless to quote a length of things in the natural world without the scale at which you measured it, which is just so different.

Human world, lengths are lengths, but in the natural world, complex, crinkly, crinulated.

Rob Bell:
Brilliant.

Jono Hey:
Oh, I do have one last thing.

Mandelbrot isn't known for fractals, which is like a pattern repeating in itself.

Yes.

So I've got my only fractal joke.

He's called Benoit B.

Mandelbrot.

Do you know what the B stands for?

Rob Bell:
No.

Jono Hey:
B stands for Benoit B.

Mandelbrot.

That's my only fractal joke.

Rob Bell:
Do you know what the B stands for?

Lovely.

Listen, thank you, chaps.

And thank you all for listening.

And just remember, as you move forward in life, your results are only as good as your method.

Is that it?

See you next time.

Go well.

Stay well.

Goodbye.

All music on this podcast series is provided by the very talented Franc Cinelli.

And you can find many more tracks at franccinelli.com.