# How long is the coastline of Britain?

## Main Question or Discussion Point

If you take a map and try to measure the length of the coastline you will obtain a value, however if you get a "better" map with a bigger scale ratio and repeat the process the length you obtain will be longer because you see more intricate details that were ignored before. Apparently you can repeat this process indefinitely with bigger and bigger maps and the value will always get larger and larger and it will never choke off!

So does that mean that the coast line is infinitely long? That seems absurd to me, there must be some physical limit to this.

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russ_watters
Mentor
Maps are scaled, but the coastline has a real physical size. So the longest value you could get would be if your map had a 1:1 scale.

Mathematically speaking - if you presuppose that the nature of the coastline is that it's a completely irregular shape - then yes, its length is mathematically infinite.

Consider a simpler example - a squiggly curve; drawn randomly. You assume that the drawing of the curve is a representation of a true mathematical object that maintains its nature as a defined, but irregular, shape - to an infinitely small length scale. Basically, your concept of the curve as a perfect 'line' IS that curve, in this abstraction.

Now consider that you can't directly measure its length - because the curve is not an object that can be defined by geometry - but you can approximate its length by lying lots of little straight lines between points on the curve and adding them all together. But you want a better estimate because you can see there are parts that don't fit the lines - so you halve the size of the straight lines you're using. But then, there are still bits outside the straight lines, and so on. Every time you decide to use smaller straight lines, the number you use increases by slightly more than 2 - because you're fitting more straight lines into smaller spaces.

Do this indefinitely, because your curve is a perfect object - where do you stop? What's the 'right' point where you can say enough, these lines are small enough?

There isn't one. So you continue to use smaller and smaller lines and each time the sum of lengths of these little lines increases a little bit - forever.

Hence the length of your curve is infinite.

D H
Staff Emeritus
Mandelbrot never claimed the coastline of Britain is infinite in his seminal 1967 paper, "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". In the first part of the paper, he noted that Britain's coastline exhibits self-similarity over a wide range of measurement scales. That does not mean Britain's coastline is infinite in length, as that would require that the coastline exhibit self-similarity over all measurement scales. This is impossible because matter is quantized.

In the second part of the paper, he investigated mathematical shapes that have a finite area but an infinite perimeter, such as the Koch snowflake. Mathematical shapes are not limited by the quantum nature of reality.

Maps are scaled, but the coastline has a real physical size. So the longest value you could get would be if your map had a 1:1 scale.
But even then if you shortened your ruler length you would get a different value.

Mandelbrot never claimed the coastline of Britain is infinite in his seminal 1967 paper, "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". In the first part of the paper, he noted that Britain's coastline exhibits self-similarity over a wide range of measurement scales. That does not mean Britain's coastline is infinite in length, as that would require that the coastline exhibit self-similarity over all measurement scales. This is impossible because matter is quantized.
Okay, I never read that paper, I might look for it. It seems obvious that there will be some kind of lower limit where the scale invariance breaks down, so that makes sense. The thing that confuses me is that I would have thought that the measurements would begin to converge as your ruler length got shorter and shorter, but they don't. So what happens? Does the fractal dimension just change suddenly and beyond that point you can't get a more accurate value?

I just measured the coastline on a map, I found that the fractal dimension was about 0.2. Does this mean that it will choke off? My lecturer said that if D<1 it will choke off.

But I also found that if you didn't convert the number of steps into an apparent length the fractal dimension is exactly 1 higher, such that D=1.2. That would imply that the reading would never choke off, but there's no change in the methodology so they can't both be right? I'm really confused now!!!

D H
Staff Emeritus
You are measuring length, which has dimensionality of 1. A non-fractal curve such as a circle has Hausdorff dimension 1. I don't know how you got 0.2; that is wrong. The value of 1.2 is right on the mark.

Well basically I just used a pair of dividers and just counted how many times it took to get around the coastline. When I plotted the 'count' number against the ruler length on a log-log graph, I found D=1.2. But then I thought it might be better to convert the 'count' into an actual length so I multiplied out by the ruler length; when I did a log-log plot of actual length against ruler length I found D=0.2.

You can't have a continuous line with a fractal dimension of less than one, except as a mathematical curiosity - this would imply that the total length of the line on average is less than the distance between arbitrary start and end points. This is clearly impossible without weird tricks of topology.

russ_watters
Mentor
But even then if you shortened your ruler length you would get a different value.
You mean if you chaged the definition of a "meter"? Sure, but that's just units of length. It isn't length itself.

D H
Staff Emeritus
You mean if you chaged the definition of a "meter"? Sure, but that's just units of length. It isn't length itself.
Think of the "ruler" as just that: a ruler (which doesn't bend). If you use a ten kilometer-long ruler to measure the length of the coastline of Britain, you will only see very coarse variations in the coastline. Your ruler will simply skip over a lot of rivers, inlets, bays, and peninsulas. Changing to smaller and smaller rulers lets you see ever finer details. The variations in the coastline that large ruler misses but can be seen by a smaller ruler will add a lot of length to the coastline length measure.

Britain's coastline exhibits something called self-similarity. Big bays and peninsulas have mid-sized bays and peninsulas in them, which in turn contain many small bays and peninsulas, which in turn contain many tiny bays and peninsulas. The length of the coastline depends on the length of the ruler and can be approximated by

$$L(r) = M r^{(1-D)}$$

where $L(r)$ is the length of the coastline as measured by a ruler of length $r$, $M$ is some constant, and $D$ is the Hausdorff dimension of the coastline. The coastline of west Britain has a Hausdorff dimension of 1.25. The coastline measured by a 1 kilometer rule is 1.78 times the length measured by a ten kilometer ruler. A 100 meter ruler yields a coastline length that is 1.78 times the length yielded by a one kilometer ruler.

This suggests that the length of Britain's coastline is infinite! This assumes that Britain's coastline exhibits self-similarity at all scales. At some point, however, the self-similarity breaks down. Nature is quantized, after all. Mathematics is not constrained by nature. It is easy to construct a mathematical shape that has an infinite perimeter but a finite area. You will get a finite perimeter length if you use a rigid ruler with some non-infinitesimal length to measure the perimeter of such an object. Changing to a smaller ruler will yield a larger value, and this growth continues without converging to any finite value as you make the ruler ever smaller.

Thanks D_H for clearing that up, I must say you've been a great help! I also figured out why I was getting the value of one less, it was basic error where I was converting Nn into Pn, that is I was converting the 'count' into the perimeter. I found some equations:

D=1+ln(Pn+1/Pn)/ln(rn/rn+1)=ln(Nn+1/Nn)/ln(rn/rn+1)

Where: D=fractal dimension; rn=ruler length

So when I corrected for size I was really just converting Nn into Pn by the relation:

Pn=Nn.rn

and not realizing that I had to add the one on again to find D.

russ_watters
Mentor
Think of the "ruler" as just that: a ruler (which doesn't bend). If you use a ten kilometer-long ruler to measure the length of the coastline of Britain, you will only see very coarse variations in the coastline. Your ruler will simply skip over a lot of rivers, inlets, bays, and peninsulas.
Oh, ok, misunderstood. That just makes the answer the same as before, then - whether you scale the map or scale the ruler to take more precise measurements, the coastline of Britain has a finite length that you'll get closer and closer to in measurement precision. It isn't a fractal.

D H
Staff Emeritus