BrightKidz Hub
The same coastline measured with a long stick and with a short one The identical wiggly coastline is drawn twice, side by side. Below each shoreline the land is shaded, so the wiggle is plainly the edge between land and sea rather than a ridge. On the left, four long straight sticks have been laid end to end along the coast, and they cut straight across every bay so most of the wiggle is skipped and the land pokes out above them. On the right the same coast has been walked with sixteen short sticks, which follow it into the bays and round the headlands, so the chain is far longer even though the coast beneath it is exactly the same shape.

A Coastline Has No Length

About 14 minutes

Spain and Portugal share a border. Both countries know exactly where it runs — there is no argument about that. Both have measured it.

Spain says it is about 987 kilometres. Portugal says it is about 1,214 kilometres.

That is a gap of 227 kilometres. Not a rounding error, and not a disagreement about where the line runs — roughly a fifth of the whole border, present in one answer and missing from the other.

Neither of them got it wrong. The question itself has no single answer, and working out why turns out to be one of the strangest things in geography.

It depends on your measuring stick

Imagine walking a coast with a rigid stick 100 kilometres long. You plant one end, swing the other round until it touches the coast again, and count. Simple.

But a 100-kilometre stick cannot get into a bay 20 kilometres wide. It steps straight over it, and every one of those bays is left out of your total.

So try a 10-kilometre stick. Now you walk into the big bays and round the far side of them — and your total goes up, because all that bay coast has joined in. But a 10-kilometre stick steps over the coves inside the bays.

Try one metre. Now you go round the coves, and round the headlands inside the coves, and your total goes up again.

One coastline measured with shorter and shorter sticks A faint wiggly coastline runs across the left of the picture with the land below it lightly shaded. Over it, a chain of straight sticks is laid end to end with a dot at every join. At first there is a single long stick that cuts straight across the whole coast. Each time the detail is increased the sticks are halved in length, so there are twice as many of them and they follow the coast into smaller and smaller bays. A table on the right fills in as this happens, giving the length of stick used and the total it produces, both in kilometres. The totals only ever go up, and they show no sign of settling on any particular number. stick total 400 400 216 431 116 462 64 508 35 559 the coast underneath never changes Halve the stick and the answer goes up again. Kilometres.

One coast, five sticks. Shorten the stick and watch the number.

The shape being measured never changed. Only the stick did.

This is not how measuring is supposed to behave

Measure a table with a metre stick and you get 1.4 metres. Measure it again with a ruler marked in millimetres and you get 1.423. Measure it with something better still and you get 1.4231.

The answer is settling down. Each better instrument nudges the number a little and the nudges get smaller. There is clearly a real length there, and you are closing in on it.

Coastlines refuse to do that. Halve the stick and the total jumps up again — and it keeps jumping, by roughly as much, however small the stick gets. There is no number it is heading for. Go all the way down to measuring round individual pebbles and grains of sand and the total climbs towards something absurd.

The coast is not being measured badly. It genuinely has no single length waiting to be found.

Because a coast looks the same at every size

Here is the reason, and it is easier to see than to say.

Look at a photograph of a coastline taken from space. Bays, headlands, a wriggling edge. Now a photograph from an aeroplane of one of those bays. Bays, headlands, a wriggling edge. Now a photograph of one metre of rock taken while standing on it. Bays, headlands, a wriggling edge.

Three views of a coast at wildly different scales Three plain boxes stand side by side, and each holds a wiggly line with the land below it shaded, standing for the edge of a coast. All three lines have the same character: long smooth stretches broken by bays, with smaller bays inside the bays and smaller notches inside those. Nothing in any of the boxes gives its size away, so there is no way to work out from the picture which one covers hundreds of kilometres, which one covers a single beach, and which one covers a stretch of rock you could step over. Five hundred kilometres, five kilometres, five metres. One of these is each. Nothing in the picture tells you which. That is exactly why the measuring never settles down.
Three pictures of the same coast at wildly different scales. With nothing in them for size, there is no way to tell which is which.

Without a boat or a person in the frame you cannot tell them apart, and that is the whole point. A coast has the same kind of wriggle at every size, because the things that make coasts — waves, weather, cracks splitting rock — work at every size. Waves carve metre-wide notches for the same reason storms carve kilometre-wide bays.

A shape that keeps showing you the same pattern however far in you zoom is called a fractal. Coastlines are the classic example, and so are river networks, clouds, mountains and the branching of a tree.

You measure a coast with a 10-kilometre stick, then again with a 1-kilometre stick. What happens to the total?

So what should an atlas print?

The honest answer is a length and the stick that produced it. Any published coastline figure really means "measured at this level of detail", and two books can print different numbers for the same country without either being wrong.

That is not a failure of measuring. It is a fact about the thing being measured, and knowing it is more useful than any of the numbers.