The Coastline Paradox


What country has the longest coastline in the world?

If you answered Canada, peel off a gold star and stick it to your shirt. Great work!

OK, now what country has the second-longest coastline? Russia? The United States? Australia? Maybe you figure the question is a trick and answered something along the lines of Indonesia? All wrong.

Number two on the list of longest coastlines is Norway! How is that possible? Check out this comparison between Norway and the United States:

The size of Norway versus the United States

How can a country whose overall size doesn’t even match the eastern seaboard of the United States have a longer coastline? You’ve got the west coast and the entirety of Alaska to add onto the U.S. tally.

The situation becomes even stranger.

Let’s look at the table of countries by longest coastline.

Countries ranked by the length of their coastlines - Wikipedia

You see Norway is, indeed, in the second spot. You might also notice that there are two sources cited: The World Factbook and World Resources Institute. Now, look at the figures just for Canada. The World Factbook shows the Canadian coast to measure 202,080 kilometers; World Resources Institute places the figure at 265,523 kilometers.

63,000 kilometers is a big spread. That’s 39,000 miles. The circumference of the Earth is just under 25,000 miles! A difference of this magnitude certainly cannot be the result of improper measurement. Gaze at the difference the two organizations give to the coast of the United States: 19,924 km and 133,312 km!

What in the world is going on here?

Lewis Fry Richardson was the first dude to notice something was awry

English mathematician and pacifist Lewis Fry Richardson theorized that the lengths of borders between countries might have something to do with the frequency of war between those nations. While collecting historical data for his idea, Richardson noticed a curiosity.

The established border lengths between countries varied! Spain and Portugal did not agree on the breadth of their border. One said 987 kilometers and the other 1,214 kilometers. The Netherlands and Belgium also provided differing opinions: 380 km and 449 km.

The ultimate reason for this seemingly illogical divergence is the Coastline Paradox.

Approximating the distance of a curve with straight measurments

Measuring something that is straight is no problem. Grab your yardstick and go to town. The issue becomes murkier when you add curves into the mix. Not to mention when you’re measuring on the surface of a sphere, such as our lovely planet.

Using straight lines, one can approximate the length of a curved section (see the image above). As you can see, the estimate will be lower than the actual length. If one employed smaller straight lines one could “hug” the curve more accurately and your estimate will inch closer to the actual length of the line. The smaller your measuring stick, the closer you’ll get to the true length of the curve. Calculus can give a very precise value of the curve, as it uses infinitesimally small distances in the calculations.

Richardson had stumbled upon the fact that different people or organizations were using different “rulers” to measure non-straight borders. The phenomenon is easy to visualize if you apply it to the coastline of the United Kingdom.

Estimating the length of the coastline with 200km sticks, 100 km sticks, and 50 km sticks

The image on left is measured with the biggest “stick.” The accuracy and the length both increase as the stick is shortened in the middle image. More accurate yet and longer still is the right image, which is measured with the shortest stick. The smaller the stick, the more accurate and longer the coastline.

Based on the curved instance above, you might believe that these smaller sticks and more accurate measurements might converge on a specific, finite value for a coastline. But Richardson demonstrated this expected outcome does not actually occur.

In what is now known as the Richardson effect, the measured length of coastlines and other natural borders increases without limit as the measurement device becomes smaller.

The first four iterations of the Koch Snowflake, a fractal figure

Did I just intimate that the length of all coastlines is essentially infinite?

Yes!

For a while, Richardson’s breakthrough mathematics were ignored as a trifle. Eventually, Benoit Mandlebrot realized that coastlines function more like fractals than regular Euclidian shapes. A fractal is a curve whose shape retains the same general pattern no matter how many times it is magnified. Zoom in on a fractal and you’ll see the same “shape” over and over. Though Mandlebrot had not yet created the term fractal, he nonetheless demonstrated that coastlines essentially behave like them because, even at an atomic level, their folds are random due to the constantly ethereal nature of their “edges.” Water is always remaking the boundaries of a coastline.

Fractals have infinite length; coastlines, unintuitively, have no finite length that can be measured. Though it boggles the mind, technically, we cannot know how long a nation’s borders are! Though odd, this reality explains why scientists measured Canada’s coastline so differently (they used different measurement sizes) and why Norway, despite being much smaller than other nations, has the second-longest ocean border. Zoom in on Norway and you’ll see folds and angles on those coasts that seem to go on to infinity (and they do!).

If this deal still seems confusing to you, try watching this video, which does a good job of showing the Coastline Paradox with visuals.

The philosophical implications of this mathematical oddity that bleeds into the real world are widespread. We can say that all coastlines are technically infinite in length, yet we still have a definitive ranking of the coastlines of the world’s nations. On a practical scale, we know that Canada has more coast than Panama. How much more? It’s literally impossible to say. If you use the same measuring stick for both, you could get a good approximation. But we know that the precision of the measurement increases with the shortness of the stick. At some point, though, it becomes impractical to use a measurement small enough to be extremely accurate.

Are these fractal coasts really infinite? Can they really be infinite on our finite planet? You tell me.

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