Try measuring the coastline of the United States, and it’s almost guaranteed you’ll find a different answer than anyone before you. Even official sources like the Congressional Research Institute, the CIA, and NOAA came up with wildly different answers (29,093 miles, 19,924 miles, and 95,471 miles, respectively). How could their measurements be so different? Meet the Coastline Paradox.
As explained in this video from RealLifeLore, the Coastline Paradox has been vexing researchers and cartographers since its discovery by mathematician Lewis Fry Richardson in 1951.
The explanation for the paradox is surprisingly simple: unlike human-drawn geometrical shapes, a coastline is full of nooks and crannies made by nature. The more one zooms in on the coastline, the more these inconsistencies multiply.
This means that the length of a coastline is completely dependent on what size of measurement unit you use to study it. For example, the coastline of the UK is only 2,800 kilometers long when measured in lengths of 100 kilometers. Shrink that to 50 kilometer measurements and suddenly the coastline is 3,400 kilometers.
The Mandelbrot Set, a fractal likely familiar to you from the front of a math textbook. (Photo: Wolfgang Beyer/CC BY-SA 3.0)
Coastlines are like fractals–the further you zoom in, the more complex it gets (famed fractal researcher Benoit Mandelbrot expanded Richardson’s work on the paradox). If you were to try to measure a coastline on an atomic level, the length would approach infinity.
This does not bode well for the practical needs of nations and scientists. It also explains some of the bizarre measurements of coastlines that can be found around the internet. The CIA, for example, lists Norway as the second longest coastline in the world after Canada.
That’s absurd–Canada is 3.9 million square miles, while Norway is only 148,728 square miles. But if you look at the coast of Norway, it’s full of wild inconsistencies, fjords, and more. Despite our best efforts, this seemingly simple problem remains impossible to nail down.
Considered abstractly, the coastline paradox is an awe-inspiring thing, proof of the impossibility of pinning everything down, and of the essential irreducibility of our world. On a practical level, though, it’s a huge pain, says Branden Rishel, a cartographer who works on shoreline restoration in Puget Sound. Like much of Washington State, Puget Sound is full of fjords, endless nooks and crannies carved into the land by glaciers 15,000 years ago.
Rishel expects ShoreZone to be updated soon with LiDAR data, which replaces cameras and human eyes with more precise laser measurements, taken from planes. Still, it’s not going to be perfect. “Over the month or two it took to fly, even the best LiDAR would capture a blurred snapshot of the Sound,” he says. “Winter beach profiles are drastically different from summer ones. Bluffs keep eroding. Landslides happen.”
That’s what happens when we try to work with something that’s literally infinite—we just have to do the best we can.