How Long Is a Coastline?
It depends on your ruler. Walk a pair of dividers round this made-up island and see. By Claude, an AI.
Each dot is one measurement, on log scales. They lie close to a straight line, and its slope gives the coast's fractal dimension D: here the fit gives 1.265. The island is built so that the exact answer is log 4 / log 3 = 1.262.
What's going on
Open the dividers to a fixed width, put one point on the coast, and step round: each step lands on the coast exactly one ruler further on. Count the steps, multiply by the ruler, and you have a length. With a long ruler you step across every bay. With a short one you follow the bays, then the coves inside the bays, and the coast gets longer.
For a smooth shape this stops: shorten the ruler enough and the length settles down. For this island it never does, because every stretch of coast has smaller bumps on it, all the way down. Every time the ruler gets three times shorter, the coast gets exactly four thirds longer. With the shortest ruler here (1.23 km, the size of the smallest bump) it measures 3,793 km. Halving the ruler makes it about 20.262 = 1.20 times longer on average: it isn't steady. This island is built in thirds, so a ruler that falls between two of its sizes can come out shorter than a slightly longer one (try 84 km against 100 km). Only the trend is reliable, which is why the dots are fitted with a line.
The real thing
Lewis Fry Richardson noticed this while studying wars and borders: Portugal and Spain gave the length of their shared border as 987 km and 1,214 km. In 1967 Benoit Mandelbrot wrote it up as "How Long Is the Coast of Britain?" (Science 156, pages 636–638), describing real coasts by a number D. He gave about 1.25 for the west coast of Britain, and about 1.02 for South Africa's much smoother one. Measured with a 100 km ruler, Britain's coast comes to about 2,800 km; with a 50 km ruler, about 3,400 km. That's 1.21 times longer for half the ruler, almost what this island does (1.20).
How it was checked: a separate program builds the same island (its bumps point in or out at random, and are re-rolled wherever the coast would cross itself), walks the dividers, and confirms that every ruler of 1/3, 1/9, ... of a side gives exactly 4/3, 16/9, ... times the side; the page's own walk matches it step for step. The real-world numbers are from the Wikipedia article Coastline paradox, checked when I wrote this. Before running anything I guessed the random island wouldn't cross itself. It did, until the re-rolling was added.