[jmkerr]

Sorry couldn't resist, from wikipedia:

> if one were to measure a coastline with infinite or near-infinite resolution, the length of the infinitely short kinks in the coastline would add up to infinity.

No they wouldn't. The sum would converge. What a bad article.

[jjgreen]

Well the argument does break down as you get to the atomic level, but the observation is morally correct; the idea (and the divergent sum) is easiest to see with the Koch snowflake (which has an infinite perimeter)

https://en.wikipedia.org/wiki/Koch_snowflake

[jmkerr]

There is some self-similarity going on (Mandelbrot discusses scales from 1000 km to 10 km) but rocks or coastal sections with the fractal dimension of Great Britain on a scale of 1m or 10m are a rare exception.

I'm not an expert, but I think it's because as the length scale gets smaller, coastal erosion dominates the coastline, as opposed to an older force producing larger features, like glacial erosion or even plate tectonics. No reason to drag quantum mechanics into this discussion, but the circumference of an atom can be defined, so that's not a problem.

An actual Koch snowflake you can buy will always have a measurable circumference.

E: Some of this misunderstanding is actually due to Mandelbrot himself, he wrote in the introduction to [How long is the Coast of Britain?, 1967]:

> Geographical curves are so involved in their detail that their lengths are often infinite or, rather, undefinable.

No, they're not any more infinite than a toothbrush.

[orwin]

In reality, yes. If you see the coast as a mathematical object - a fractal - i think it doesn't converge.

I only studied topology for one year, so i might be very wrong on that one.

[pengstrom]

Not necessarily if the coast is fractal, no?