[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.