Beauty is as relative as light and dark. Thus, there exists no beautiful woman, none at all, because you are never certain that a still far more beautiful woman will not appear and completely shame the supposed beauty of the first. - Paul Klee, diary, 1910
(And yet, there is light, dark, and beautiful women..)
> 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.
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.
https://en.wikipedia.org/wiki/Coastline_paradox
Most maps fairly accurately measure the coastline of a place. Their granularity of topographical measurement is good enough.
In other words, it's not if you understand a concept but how deeply. Also, it's hard to sit down and list all the complexities of concepts. You could be understanding a lot more complexity than you realize.