4 ms·

I think it's a bit like the coastline paradox: https://en.wikipedia.org/wiki/Coastline_paradox https://en.wikipedia.org/wiki/Coastline_paradox Most maps fairl

I think it's a bit like the coastline paradox: https://en.wikipedia.org/wiki/Coastline_paradox 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.

yesenadam 6y ago
Yeah...What one person considers deep understanding about something, might seem to another the most basic, superficial knowledge. It's all relative. It's like: 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..)
jmkerr 6y ago
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.
- pengstrom 6y ago
  Not necessarily if the coast is fractal, no?
- jjgreen 6y ago
  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 https://en.wikipedia.org/wiki/Koch_snowflake
  - jmkerr 6y ago
    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 6y ago
  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.

yesenadam 6y ago

Yeah...What one person considers deep understanding about something, might seem to another the most basic, superficial knowledge. It's all relative. It's like: 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..)

jmkerr 6y ago

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.

pengstrom 6y ago
Not necessarily if the coast is fractal, no?
jjgreen 6y ago
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 https://en.wikipedia.org/wiki/Koch_snowflake
- jmkerr 6y ago
  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 6y ago
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.