But if you shrink the “yardstick” down to an infinitesimally small size, the length, effectively, becomes infinite… and it’s the same for all coastlines. They’re all infinitely long.
Limits can resolve to infinity. The coastline paradox is just the observation that the (semi-reasonable) assumption that landmasses are fractal shaped implies the coastline tends towards infinity with smaller yardsticks.
You can’t shrink the yardstick down to an infinitesimal size.
Coastlines are not well defined. They change in time with tides and waves. And even if you take a picture and try to measure that, you still have to decide at what point exactly the sea ends and the land starts.
If the criteria for that is “the line is where it would make a fractal” then sure, by that arbitrary decision, it is infinite. However, a way better way to answer the question “where is the line” is to just decide on a fixed resolution (or variable if you want to get fancy), which makes the distinction between sea and land clearer.
It is like saying that an electron is everywhere in the universe, because of Heisenberg’s uncertainty principle. While it is very technically true, just pick a resolution of 1mm^3 and you know exactly where the electron is.
But if you shrink the “yardstick” down to an infinitesimally small size, the length, effectively, becomes infinite… and it’s the same for all coastlines. They’re all infinitely long.
… but some are longer than others. ;)
Literally no. Very hard to measure, but strictly still a finite length. Limits and all that jazz.
Limits can resolve to infinity. The coastline paradox is just the observation that the (semi-reasonable) assumption that landmasses are fractal shaped implies the coastline tends towards infinity with smaller yardsticks.
They can… I wasn’t saying they couldn’t… I meant that as to point to the logic you’d use to prove it finite
My bad for the poor wording though.
Max Planck says no…
You can’t shrink the yardstick down to an infinitesimal size.
Coastlines are not well defined. They change in time with tides and waves. And even if you take a picture and try to measure that, you still have to decide at what point exactly the sea ends and the land starts.
If the criteria for that is “the line is where it would make a fractal” then sure, by that arbitrary decision, it is infinite. However, a way better way to answer the question “where is the line” is to just decide on a fixed resolution (or variable if you want to get fancy), which makes the distinction between sea and land clearer.
It is like saying that an electron is everywhere in the universe, because of Heisenberg’s uncertainty principle. While it is very technically true, just pick a resolution of 1mm^3 and you know exactly where the electron is.
Didn’t calculus solve this stuff?
Surely the distance approaches some finite value.