3 ms·

> As you "zoom in" on the coast this is what happens to the length: 100km > 120km > 128km > 129.5km > 129.52km > 129.528km > 129.6281km > ... I don't think tha

by Osmium 11y ago

> As you "zoom in" on the coast this is what happens to the length: 100km > 120km > 128km > 129.5km > 129.52km > 129.528km > 129.6281km > ... I don't think that's true. In real life this series does not converge unless you specify a minimum feature size, which you hinted at. This isn't just a mathematical curiosity; real life coastlines are fractal and have no well-defined length, as the parent poster's link explains. [Edited wording.]

zamalek 11y ago
> To measure a coastline you have to specify a minimum feature size. Not coastlines: only fractals. The name of the paradox is really unfortunate because it doesn't apply the coastlines "all the way down." You can't keep subdividing a coastline because eventually you start working with curves meaning that you can use calculus and meaning that you can work out limits. The actual issue with real-life coastline (and river) length is that it's continuously changing. [Edit to your edit]: yep. However the important result is that we can actually arrive at a length for a river, regardless of the mathematical thought experiment.
- Osmium 11y ago
  Agreed that the fact it's continuously changing is an additional issue, but I don't see the "eventually you start working with curves" argument. Where are these curves on a typical coastline? If you're measuring the coastline on a map, that argument holds, but the map is only one representation of the reality, and they've already made decisions regarding minimum feature size implicit in the construction of the map. But yes, also agreed that measuring rivers should be fine, because you can represent it as a one-dimensional line along the 'centre of mass' of each segment of the river, which should give well-defined values regardless of what the 'edges' of the river look like.
  - zamalek 11y ago
    > Where are these curves on a typical coastline? Between atom nuclei.
    - Osmium 11y ago
      How so? Do you fit the positions of your nuclei to a spline to get your curve? How about quantum uncertainty in its position? How do you assign whether a given nuclei belongs to the 'coast' or to the 'sea', etc. I don't think it holds.
      - zamalek 11y ago
        > How about quantum uncertainty in its position? Aw crap, really good point. You're right.

by Osmium 11y ago

zamalek 11y ago

> To measure a coastline you have to specify a minimum feature size. Not coastlines: only fractals. The name of the paradox is really unfortunate because it doesn't apply the coastlines "all the way down." You can't keep subdividing a coastline because eventually you start working with curves meaning that you can use calculus and meaning that you can work out limits. The actual issue with real-life coastline (and river) length is that it's continuously changing. [Edit to your edit]: yep. However the important result is that we can actually arrive at a length for a river, regardless of the mathematical thought experiment.

Osmium 11y ago
Agreed that the fact it's continuously changing is an additional issue, but I don't see the "eventually you start working with curves" argument. Where are these curves on a typical coastline? If you're measuring the coastline on a map, that argument holds, but the map is only one representation of the reality, and they've already made decisions regarding minimum feature size implicit in the construction of the map. But yes, also agreed that measuring rivers should be fine, because you can represent it as a one-dimensional line along the 'centre of mass' of each segment of the river, which should give well-defined values regardless of what the 'edges' of the river look like.
- zamalek 11y ago
  > Where are these curves on a typical coastline? Between atom nuclei.
  - Osmium 11y ago
    How so? Do you fit the positions of your nuclei to a spline to get your curve? How about quantum uncertainty in its position? How do you assign whether a given nuclei belongs to the 'coast' or to the 'sea', etc. I don't think it holds.
    - zamalek 11y ago
      > How about quantum uncertainty in its position? Aw crap, really good point. You're right.