4 ms·

Do rivers actually have a well-defined length? I know coastlines do not, and rivers seem similar. https://en.wikipedia.org/wiki/Coastline_paradox https://en.wi

by oofabz 11y ago

Do rivers actually have a well-defined length? I know coastlines do not, and rivers seem similar. https://en.wikipedia.org/wiki/Coastline_paradox https://en.wikipedia.org/wiki/Coastline_paradox

placeybordeaux 11y ago
The complexities there don't seem to have a correlary to the rivers.
unholiness 11y ago
Yes. With a coastline, the closer you measure, the more length-increasing features you observe, causing the measured length to diverge. Rivers have a finite width, so once the distance between your measurements is smaller than the width of the river, the measured length will converge to the true length.
- pmalynin 11y ago
  Indeed, one would reach the plank length -- and hence the distance simply cannot be infinite.
  - thedufer 11y ago
    The planck length is simply the base length when you set up your units such that c, G, and h are all 1. There's currently no reason to believe it has any sort of physical meaning beyond that.
    - iaw 11y ago
      I think wikipedia disagrees with your assessment: "According to the generalized uncertainty principle (a concept from speculative models of quantum gravity), the Planck length is, in principle, within a factor of 10, the shortest measurable length – and no theoretically known improvement in measurement instruments could change that."[1] [1] https://en.wikipedia.org/wiki/Planck_length https://en.wikipedia.org/wiki/Planck_length
      - thedufer 11y ago
        "within a factor of 10" is a curiosity rather than anything of real significance. And even that is based on untested theories. It is possible that the Planck length will end up being significant in the way described upthread, but given our scientific record it seems somewhat more likely that such an idea will end up looking similar to the theory about atoms being the smallest unit of matter.
Strilanc 11y ago
Suppose you expanded every point of a coastline into a small disk (i.e. take the minkowski sum [1] of the coastline and a disk). Is the perimeter of the resulting shape still infinite? The coastline had infinite perimeter because as you zoomed it got jankier and jankier. But once the janks get smaller than the disk, they start being hidden by disks from the surrounding points. You can't jump in and out anymore on tiny scales, because you run into the adjacent disks. This smooths out the noise and things converge instead of diverging. ... I think. A river's length would smooth out for essentially the same reason. The set of points equidistant from both sides of the river is smoothed out compared to the sides because points on the side that are closer to the center of the river will hide jank from the nearby further points. That's my intuition anyways. Not sure how it actually plays out. Edit And if you go by "shortest path within the river from source to sink" as the length then it's definitely finite. 1: https://en.wikipedia.org/wiki/Minkowski_addition https://en.wikipedia.org/wiki/Minkowski_addition
zamalek 11y ago
The coastline paradox is a matter of accuracy (significant digits). 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 > ... If you have to "zoom in" to see a length feature it means that the length feature is small and therefore the contribution to the overall length that these features provide diminishes. Something like a sigmoid[1], it would approach a limit (except in the case of some fractals, but not coastlines) - it can increase infinitely but at tinier and tinier increments. Eventually you reach the size of atoms and you are now talking about fractals and not real-life coastlines. While it would technically apply to rivers as well, the website seems to be using a single significant digit. The above example becomes: 100km > 120km > 128km > 129.5km > 129.5km > 129.5km > 129.6km > ... So we can accurately measure a coast/river length up to a specific significant digit. [1]: https://upload.wikimedia.org/wikipedia/commons/5/55/Sigmoid_curve_for_an_autocatalytical_reaction.jpg https://upload.wikimedia.org/wikipedia/commons/5/55/Sigmoid_...
- 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.

by oofabz 11y ago

Do rivers actually have a well-defined length? I know coastlines do not, and rivers seem similar. https://en.wikipedia.org/wiki/Coastline_paradox https://en.wikipedia.org/wiki/Coastline_paradox

placeybordeaux 11y ago

The complexities there don't seem to have a correlary to the rivers.

unholiness 11y ago

Yes. With a coastline, the closer you measure, the more length-increasing features you observe, causing the measured length to diverge. Rivers have a finite width, so once the distance between your measurements is smaller than the width of the river, the measured length will converge to the true length.

pmalynin 11y ago
Indeed, one would reach the plank length -- and hence the distance simply cannot be infinite.
- thedufer 11y ago
  The planck length is simply the base length when you set up your units such that c, G, and h are all 1. There's currently no reason to believe it has any sort of physical meaning beyond that.
  - iaw 11y ago
    I think wikipedia disagrees with your assessment: "According to the generalized uncertainty principle (a concept from speculative models of quantum gravity), the Planck length is, in principle, within a factor of 10, the shortest measurable length – and no theoretically known improvement in measurement instruments could change that."[1] [1] https://en.wikipedia.org/wiki/Planck_length https://en.wikipedia.org/wiki/Planck_length
    - thedufer 11y ago
      "within a factor of 10" is a curiosity rather than anything of real significance. And even that is based on untested theories. It is possible that the Planck length will end up being significant in the way described upthread, but given our scientific record it seems somewhat more likely that such an idea will end up looking similar to the theory about atoms being the smallest unit of matter.

Strilanc 11y ago

Suppose you expanded every point of a coastline into a small disk (i.e. take the minkowski sum [1] of the coastline and a disk). Is the perimeter of the resulting shape still infinite? The coastline had infinite perimeter because as you zoomed it got jankier and jankier. But once the janks get smaller than the disk, they start being hidden by disks from the surrounding points. You can't jump in and out anymore on tiny scales, because you run into the adjacent disks. This smooths out the noise and things converge instead of diverging. ... I think. A river's length would smooth out for essentially the same reason. The set of points equidistant from both sides of the river is smoothed out compared to the sides because points on the side that are closer to the center of the river will hide jank from the nearby further points. That's my intuition anyways. Not sure how it actually plays out. Edit And if you go by "shortest path within the river from source to sink" as the length then it's definitely finite. 1: https://en.wikipedia.org/wiki/Minkowski_addition https://en.wikipedia.org/wiki/Minkowski_addition

zamalek 11y ago

The coastline paradox is a matter of accuracy (significant digits). 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 > ... If you have to "zoom in" to see a length feature it means that the length feature is small and therefore the contribution to the overall length that these features provide diminishes. Something like a sigmoid[1], it would approach a limit (except in the case of some fractals, but not coastlines) - it can increase infinitely but at tinier and tinier increments. Eventually you reach the size of atoms and you are now talking about fractals and not real-life coastlines. While it would technically apply to rivers as well, the website seems to be using a single significant digit. The above example becomes: 100km > 120km > 128km > 129.5km > 129.5km > 129.5km > 129.6km > ... So we can accurately measure a coast/river length up to a specific significant digit. [1]: https://upload.wikimedia.org/wikipedia/commons/5/55/Sigmoid_curve_for_an_autocatalytical_reaction.jpg https://upload.wikimedia.org/wikipedia/commons/5/55/Sigmoid_...

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.