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:
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.
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.
Why on earth would the rivers of the world have an average sinuosity of pi? Rivers are super dynamic and are effectively a side effect of localized water cycles and geology. This seems like Music of the Spheres... Aka looking for harmony in a chaotic universe.
An explanation of this is put forth in both the video and paper linked to in the opening paragraph. The principle is that bends in rivers tend to grow as erosion happens on the outside of the bend and soil deposition on the inside. This increases sinuosity until the point at which the bend comes full circle, forms an oxbox lake, and returns the local region of the river to a straight line with sinuosity of 1. The value of pi is supposed to come out when you consider all of a river's curves and wiggles on all length scales.
The right answer may not be pi, but the data shown make a compelling case that rivers do tend to some average value.
As a sidenote, there are active human efforts to keep certain rivers, like the Mississippi, from meandering too far from their current locations. I don't know how many of the world's rivers have such efforts being applied to them, but it's not unreasonable to think that this could have some effect.
They could tend towards that value though, if it is somehow more efficient in the general case, with variation being due to local conditions which cancel out if you consider a large enough data set of locations.
Or I could just be babbling and made that sentence up off the top of my head!
I suspect there is an error in the data. The project lists the Lukuga River as being 1904 km long. But Wikipedia[1] only has 320 km, which would give it a much more ordinary sinuosity of 1.27.
For a river that might actually have a very high sinuosity, take a look at the Fraser River -- #20 on the list, with sinuosity listed as 5.12.
What's going on with the HN title font? That square-with-the-bottom-missing character seems to be the proper codepoint for lowercase-pi, but it's definitely a not a recognizable rendering of lowercase-pi. Even in sans serif, the top bar should extend past the corners on the left and right.
Great project! Seems like one could scrape the data from Wikipedia. I entered a few rivers by just copying data out of the Wikipedia side bars for them. Any reason that wasn't done?
tl;dw: when rivers are very sinuous, the kinks turn into oxbow lakes and separate from the river, so that limits how sinuous they can get. On the other hand, rivers' bends are constantly amplified by erosion. Researchers modeling these phenomena showed that under certain assumptions, these forces are in equilibrium at a sinuosity of ᴨ.
This is somewhat pedantic, but the average anything of anything can never be equal to π; it's an irrational number, and averages (arithmetic means) are ratios.
What should be said is that the average might approach π.
The average of 0 and 2π is π, so an average can certainly be π. Yes, the average of a finite number of rational numbers cannot be π since π is irrational. But why would the sinuosity of any river be rational? The sinuosity of a circle is exactly π, for example. The true sinuosity of any given river is almost certainly irrational as well, since irrational numbers vastly outnumber rational ones in a very relevant sense: if you pick a real number uniformly at random between 0 and 1, there is literally zero chance that you will pick a rational number. Similarly, there is zero chance that the sinuosity of any river will be π or that the average of any finite number of rivers will be π. However, what they mean when they say this is this more precise fact: if there were an infinite number of rivers formed like those on Earth, the average sinuosity of that infinite collection of rivers would be exactly π.
[+] [-] oofabz|10 years ago|reply
https://en.wikipedia.org/wiki/Coastline_paradox
[+] [-] zamalek|10 years ago|reply
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_...
[+] [-] Strilanc|10 years ago|reply
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
[+] [-] unholiness|10 years ago|reply
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.
[+] [-] placeybordeaux|10 years ago|reply
[+] [-] reilly3000|10 years ago|reply
[+] [-] brittonsmith|10 years ago|reply
The right answer may not be pi, but the data shown make a compelling case that rivers do tend to some average value.
As a sidenote, there are active human efforts to keep certain rivers, like the Mississippi, from meandering too far from their current locations. I don't know how many of the world's rivers have such efforts being applied to them, but it's not unreasonable to think that this could have some effect.
[+] [-] dspillett|10 years ago|reply
Or I could just be babbling and made that sentence up off the top of my head!
Or both the above could be true...
[+] [-] hobarrera|10 years ago|reply
The project is still interesting, since it seems to indicate a strong convergence to a certain sinuosity (~1.5?).
[+] [-] bbcbasic|10 years ago|reply
[+] [-] te|10 years ago|reply
[+] [-] aaron695|10 years ago|reply
(Although I'm not sure the paper actually said it was true, just the hundreds of articles that reported on it)
[+] [-] gus_massa|10 years ago|reply
Also, there is a clear outlier with sinuosity 7.6. Which river is it?
[+] [-] ggchappell|10 years ago|reply
I suspect there is an error in the data. The project lists the Lukuga River as being 1904 km long. But Wikipedia[1] only has 320 km, which would give it a much more ordinary sinuosity of 1.27.
For a river that might actually have a very high sinuosity, take a look at the Fraser River -- #20 on the list, with sinuosity listed as 5.12.
[1] https://en.wikipedia.org/wiki/Lukuga_River
[+] [-] votingprawn|10 years ago|reply
[+] [-] PhasmaFelis|10 years ago|reply
[+] [-] tgb|10 years ago|reply
[+] [-] lsjroberts|10 years ago|reply
Thanks for the feedback, I'm working on expanding the project to import data from a couple of sources, you can follow it at http://github.com/lsjroberts/pi-me-a-river
[+] [-] IsaacL|10 years ago|reply
[+] [-] mistercow|10 years ago|reply
tl;dw: when rivers are very sinuous, the kinks turn into oxbow lakes and separate from the river, so that limits how sinuous they can get. On the other hand, rivers' bends are constantly amplified by erosion. Researchers modeling these phenomena showed that under certain assumptions, these forces are in equilibrium at a sinuosity of ᴨ.
[+] [-] tofupup|10 years ago|reply
[+] [-] spacehome|10 years ago|reply
[+] [-] plonh|10 years ago|reply
[+] [-] matt_kantor|10 years ago|reply
What should be said is that the average might approach π.
EDIT: I'm wrong, read the replies.
[+] [-] StefanKarpinski|10 years ago|reply
The average of 0 and 2π is π, so an average can certainly be π. Yes, the average of a finite number of rational numbers cannot be π since π is irrational. But why would the sinuosity of any river be rational? The sinuosity of a circle is exactly π, for example. The true sinuosity of any given river is almost certainly irrational as well, since irrational numbers vastly outnumber rational ones in a very relevant sense: if you pick a real number uniformly at random between 0 and 1, there is literally zero chance that you will pick a rational number. Similarly, there is zero chance that the sinuosity of any river will be π or that the average of any finite number of rivers will be π. However, what they mean when they say this is this more precise fact: if there were an infinite number of rivers formed like those on Earth, the average sinuosity of that infinite collection of rivers would be exactly π.
There. That's how to be pedantic.
[+] [-] IvyMike|10 years ago|reply