5 ms·
Is the average sinuosity of the world's rivers equal to pi?
- gus_massa 11y agoIt would be nice if the distribution graphic of the sinuosity has units. Also, there is a clear outlier with sinuosity 7.6. Which river is it?
- ggchappell 11y agoThe Lukuga -- #40 on the list. 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 https://en.wikipedia.org/wiki/Lukuga_River
- votingprawn 11y agoAnd also numerous rivers with sinuosity < 1, i.e. the river is shorter than the straight line distance between the source and mouth.
- tgb 11y agoGreat 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?
- oofabz 11y agoDo 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 agoThe complexities there don't seem to have a correlary to the rivers.
- unholiness 11y agoYes. 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 agoIndeed, one would reach the plank length -- and hence the distance simply cannot be infinite.
- thedufer 11y agoThe 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 agoI 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 agoSuppose 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 agoThe 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.
- spacehome 11y agoI guess the answer is no.
- IsaacL 11y agoWhat's the reasoning behind why this should be the case? Something to do with the curvature of the Earth?
- mistercow 11y agoThere's a link to a video that explains it. 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 11y agofrom visual inspection the histogram seems to peak at 1.6 the golden mean
- PhasmaFelis 11y agoWhat'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.
- matt_kantor 11y agoThis 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. EDIT: I'm wrong, read the replies.
- IvyMike 11y agoWhat's the average of 0 and 2pi?
- StefanKarpinski 11y agoIf you're going to be pedantic, do it right. 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.
- matt_kantor 11y agoGood call, I'm not sure why I hadn't considered that. I blame the whiskey!
- StefanKarpinski 11y agoNo worries. You gave me an excellent excuse to exercise my pedantry in a way that I usually try to avoid so as not to be a social pariah :-)
- plonh 11y ago
- bbcbasic 11y agoOne thought - should it be a weighted average? I.e. the longer the river, the more bearing it's ratio has on the result.
- te 11y agoMaybe the harmonic mean? https://en.wikipedia.org/wiki/Harmonic_mean https://en.wikipedia.org/wiki/Harmonic_mean
- plonh 11y agoNeed a really good reason to read past "equal to π (3.141593)".
- aaron695 11y agoCrowd sourcing to disprove a published paper is the real story here. (Although I'm not sure the paper actually said it was true, just the hundreds of articles that reported on it)
- reilly3000 11y agoWhy 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.
- dspillett 11y agoThey 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! Or both the above could be true...
- hobarrera 11y agoIt would have been quite interesting if it had been, and probably something we'd spend the next few centuries banging our heads trying to understand. The project is still interesting, since it seems to indicate a strong convergence to a certain sinuosity (~1.5?).
- brittonsmith 11y agoAn 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.
- eitally 11y agoI suspect damming (and other human intervention as you note) in general causes restrictions in sinuosity, artificially either preventing on creating local regions of sinuosity = 1.
- lsjroberts 11y agoAuthor here, feel free to contact me if you're interested in this or a scientist in a related field - lsjroberts [at] outlook {dot} com 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 http://github.com/lsjroberts/pi-me-a-river