Is the average sinuosity of the world's rivers equal to pi?
pimeariver.com
pimeariver.com
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.
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.
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_...
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.]
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.
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.
Between atom nuclei.
Aw crap, really good point. You're right.
Or I could just be babbling and made that sentence up off the top of my head!
Or both the above could be true...
The project is still interesting, since it seems to indicate a strong convergence to a certain sinuosity (~1.5?).
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.
(Although I'm not sure the paper actually said it was true, just the hundreds of articles that reported on it)
Also, there is a clear outlier with sinuosity 7.6. Which river is it?
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.
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
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.
What should be said is that the average might approach π.
EDIT: I'm wrong, read the replies.
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.
Let's do it.
> 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.
Apparently you have access to measuring devices that can spit out irrational numbers. I'm impressed. No other scientist has ever seen such a thing. Unfortunately, because the computable numbers are countable, the set of irrational numbers that you will almost surely see in your setup will almost surely be uncomputable. In other words, not only will you almost surely select a number that cannot be the result of a measurement of finite precision, but you will almost surely select one that has no finite description at all.
So you will almost surely never get a measurement of the length of a river. The average of an empty set is undefined, and oofabz's objection above pertains: rivers almost surely lack lengths.
(Less pedantically: matt_kantor's pedantry was essentially correct, even if in an unintended way. Irrational numbers do not exist in the world of physical measurements. Abstracting from this reality, as StefanKarpinski did, can lead to ridulous models, because the real numbers are wholly artificial. Make probabilistic assertions about physical realities modeled with real numbers at your own peril.)
> Unfortunately, because the computable numbers are countable, the set of irrational numbers that you will almost surely see in your setup will almost surely be uncomputable.
That only applies if you assume some continous distribution. The world may for all we know be discrete and finitary, and so it may be that any real number which pops up is computable.
And surely there is nothing preventing a measuring instrument to produce irrational numbers. It is just that it will still be an approximation, and thus there are rational numbers which are just as close to the real value.
First, it would be extremely straightforward to make a measuring device that spits out irrational numbers. Take the output, truncate at half the accuracy, and append an irrational to the output.
Second, outputting an irrational number as a measurement does not imply that it's able to output any member of the complement of the rationals in the reals. You also conflate the computable numbers with the describable numbers -- but I'll give you the benefit of the doubt and assume you believe strong Church-Turing and aren't just committing an elementary error.
There are ways in which set theoretic concerns apply to the real world, but they are few and far between, and this is not among them. You're essentially in line with people who attempt to use Goedel's proofs to make grandiose pronouncement about human thought. It. Does. Not. Apply.
It's my fault, I'm sure, but you have missed the point entirely. The argument to which I responded, and which I extended absurdly, does imply that every real number is a valid output of a chance setup, and further assumes that each real number, including "any member of the complement of the rationals in the reals", is an equally likely outcome of that setup. That is anyway the conventionally understood definition of "pick a real number uniformly at random between 0 and 1".
I would attempt to clarify the rest of what I wrote, but your condescension dissuades me.
There are devices that indirectly measure something, and relate it to the desired measurement involving pi.
Stupid example: Take a measuring wheel that counts encoder clicks. Say the encoder has 300 clicks/revolution. The device is quite likely calibrated to output clicks(2pi*radius/300).
Clearly the existence of describable irrational numbers implies that we can describe a measurement using an irrational number. That does not make a finite measurement essentially irrational in any meaningful sense.
Though, at these levels of precision any physical quantity stops being a number, and becomes more like a distribution that varies over time.