One thought - should it be a weighted average? I.e. the longer the river, the more bearing it's ratio has on the result.
Maybe the harmonic mean?
https://en.wikipedia.org/wiki/Harmonic_mean
Is the average sinuosity of the world's rivers equal to pi?
31–40 of 48 posts
Re: Is the average sinuosity of the world's rivers equal to pi?
#32Earlier quoted context omitted.
If 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 out…
> If you're going to be pedantic, do it right. 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…
> measuring devices that can spit out irrational numbers. ... No other scientist has ever seen such a thing.
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).
Re: Is the average sinuosity of the world's rivers equal to pi?
#33Why 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.
Re: Is the average sinuosity of the world's rivers equal to pi?
#34Earlier quoted context omitted.
> If you're going to be pedantic, do it right. 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…
This is a trivially absurd argument for two reasons: 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 confla…
> ... 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.*
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.
Re: Is the average sinuosity of the world's rivers equal to pi?
#35Earlier quoted context omitted.
> If you're going to be pedantic, do it right. 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…
> measuring devices that can spit out irrational numbers. ... No other scientist has ever seen such a thing. 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 (2 pi*radius/300).
This sort of thing exists and is apparently what other commenters also have in mind, but I would argue that in your example the measuring device's output is a natural number n in the range [0, 300] and that its interpretation as a fraction of a circle's circumference is just that.
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.
Re: Is the average sinuosity of the world's rivers equal to pi?
#36Why 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.
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!
Or both the above could be true...
Re: Is the average sinuosity of the world's rivers equal to pi?
#37Why 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.
It 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?).
Re: Is the average sinuosity of the world's rivers equal to pi?
#38Why 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.
Re: Is the average sinuosity of the world's rivers equal to pi?
#39Do rivers actually have a well-defined length? I know coastlines do not, and rivers seem similar. https://en.wikipedia.org/wiki/Coastline_paradox
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_...
Re: Is the average sinuosity of the world's rivers equal to pi?
#40Do rivers actually have a well-defined length? I know coastlines do not, and rivers seem similar. https://en.wikipedia.org/wiki/Coastline_paradox
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],…
> 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.]