
Is the average sinuosity of the world's rivers equal to pi? - curtis
http://pimeariver.com/
======
oofabz
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)

~~~
zamalek
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_...](https://upload.wikimedia.org/wikipedia/commons/5/55/Sigmoid_curve_for_an_autocatalytical_reaction.jpg)

~~~
Osmium
> 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
> 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.

~~~
Osmium
Agreed that the fact it's continuously changing is an additional issue, but I
don't see the "eventually you start working with curves" argument. Where are
these curves on a typical coastline?

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.

~~~
zamalek
> Where are these curves on a typical coastline?

Between atom nuclei.

~~~
Osmium
How so? Do you fit the positions of your nuclei to a spline to get your curve?
How about quantum uncertainty in its position? How do you assign whether a
given nuclei belongs to the 'coast' or to the 'sea', etc. I don't think it
holds.

~~~
zamalek
> How about quantum uncertainty in its position?

Aw crap, _really_ good point. You're right.

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reilly3000
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.

~~~
brittonsmith
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.

~~~
eitally
I 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.

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bbcbasic
One thought - should it be a weighted average? I.e. the longer the river, the
more bearing it's ratio has on the result.

~~~
te
Maybe the harmonic mean?
[https://en.wikipedia.org/wiki/Harmonic_mean](https://en.wikipedia.org/wiki/Harmonic_mean)

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aaron695
Crowd 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)

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gus_massa
It 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
The 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)

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PhasmaFelis
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.

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tgb
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?

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lsjroberts
Author 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)

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IsaacL
What's the reasoning behind why this should be the case? Something to do with
the curvature of the Earth?

~~~
mistercow
There'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.

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tofupup
from visual inspection the histogram seems to peak at 1.6 the golden mean

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spacehome
I guess the answer is no.

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plonh
Need a really good reason to read past "equal to π (3.141593)".

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matt_kantor
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.

EDIT: I'm wrong, read the replies.

~~~
StefanKarpinski
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 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.

~~~
pash
_> 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 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.)

~~~
dogecoinbase
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 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.

~~~
pash
_> ... 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.

