One thought - should it be a weighted average? I.e. the longer the river, the more bearing it's ratio has on the result.
Is the average sinuosity of the world's rivers equal to pi?
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Re: Is the average sinuosity of the world's rivers equal to pi?
#22This 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.
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 are going to be pedantic, please explain how anything in the Heisenberg Uncertainty-bounded universe can have a transcendental size.
Re: Is the average sinuosity of the world's rivers equal to pi?
#23Need a really good reason to read past "equal to π (3.141593)".
Re: Is the average sinuosity of the world's rivers equal to pi?
#24This 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.
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 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.)
Re: Is the average sinuosity of the world's rivers equal to pi?
#25Earlier quoted context omitted.
Indeed, one would reach the plank length -- and hence the distance simply cannot be infinite.
The 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.
I 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]
Re: Is the average sinuosity of the world's rivers equal to pi?
#26This 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.
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…
> 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 [...]
Though, at these levels of precision any physical quantity stops being a number, and becomes more like a distribution that varies over time.
Re: Is the average sinuosity of the world's rivers equal to pi?
#27Earlier 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…
> > 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.
> 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.
Re: Is the average sinuosity of the world's rivers equal to pi?
#28Crowd 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)
Re: Is the average sinuosity of the world's rivers equal to pi?
#29Earlier 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…
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.
Re: Is the average sinuosity of the world's rivers equal to pi?
#30Earlier quoted context omitted.
The 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.
I 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
"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.