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A wonderful coincidence or an expected connection: why π² ≈ g

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Re: A wonderful coincidence or an expected connection: why π² ≈ g

#111

Post by mistercow »

KeplerBoy wrote: post #72
karmakurtisaani wrote: post #10

Earlier quoted context omitted.

Actually no, the whole equation boils down to the definition of meter. Or rather, one of the earlier definitions.

It does not? Pi has nothing to do with our arbitrary unit system.

Can you explain what you’re taking issue with in the post, then? Because it specifically lays out how the historical relationship between the meter and the second does in fact involve pi^2 and the force of gravity on earth.

(Granted, from what I can tell, it’s waving away a few details. It was the toise which was based on the seconds pendulum, and then the meter was later defined to roughly fit half a toise.)

Re: A wonderful coincidence or an expected connection: why π² ≈ g

#112

Post by Someone »

lupire wrote: post #88
bonzini wrote: post #55

Earlier quoted context omitted.

But it's a coincidence, right? N*k=8.31 is pressure*volume/temperature for a mole of gas. Temperature has a relatively small range (100-1000) and there's no reason why the range of P*V couldn't be far from that range, for example 0.01–0.1, with a different definition of meter, second or kilogram.

Meter, second, and kilogram were all chosen to be approximately the scale of a human, and the combined multiplicative units like Pascal, m^3, and Kelvin/Celsius are also numerically 1 in these units.

> Meter, second, and kilogram were all chosen to be approximately the scale of a human,

“Approximately the scale of a human” leaves so much wiggle room that I don’t see how one can defend that claim.

> and the combined multiplicative units like Pascal

You don’t explicitly claim it, but I wouldn’t say the Pascal is “Approximately the scale of a human”. Atmospheric air pressure is about 10⁵ pascal, human blood pressure about 10⁴ pascal, and humans can very roughly produce about that pressure by blowing.

Re: A wonderful coincidence or an expected connection: why π² ≈ g

#113

Post by mistercow »

msteffen wrote: post #110
mistercow wrote: post #5

This is interesting, but I have to quibble with this: > If you express this value in any other units, the magic immediately disappears. So, this is no coincidence Ordinarily, this would be extremely indicative of a coincidence. If you’re looking for a heuristic for non-coincidences, “sticks around when you change units” is the one you want. This is just an unusual case where that heuristic fails.

It’s really the best and only way to find non-coincidences involving the definition of units, though. All such non-coincidences will have this property

All coincidences involving the definition of units will also have this property. Once you’ve narrowed to that specific domain, invariance to change of units is completely uninformative.

Re: A wonderful coincidence or an expected connection: why π² ≈ g

#115

Post by rich_sasha »

This is neat, but still something if a coincidence.

It appears the first definition of a metre is in fact around 1/4e10 the circumference of Earth, and the further coincidence is that a 1m mathematical pendulum has a period of almost exactly 2 seconds.

So there's still a neat relationship between mass/radius of Earth, its diurnal rotation period and the Babylonian division of it into 86,400 seconds.

Re: A wonderful coincidence or an expected connection: why π² ≈ g

#116

Post by ta1243 »

Waterluvian wrote: post #107
mistercow wrote: post #19

Earlier quoted context omitted.

Yeah, I read the post. What I’m saying is “this relationship vanishes when you change units, so it must not be a coincidence” is a bad way to check for non-coincidences in general. For example, the speed of sound is almost exactly 3/4 cubits per millisecond. Why is it such a nice fraction? The magic disappears if you change units… (of course, I just spammed units at wolfram alpha until I found something mildly intere…

Or the speed of light being almost a sweet 300 million m/s. Or after-atmosphere insolation being somewhat on average 1kw/m2.

I always find insolation and insulation to be such an interesting pair of words

I guess the equivelent of "change the units" is "change the language".

French: insolation et isolation

German: Sonneneinstrahlung / Isolierung

Spanish: insolación / aislamiento

Chinese: 日照 / 绝缘

I guess coincidence

Re: A wonderful coincidence or an expected connection: why π² ≈ g

#117

Post by euroderf »

mistercow wrote: post #19
karmakurtisaani wrote: post #10

Earlier quoted context omitted.

Actually no, the whole equation boils down to the definition of meter. Or rather, one of the earlier definitions.

Yeah, I read the post. What I’m saying is “this relationship vanishes when you change units, so it must not be a coincidence” is a bad way to check for non-coincidences in general. For example, the speed of sound is almost exactly 3/4 cubits per millisecond. Why is it such a nice fraction? The magic disappears if you change units… (of course, I just spammed units at wolfram alpha until I found something mildly intere…

32 meters is 35 yards, to within about an eighth of an inch. How's that grab you ?

Re: A wonderful coincidence or an expected connection: why π² ≈ g

#118

Post by mistercow »

Waterluvian wrote: post #107
mistercow wrote: post #19

Earlier quoted context omitted.

Yeah, I read the post. What I’m saying is “this relationship vanishes when you change units, so it must not be a coincidence” is a bad way to check for non-coincidences in general. For example, the speed of sound is almost exactly 3/4 cubits per millisecond. Why is it such a nice fraction? The magic disappears if you change units… (of course, I just spammed units at wolfram alpha until I found something mildly intere…

Or the speed of light being almost a sweet 300 million m/s. Or after-atmosphere insolation being somewhat on average 1kw/m2.

> Or after-atmosphere insolation being somewhat on average 1kw/m2.

I’m kind of inclined to say that this one isn’t so much of a coincidence as it is another implicit “unit” in the form of a rule of thumb. Peak insolation is so variable that giving a precise value isn’t really useful; you’re going to be using that in rough calculations anyway, so we might as well have a “unit” which cancels nicely. The only thing that’s missing is a catchy name for the derived unit. I propose “solatrons”.

Re: A wonderful coincidence or an expected connection: why π² ≈ g

#119

Post by ithkuil »

fouronnes3 wrote: post #17
BrandoElFollito wrote: post #7

As a physicist, this makes sense. Pi = 3, pi^2 = 10, which is g Not sure why everyone is surprised. Ah, and a year is pi*10e9 seconds (IIRC)

As a computer scientist this is not surprising either. After all there are only three numbers: 0 1 and n.

And -0 and NaN
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