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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

#121

Post by arcastroe »

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.

I'm surprised at the number of people disagreeing with your quibble. I had the exact same thought as you!

If pi^2 were _exactly_ g, and the "magic" disappeared in different units, THEN we could say "so this is no coincidence" and we could conclude that it has to be related to the units themselves.

But since pi^2 is only roughly equal to g, and the magic disappears in different units, I would likely attribute it to coincidence if I hadn't read the article.

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

#122

Post by mannykannot »

mistercow wrote: post #99
mikequinlan wrote: post #83

Earlier quoted context omitted.

Pi is related to the circumference of a circle; the meter was originally defined as a portion of the circumference of the Earth, which can be approximated as a circle. "The meter was originally defined as one ten-millionth of the distance between the North Pole and the equator, along a line that passes through Paris."

But that connection actually is a coincidence. From what I can tell, when they standardized the meter, they were specifically going for something close to half of a toise, which was the unit defined as two pendulum seconds. So they searched about for something that could be measured repeatably and land on something close to a power of ten multiple of their target unit. The relationship to a circle there doesn’t have…

It was news to me, but that's what the article says, and it is supported by by Wikipedia, at least. [1]

In addition, I feel the article glosses over the definition of the second. At the time, it was a subdivision of the rotational period of the earth (mostly, with about 1% contribution from the earth's orbital period, resulting in the sidereal and and solar days being slightly different.) Clearly, the Earth's rotational period can (and does) vary independently of the factors (mass and radius) determining the magnitude of g.

The adoption of the current definition of the second in terms of cesium atom transitions looks like a parallel case of finding a standard that could be measured repeatably (with accuracy) and be close to the target unit - though it is, of course, a much more universal measure than is the meridional meter.

[1] https://en.wikipedia.org/wiki/History_of_the_metre

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

#123

Post by imoverclocked »

jeffwass wrote: post #57
fouronnes3 wrote: post #17

Earlier quoted context omitted.

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

One of my old physics professors said something similar - there are only three numbers in the world - 0, 1, and infinity. No wait, zero is just one divided by infinity, so there are only two numbers, zero and one. So if the answer is not zero, it must be one. (ie, how to justify dimensional analysis and ignore any dimensionaless constant). Hysterical, especially for the fact that he quotes 'two' and 'three' in the se…

He already got rid of "three" and just needed a little help to get rid of "two." Since we already have 0 and (almost) everything else can just be "one more" than something else, we only need 0 and one more ... or 1.

One and Done!

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

#124

Post by setopt »

HPsquared wrote: post #37
blablabla123 wrote: post #15

Earlier quoted context omitted.

Changing units in Electrodynamics for instance comes with unexpected factors in formulas though, indeed containing π. (CGS SI)

Isn't that just the change between rad/s and Hz?

It’s more precisely the difference between “rationalized” and “unrationalized” units.

You need a factor 4pi in either Gauss’ law or Coulomb’s law (because they are related by the area 4pi*r^2 of a sphere), and different unit systems picked different ones.

It’s more akin to how you need a factor 2pi in either the forward or backward Fourier transform and different fields picked different conventions.

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

#125

Post by owisd »

rich_sasha wrote: post #115

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.

According to the article the 1/4e10 circumference definition came second

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

#127

Post by shermantanktop »

jds-67 wrote: post #103
gpvos wrote: post #29

Earlier quoted context omitted.

Have you read the article? The point is that the definition of the metre, which is used in g, originates from the length of a pendulum that swings once per second in the gravity field around Paris. So it is a matter of definitions, and the length of the metre originates from the duration of the second and the Earth's gravity field. The definitions of 1/40.000 of the Earth's circumference or ~1/300.000.000 of a light…

I have to admit I only read half of the article. Even if there is some historical fact there (but it was not mentioned at the beginning of the article), from a physical standpoint this comparison is already dimensionally wrong and also coincidentally only correct if you choose appropriate units. That was the point I was trying to make. There is not anything "deep" here.

I’d suggest fully reading the article.

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

#128

Post by GoldenRacer »

euroderf wrote: post #117
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…

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

My favorite is 1 mile = phi kilometers with <1% error

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

#129

Post by Sharlin »

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)

The pi in pi*10^9 seconds clearly comes from the fact that Earth’s orbit is circular.

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

#130

Post by imoverclocked »

xxmarkuski wrote: post #82

I remember in mechanical engineering class we would often use this for exercise sheets. On our calculator we could directly enter π and ², thus it was equally as fast to entering 10.

That's one way of setting up a standard deviation!
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