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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

#271

Post by ycui1986 »

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 think what the author want to convey is that the metric system was designed based on the assumption that pi^2 = g. The assumption pi^2 = g is one of the source of the metric system (at least for the relationship between meter and second). The deviation was due to the size of earth being incorrectly measured by French in the original expedition.)

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

#272

Post by bonzini »

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.

Pascal is pretty small and m^3 is pretty large though. That's why we still often use millibar and liter. They just happen to largely cancel out.

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

#273

Post by Tainnor »

TheRealPomax wrote: post #185
panarky wrote: post #174

Earlier quoted context omitted.

Another bad way to check for non-coincidences is to use a value like g which changes depending on your location. Pi is the same everywhere in the universe. g on Earth: 9.8 m/s² g on Earth's moon: 1.62 m/s² g on Mars: 3.71 m/s² g on Jupiter: 24.79 m/s² g on Pluto: 0.62 m/s² g on the Sun: 274 m/s² (Jupiter's estimate for g is at the cloud tops, and the Sun's is for the photosphere, as neither body has a solid surface.)

Fun fact: pi is both the same, and not the same, in all of those places, too. Because geometry. If you consider pi to just be a convenient name for a fixed numerical constant based on a particular identity found in Euclidean space, then yes: by definition it's the same everywhere because pi is just an alias for a very specific number. And that sentence already tells us it's not really a "universal" constant: it's a m…

Modern mathematics is more likely than not going to define pi as twice the unique zero of cos between 0 and 2, and cos can be defined via its power series or via the exp function (if you use complex numbers). None of this involves geometry whatsoever.

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

#274

Post by kelnos »

koolala wrote: post #225
lutusp wrote: post #164

Another "wonderful coincidence" is that the conversion between miles and kilometers involves this constant of conversion : kilometers = miles * 1.609344. Let's call 1.609344 the "km" constant. As it happens, km is very close the the Golden Ratio (sqrt(5)+1)/2 = 1.618033989... (call this "gr"). In fact they only differ by about 1/2 of one percent (100 * (gr/km - 1) = 0.54%)! As the author of the original article says,…

Do miles connect to feet and feet vs. gravity connect to the Golden ratio? 6 feet is like 2 pi?

Not sure, but I am indeed often connected to the earth via gravity by my feet.

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

#276

Post by [deleted] »

Thorrez wrote: post #264
rat_on_the_run wrote: post #258

Earlier quoted context omitted.

Yeah, I saw the author says it depends on the units. But like, why is this interesting? This is not physics, just some number coincidence in the metric unit system, and I'm sure one can find many more these kind of things by playing around with the constants. The fact the author calls this a "wonderful coincidence" is just... Like, a simple energy conservation or momentum conservation, taught in middle school, is inf…

>This is not physics, just some number coincidence in the metric unit system It's not a coincidence. The meter was (historically) intentionally defined as how long a pendulum is that swings in 2 seconds. When you do that, g = pi^2. >The fact the author calls this a "wonderful coincidence" The author doesn't call it a wonderful coincidence. The author asks the question of whether it's a wonderful coincidence or not, a…

[deleted]

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

#277

Post by jjk166 »

mistercow wrote: post #210
jjk166 wrote: post #198

Earlier quoted context omitted.

Reasonably accurate values for both M_earth and G were known at the time the SI meter was defined. Also it's not too hard to extend this. M_earth is a function of Earth's radius which goes into the definition of the meter. G is a function of earth's orbital period, which goes into the definition of the second. Further our definition of mass is based on the density of water, which is chosen because it is a stable liqu…

As far as I can tell, the most recent experiment to measure the mass of the Earth by 1790, when they decided on the definition of the meter, was the 1772 Schiehallion experiment, which gave a value 20% below the actual value. So if pi^2 were to somehow fall out of that it would likely be so far off as to be unrecognizable. But even that doesn’t matter, because the mass of the Earth didn’t play a direct role in the de…

Well first of all it is an adulterated pi^2 so the odds of getting something close are substantially higher.

Second, we'd be having exactly the same conversation if it happened to be g = 2pi^2, or 4pi^2, or any other reasonably artificial number.

Third, if you do the math, the mass of the earth and gravitational constant do conspire.

g = G * M_earth / R_earth^2

M_earth is approximately (4/3) * pi * R_earth^3 * Density_earth

G is approximately (2/3) * 10^-10 m^3.kg^-1.s^-2

We can eliminate our human units and rewrite it as

G = 2/3 * 10^8 / ( Density_water * Earth_orbital_period^2)

Put this all together and you get

g = 8/9 * pi * 10^8 * (Density_earth / Density_water) * ( R_earth / Earth_orbital_period^2 )

the ratio of the densities of earth and water is a dimensionless number that is independent of our units of measurement, and is approximately 5.5. With a little rearrangement we get

g = 88/9 * (pi/2) * 10^8 * R_earth / Earth_orbital_period^2

That 88/9 happens to be equal to pi^2 to within 1% error. This comes purely from nature.

1 m/s^2 is defined to be (pi/2) * 10^8 * R_earth / Earth_orbital_period^2 and thus we get the nice and neat g = pi^2 in metric units, but getting (pi^3)/2 * 10^8 in natural units is just as remarkable.

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

#278

Post by _a_a_a_ »

mr_mitm wrote: post #229
_a_a_a_ wrote: post #212

Earlier quoted context omitted.

I don't agree, I thought what he said was very interesting. It never occurred to me that pi might vary, and over a non-flat space I can see what they're saying. I think it's intrinsically interesting simply because it breaks one of my preconceptions, that pi is a constant. Talking about it being 'not very useful' just seems far too casually dismissive.

Pi doesn't vary. The ratio of circumference to diameter of a circle may vary depending on the geometry. Clearly everyone means euclidean space unless specified otherwise. Any other interpretation will only lead to problems, which is why it's not useful. There is really no ambiguity about this in mathematics. Mathematicians still use the pi symbol as a constant when they compute the circumference of a circle in a give…

Ur being rather snotty about this. I've just realised something important which is so obvious to you that you consider it trivial, but it's not. I realised something important today, you might just want to feel pleased for me, and a bit pleased that the world is a little less ignorant today...? Or not?

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

#279

Post by aydyn »

JadeNB wrote: post #246
jvanderbot wrote: post #232

Earlier quoted context omitted.

Well, a second is also a pretty good approximate resting heart rate (60 bpm)

> Well, a second is also a pretty good approximate resting heart rate (60 bpm) I'm sorry to be the kind of person who feels compelled to make this comment, but you mean a Hertz, not a second.

Youre right, you should be sorry. Dont be that guy.

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

#280

Post by aydyn »

Tainnor wrote: post #273
TheRealPomax wrote: post #185

Earlier quoted context omitted.

Fun fact: pi is both the same, and not the same, in all of those places, too. Because geometry. If you consider pi to just be a convenient name for a fixed numerical constant based on a particular identity found in Euclidean space, then yes: by definition it's the same everywhere because pi is just an alias for a very specific number. And that sentence already tells us it's not really a "universal" constant: it's a m…

Modern mathematics is more likely than not going to define pi as twice the unique zero of cos between 0 and 2, and cos can be defined via its power series or via the exp function (if you use complex numbers). None of this involves geometry whatsoever.

What do you mean? Cos is an inherently geometric function.
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