Having pi^2 = g would annoy me a bit as g is fundamentally a measured value. Depending on the required accuracy of the calculation, you can't even use the idealized value.
A wonderful coincidence or an expected connection: why π² ≈ g
191–200 of 352 posts
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#192Earlier 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.
A human is ~1 meter, 1e2 kilograms and 1e9 seconds.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#193Philosophy time: Does this mean in 400 years it's possible we no longer disagree about how to evaluate things? i.e. we converge on one totalitarian utility function that everyone basically accept answers every possible trolley dilemma? In 1600, people just took the world as that: measurements are sloppy, and vary culturally and based upon location etc. But we eventually came upon tools and techniques that are broadly…
Personal preferences continue to exist. I don't see that happening.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#194Earlier quoted context omitted.
No. The other way around. Two seconds is the period of a pendulum with a length of two Sumerian cubits. (One meter is thus two Sumerian cubits, but that's an artifact due to us still using Sumerian time measurements.) P.S. I don't know why Sumerians used a factor of two. Americans still divide the day into two 12 hour spans, according to Sumerian fashion. P.P.S. One second is 1/(2*12*60*60) of a solar day. 12 and 60…
Probably because 12 is a much better base than 10. 12 can be divided by 2, 3, 4 and 6 and still results in whole numbers, which helps a lot when doing rounding and fraccional numbers. The only reason we use base 10 is because is much easier to count with our fingers.
There's also a method of counting where you touch your thumb to the sections of your fingers on the same hand, which naturally lends itself to base 12. This can be extended by keeping track of how many times you've counted to 12 on the other hand, which lends itself to either base 60 or base 144.
Interestingly, the Sumerians did not seem to employ this method, they would count 6 instances of counting to 10.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#195Earlier quoted context omitted.
Or the speed of light being almost a sweet 300 million m/s. Or after-atmosphere insolation being somewhat on average 1kw/m2.
Usefully, the speed of light is extremely close to one foot per nanosecond. This makes reasoning about things like light propagation delays in circuits much easier.
I really wish we had known this back before it was way too late to seriously change our units around. It would mean that our SI length units wouldn't have to have some absolutely ridiculous denominator to derive them from physical constants, and also the term "metric foot" is pretty fun.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#196Earlier quoted context omitted.
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…
Not a coincidence. They defined the meter from the second using the pendulum formula, and the pandulum formula has a pi in it, so pi is going to appear somewhere. The reason there is pi is probably because a pendulum is defined by its length and follows a circular motion that has the length as its radius. We could imagine removing pi from the pendulum equation, but that would mean putting it back elsewhere, which wou…
> The reason there is pi is probably because a pendulum is defined by its length and follows a circular motion that has the length as its radius.
It’s not quite that easy: For small excursions x the equation of motion boils down to x’’+(g/L)x=0. There is not a π in sight there! But the solution has the form x=cos(√(g/L)t+φ), with a half period T=π√(L/g), thus bringing π back in the picture. So indeed not a coincidence.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#197Re: A wonderful coincidence or an expected connection: why π² ≈ g
#198g is related to the radius of the earth; the meter is related to the circumference of the earth; and pi is the relationship between the radius and the circumference.
Aside from the fact that the post already explained what the actual historical connection is, your explanation requires some serious hand-waving about the mass of the Earth and the gravitational constant, neither of which were known when the meter was first defined.
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 liquid at this particular orbital distance from a star of our sun's mass.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#199As 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 mechanical engineer, can confirm. also, e ≈ pi ≈ 3
in fact, e = 2, made rather abundantly clear in "finite difference" calculus, and also that in computer science, the "natural" log base 2.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#200This 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.
Not necessarily. One of the things I was taught when studying astronomy is that if you observe periodicity that is similar to a year or a day, that's probably not a coincidence, you probably failed to account for the earth's orbit or rotation.