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A wonderful coincidence or an expected connection: why π² ≈ g
171–180 of 352 posts
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#172Another "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,…
ln(5) ~ 1.6094379 is much closer, differing by about half of a percent of a percent.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#173Underlying idea of whole metric and SI system is the real start point. You want to define some units that you can easily replicate. Time is reasonable enough one, measure and track length of day and then length of second. Now based on this figure out way to come up with replicable definition for distance, pendulum is good enough gravity is constant enough. Thus linking gravity and meter arises.
From here you can define lot of other units like mass and Volt and Ampere... Everything comes from second which is weird, but does make lot of sense.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#174Earlier 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…
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.)
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#175This 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.
you can rule that heuristic out immediately because pi is unitless, surely?
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#176Another wonderful coincidence:
- Speed of light: 299,792,458 m/s
- Great Pyramid of Giza: 29.9792458°N
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#177Another interesting coincidence (or perhaps a decades-long dad-joke troll perpetrated by German-speaking scientists) is that 1 hertz is roughly equal to the frequency at which a human heart (“Herz” in german, with a nearly indistinguishable pronunciation to “Hertz”) beats.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#178Earlier quoted context omitted.
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.
> 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.
Some fields even use the unitary transform -- they split the difference and just throw in a 1/sqrt(2pi) in both directions.
https://en.wikipedia.org/wiki/Fourier_transform#Angular_freq...
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#179Earlier 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.
How strange.
"I only ran the first half of the program, but it didn't seem to give the correct answer, so it's obviously broken."
"I only read the first half of the proof, but the answer wasn't contained there, so I'm forced to conclude the proof is worthless."
You simply gave up before encountering the mathematical reason the relationship exists, why the units are different, and so on. You just ran with your incorrect initial assumption.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#180Earlier quoted context omitted.
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
I use that approximation, via the Fibonacci sequence, to translate between miles and km. 13 miles ~ 21 km (actually 20.921470).
My favorite approximation is π·E7 = 31415926.5... , which is a <1% error from the number of seconds in a year.