A wonderful coincidence or an expected connection: why π² ≈ g • Hacker News Board

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

#301

Post by _a_a_a_ » Sun, Aug 11, 2024, 1:09 PM UTC

jcelerier wrote: post #298

mr_mitm wrote: post #229

Earlier quoted context omitted.

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…

> Pi doesn't vary. The ratio of circumference to diameter of a circle may vary depending on the geometry. Can you share some place where Pi isn't defined exclusively as being "the ratio of circumference to diameter of a circle"? I have never heard any other definition in my life, and couldn't find any other through the first few Google results

"This definition of π implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curve (non-Euclidean) geometry, these new circles will no longer satisfy the formula π = C / d"

https://en.wikipedia.org/wiki/Pi

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

#302

Post by wongarsu » Sun, Aug 11, 2024, 1:11 PM UTC

TheOtherHobbes wrote: post #292

stkdump wrote: post #187

Earlier quoted context omitted.

My physics prof said g is actually a vector field. Because the acceleration has a direction and both magnitude and direction vary from point to point.

Absolutely true on astronomical scales. An unnecessary complication if you're dropping a brick out of a window.

It's funny how much of physics we do assuming a flat earth.

If you did it "properly" you would calculate the orbit of the brick (assuming earth was a point mass), then find the intersection between that orbit and earth's surface. But for small speeds and distances you can just assume g points down as it would in a flat earth

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

#303

Post by mistercow » Sun, Aug 11, 2024, 1:12 PM UTC

ordu wrote: post #261

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.

> This is just an unusual case where that heuristic fails. I don't have this heuristic drilled into me, so I saw the point immediately. To be frank, I suspected the general direction of the answer after reading the headline, and this general direction, probably, can be expressed the best by pointing at the sensitivity of the approx. equation from the headline to the choice of units. So, I think, the reaction to this…

No, like objectively, a dimensionless number lining up with a meaningful constant is more likely to be because of some underlying mathematical connection, and a dimensioned number lining up is more likely to be a coincidence. There are only a handful of ways for a unit’s heritage to have a connection to a local physical phenomenon like the post describes, and that’s what it takes to have a unit-dependent non-coincidence. That’s not dependent on your perspective.

The thing that’s interesting in this case is that the meter’s connection to g is obscured by history, whereas most of the time a unit’s heritage is well known. Nobody is going to be surprised by constants coming out of amps, ohms, and volts, for example, because we know that those units are defined to have a clean relationship.

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

#304

Post by School-Cotton » Sun, Aug 11, 2024, 1:30 PM UTC

OJFord wrote: post #296

The pendulum equation isn't progress at all, it's just another observation of it? Driving it 'backwards' as it were with known values for the parameters it would determine, and seeing that pi squared is roughly g without having to know the actual values of those constants. And now I've finished the article, nothing more is really offered. Am I missing something? That doesn't explain it/answer the question at all afai…

The point is that the original definition of the meter was the length such that, when g is expressed in meters, pi^2 = g

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

#305

Post by randallsquared » Sun, Aug 11, 2024, 1:34 PM UTC

jvanderbot wrote: post #218

thayne wrote: post #200

Earlier quoted context omitted.

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.

This is a good example, but actually this is exactly what GP was referring to. It is a coincidence that the thing you're observing is periodic with earth's rotation. Observing a similar thing from a satellite (allegorically the same as "changing bases") would remove the interesting periodicity. The earths rotation coincides with the phenomenon, so it's likely a coincidence .

In the example case, the earth's rotation is producing the apparent observation: it's the cause, not a separate phenomenon that happens to coincide, or that might be indicative of a deeper relationship. For something to be a coincidence, it must be otherwise unconnected causally, which is not the case if the reason you found a ~24 hour period is that you forgot to account for the earth's rotation.

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

#306

Post by School-Cotton » Sun, Aug 11, 2024, 1:36 PM UTC

rat_on_the_run wrote: post #258

aaplok wrote: post #252

Earlier quoted context omitted.

This is addressed in the article, including the fact that you can change the unit system to change the value of g. The article explains that the coincidence comes from the fact that the meter, as a unit, was defined (by Huygens) based on g and π. It was later redefined several times and the link between the two values became anecdotal. In other words, on another planet the gravitational constant would still have had…

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…

Consider this to be an article about physics, not history. The article can be boiled down to one sentence: “the meter was originally defined to make pi^2=g”. This was a fun fact I didn’t know.

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

#307

Post by ValentinA23 » Sun, Aug 11, 2024, 1:46 PM UTC

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…

There is relationship between the metric system and the French royal system. The units used in this system have a fibonacci-like relationship where unit n = unit n-1 + unit n-2.

    palm : 7,64 cm
    span : 12,36 cm
    handspan : 20 cm
    foot : 32,36 cm
    cubit: 52,36 cm

cubit/foot =~ 1,618 =~ phi, el famoso Golden ratio. foot/handspan =~ phi too. And so on.

From this it turns out that 1 meter = 1/5 of one handspan = 1/5 x cubit/phi^2

Another way to get at it is to define the cubit as π/6 meters (= 0.52359877559). From this we can tell that

1cbt = π/6m

π meters = 6 cubits

Source: https://martouf.ch/crac/index.php?title=Quine_des_b%C3%A2tis...

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

#308

Post by karmakaze » Sun, Aug 11, 2024, 2:20 PM UTC

mistercow wrote: post #184

rvbissell wrote: post #93

Earlier quoted context omitted.

No, Tau is fundamental. Pi only exists because someone mistakenly thought the formula for circumference involved diameter, when in fact it involves radius. ("Quit factoring a 2 out of Tau!" I tell them.)

Eh, you can find plenty of cases where tau is just as awkward as pi is elsewhere. Right off the bat, the area of a circle becomes more awkward with tau, becoming (tau*r^2)/2, and in general, the volume of an n-ball gains weird powers and roots of two in its denominator as n increases if you switch to tau. In general, I don't think you can really claim either one is "more fundamental". It's just a matter of framing.

It's not really about being awkward (that's a tell not the motivation), it's about basing on a radius or diameter: which is more fundamental? Or the arc length of a unit circle or half a circle, which isn't an arbitray formula it's the definition.

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

#309

Post by karmakaze » Sun, Aug 11, 2024, 2:24 PM UTC

rvbissell wrote: post #93

karmakaze wrote: post #64

Might be interesting if were true in Planck units. But also 2Pi is fundamental, who defines a ratio of something to 2 of something (radius)?

No, Tau is fundamental. Pi only exists because someone mistakenly thought the formula for circumference involved diameter, when in fact it involves radius. ("Quit factoring a 2 out of Tau!" I tell them.)

That was (obliviously?) my point.

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

#310

Post by simpaticoder » Sun, Aug 11, 2024, 2:28 PM UTC

andrei-akopian wrote: post #300

You could not have done a worse job explaining it. This is written for which audience. For a person who doesn't know physics this is a very long and confusing explanation. Explaining that some units depend on others, and the importance of the ability to reproduce the metric system on your own, is much more important than the whole pre-story of length standards. There are lots of unanswered questions. What was the sec…

The meter is now (as of 2019) defined as the distance light travels in vacuum during N cycles of an atomic clock[1]. Note that to take into account GR effects you'd need to specify where on Earth you do the measurement, since gravity affects the clock rate. The velocity of light is defined, not measured, now. This is actually quite profound, because our system of units is now based on the validity of special relativity.

1 - https://en.wikipedia.org/wiki/Metre