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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

#261

Post by ordu »

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 quote says more about the person reacting, then about this quote. If the person tends to look answers in a physics (a popular approach for techies), then this quote feels wrong. If the person thinks of physics as of an artificial creation filled with conventions and seeking answers in humans who created physics (it is rarer for techies and closer to a perspective of humanities and social sciences), then this quote is the answer, lacking just some details.

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

#262

Post by minitech »

soulofmischief wrote: post #85
ahmedfromtunis wrote: post #33

[flagged]

Your comment is much more rage-bait than the article. Universal isn't a way we describe numbers. You meant to say dimensionless . Pi is dimensionless constant because it describes a relationship between two measurements of a dimensionless unit circle. Pi is expressed as a pure ratio between two other dependent numbers. Dimensionless values are special because they don't rely on any particular measurement in any parti…

> Universal isn't a way we describe numbers. You meant to say dimensionless.

They probably really meant to say “universal”, since dimensionless values are a less interesting category that includes… well, every number. Pi shows up in math without having to parameterize anything, making it universal in a way that even physical constants of our universe aren’t.

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

#263

Post by DistractionRect »

stkdump wrote: post #187
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.)

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.

This is correct, gravitational constants are a good approximation/simplification since the mass of solar bodies is usually orders of magnitude greater than the other bodies in the problem, and displacement over the course of the problem is usually orders of magnitude smaller than absolute distance between them.

In other words, we assume spherical cows until that approximation no longer works.

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

#264

Post by Thorrez »

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…

>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, and comes to the conclusion: no.

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

#265

Post by smallnamespace »

withinboredom wrote: post #247
TheRealPomax wrote: post #236

Earlier quoted context omitted.

Yes. That's what makes it a fun fact. Most people never even learn about non-euclidean math, and this is the kind of "wow I never even thought about this" that people should be able to learn about in a comment thread. Calling it painful to read is downright weird. Pi, the constant, has one value, everywhere. So now let's learn about what pi can also be and how that value is not universal.

It isn't a "fun fact" ... it's plainly incorrect. π never changes its value. Ever. It is a constant in mathematics, no matter the geometry. However, π can have different ratios in other geometries, but it will still be ~3.14. It is painful because this statement: > draw a circle on a sphere. That circle has a curved diameter that is bigger than if you drew it on a flat sheet of paper. The ratio of the circle circumfe…

It doesn't have anything to do with projections.

Let's say you live in a non-flat space. You come up with the idea of a "circle" with the usual definition: the set of points on the same plane equidistant from a central point.

You then trace along the circle and measure the length, and compare it to the length of the diameter. It turns out that this ratio changes as a function of the diameter. This truth is inherent to curved spaces themselves, and is not an artifact of choosing to describe the space using a projection.

The definition of pi in this world is no longer the invariant ratio of diameter to circumference. You can still recover pi by taking the limit of this ratio as the diameter length goes to zero. But perhaps mathematicians in this world would (justifiably) not see pi as such a fundamental number.

Now back to GP's example: people living on a sphere (like us) are analogous to inhabitants of a non-Euclidean space. The surface of Earth is analogous to 3-space, and the curvature of Earth is analogous to the curvature of space.

And indeed, if you draw real-life larger and larger circles on our planet, you will find that the ratio of circumference to diameter is smaller than pi. For example, if you start at some point on the Earth (say, the North pole) and trace out a circle 100 miles distant from it, you will find that the circumference of that circle (as measured by walking around that circle) is a little bit _less_ than pi x 100 miles.

Again, we have done no "projection" here. We've limited ourselves to operations that are fairly natural from the perspective of a mathematician living in that space, such as measuring lengths.

The fact that large circle circumferences measure less than pi x diameter on Earth did not change how we developed math, likely because you only notice start to notice this effect with extremely large (relative to us) circles.

But perhaps the inhabitants of a non-Euclidean space that was much more highly curved would notice it much earlier, and it would affect their development of maths, such that the number pi is held in lower regard.

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

#266

Post by trainyperson »

This article reasons that it is not a coincidence because of the “seconds pendulum” definition of the meter, which would necessitate the values being equal because of the pendulum time period equation.

That all makes sense to me, and I agree.

But here’s what’s odd to me:

We ended up not choosing the seconds pendulum approach (for reasons mentioned in the article). Instead they chose to use “1 ten-millionth of the Earth’s quadrant”. Now, how is it that that value is so close to the length of the seconds pendulum? Were they intentionally trying to get it close to seconds pendulum length, and it just happened to be a nice round power of ten? Is that a coincidence?

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

#267

Post by 8note »

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…

Similarly, g depends on the geometry, and g is a constant 0 for Euclidean space

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

#268

Post by [deleted] »

jvanderbot wrote: post #232
dr_dshiv wrote: post #152

Earlier quoted context omitted.

Alpha brainwaves are almost exactly 10hz, in humans and mice. The typical walking frequency (for humans) is almost exactly 2hz (2 steps per second). And the best selling popular music rhythm is 2hz (120bpm) [1]. Perhaps seconds were originally defined by the duration of a human pace (i.e. 2 steps). These are determined by the oscillations of central pattern generators in the spinal cord. One might suspect that these…

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

[deleted]

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

#269

Post by tymscar »

imoverclocked wrote: post #123
jeffwass wrote: post #57

Earlier quoted context omitted.

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!

Very much like lambda calculus

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

#270

Post by juped »

Why is this comment section, specifically, such an embarrassing dumpster fire?

It's a serious question; this is the sort of neat derivation that makes for a popular Youtube video, and despite Youtube comments being famously... variable in quality, the comment section on videos about things like this is vastly more literate than the threads here right now.

Is it just a coincidence, the chaotic behavior of uninformed sneer comments (which exist on every post; I've certainly been guilty, to my shame) meaning that some post is going to end up being the one with no other type of comment? Or is there some surprising reason why?

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