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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

#241

Post by travisjungroth »

TheRealPomax wrote: post #238
travisjungroth wrote: post #208

Earlier quoted context omitted.

Just sounds like you’ve confused yourself. It’s like spinning in circles and acting like no one else knows which way is up. That isn’t a different pi. That’s a different ratio. Your hint is that there are ways to calculate pi besides the ratio of a circle’s circumference to its diameter. This constant folks have named pi shows up in situations besides Euclidean space.

Good job, you completely missed the point where I explain that pi, the constant, is a constant. And that "pi, if considered a ratio" (you know, that thing we did to originally discover pi) is not the same as "pi, the constant". Language skills matter in Math just as much as they do in regular discourse. Arguably moreso: how you define something determines what you can then do with it, and that applies to everything f…

I haven’t missed your point. I’ve criticized it.

You’re treating your non-consensus definition not as a hypothetical, but as a fact. Your comment started with “fun fact”.

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

#242

Post by travisjungroth »

TheRealPomax wrote: post #235
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…

There is no "clearly" in Math. The fact that pi is a constant while at the same time not being "the same constant" in all spaces, and not even being "a single value, even if we alias it as the symbol pi" is what makes it a fun fact. Heaven forbid people learn something about math that extends beyond the obvious, how dare they!

> There is no "clearly" in Math.

Sure there is. You don’t expect a paper to explain that the numbers are in decimal and not hexadecimal.

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

#243

Post by chrz »

karmakurtisaani wrote: post #10
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.

Actually no, the whole equation boils down to the definition of meter. Or rather, one of the earlier definitions.

Im wondering is there connection or not? We use distance unit to get to π number, whatever the distance unit is right? We get π from circumference to diameter ratio, so however long the meter is the π in your distance unit is same ratio

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

#244

Post by TheRealPomax »

travisjungroth wrote: post #242
TheRealPomax wrote: post #235

Earlier quoted context omitted.

There is no "clearly" in Math. The fact that pi is a constant while at the same time not being "the same constant" in all spaces, and not even being "a single value, even if we alias it as the symbol pi" is what makes it a fun fact. Heaven forbid people learn something about math that extends beyond the obvious, how dare they!

> There is no "clearly" in Math. Sure there is. You don’t expect a paper to explain that the numbers are in decimal and not hexadecimal.

I don't know which ones you read so I can't comment on that, but the ones I read most definitely specify which fields of math they apply to, and which axioms are assumed true before doing the work, because the math is meaningless without that?

Papers on non-Euclidean spaces always call that out, because it changes which steps can be assumed safe in a proof, and which need a hell of a lot of motivation.

And of course, that said: yeah, there are papers for proofs about things normally associated with decimal numbers that explicit call out that the numbers they're going to be using are actually in a different base, and you're just going to have to follow along. https://en.wikipedia.org/wiki/Conway_base_13_function is probably the most famous example?

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

#245

Post by SAI_Peregrinus »

koolala wrote: post #213
SAI_Peregrinus wrote: post #204

Earlier quoted context omitted.

π^2 ≈ 32 to you?

Replace s in your calculation with imperial s instead of metric s and it isn't imperial feet per metric seconds.

Imperial seconds were very, very close to metric seconds.

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

#246

Post by JadeNB »

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)

> 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.

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

#247

Post by withinboredom »

TheRealPomax wrote: post #236
withinboredom wrote: post #216

Earlier quoted context omitted.

This whole conversation is painful to read: 1. Your parent was talking about projections from one space to another and getting it confused. 2. Pi is pi and their non-Euclidean pi is still pi (unless you want to argue that a circle drawn on the earth’s surface has a different value of pi). The problem comes down to projections, then all bets are off.

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 circumference to its diameter is less than 3.1415etc, so is that a different pi? You bet it is: that's the pi associated with that particular non-Euclidean, closed 2D plane.

The problem here is *projections*. If you project non-Euclidean space onto Euclidean space, you end up with some seemingly nonsensical things, like straight lines that get projected into arcs. This is your problem. You projected a curved line from non-Euclidean space onto Euclidean space and got an arc, but didn't account for the curvature of your "real non-Euclidean space" and thus ended up with an invalid value for π. If you got something that isn't ~3.14, then you did the math wrong somewhere along the way.

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

#248

Post by theginger »

koolala wrote: post #214
lupire wrote: post #109

Earlier quoted context omitted.

A more natural way to say it is that equality requires that the unit of length is the length of an arbitrary pendulum and the unit of time is the half-period of the same pendulum. The pendulum is a device that relates pi to gravity.

Sounds universal. Get a different value on the Moon? Of course... pi squares differently on the moon :)

The arbitrary length pendulum with a period of 2 seconds which is your unit of length, (or 1 Catholic meter) is much shorter on the moon. In local Catholic meters gravity would be pi squared Catholic meters / second. As it would on any planet.

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

#249

Post by anigbrowl »

How do mathematicians handle it when there are tantalizingly close relationships between purely mathematical numbers? I fell down this particular rabbit hole through the musical entrance (just intonation intervals), but I mean look at all this, it's clearly a setup.

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

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