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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

#341

Post by aydyn »

Tainnor wrote: post #340
aydyn wrote: post #334

Earlier quoted context omitted.

> Lean's mathlib defines pi precisely the way I've described it - cos is defined via exp, and pi is defined as the unique zero in [0,2] Which, _again_ is a taylor series. A function having a taylor series does not argue against the geometric nature of cos. > Second of all, sin and cos appear in all sorts of contexts that are not primarily geometric (such as harmonic analysis). I'm not sure how you can say that, there…

> Which, _again_ is a taylor series. Mathlib defines cos x = (exp(ix)+exp(-ix))/2, which is not a definition via Taylor series (even though you can derive the Taylor series of cos quite easily from it). exp is defined as a Taylor series in mathlib, but it might just as easily be defined as the unique solution of a particular IVP, etc. Regardless of this, I have no idea why to you seemingly a definition via Taylor ser…

> Mathlib defines cos x = (exp(ix)+exp(-ix))/2

I am not understanding why you think this is at all relevant.

Its like saying a^2 + b^2 = c^2 is not geometric because it doesnt make reference to triangles. But at the end of the day, everyone understands Pythagorean theorem to be an inherently geometric equation, because geomtry is just equations and numbers.

I realize to some extent this is all subjective, but to me its insane to claim that cos is not an inherently geometric function. Agree to disagree.

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

#342

Post by spotplay »

smallnamespace wrote: post #265
withinboredom wrote: post #247

Earlier quoted context omitted.

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 t…

Thanks for the original comment—I picked up something new that I hadn't considered before.

That said, you could spin this by suggesting that the mathematician living in that non-Euclidean space might also have a different perspective on numbers. If we assume pi is still constant for him, then the numbers he's always known could be shifting in value but maybe that's a stretch.

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

#343

Post by jncfhnb »

mistercow wrote: post #75
jncfhnb wrote: post #56

Earlier quoted context omitted.

X^2 is a lot more interesting than x*0.0000743 or whatever it is

Ok, then by that thinking, you should find it really interesting that Earth escape velocity is almost exactly ϕ^4 miles per second. In fact, adding exponents here objectively makes it less interesting, because it increases the search space for coincidences. What makes the case in the post most interesting to me is that it looks at first glance like it must be a coincidence, and then it turns out not to be.

You would need a compelling argument why x^4 is interesting. Especially something like the golden ratio.

Not a long of things go to the fourth power in equations we need to use. Pi^2 directly features in periodicity

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

#344

Post by Tainnor »

aydyn wrote: post #341
Tainnor wrote: post #340

Earlier quoted context omitted.

> Which, _again_ is a taylor series. Mathlib defines cos x = (exp(ix)+exp(-ix))/2, which is not a definition via Taylor series (even though you can derive the Taylor series of cos quite easily from it). exp is defined as a Taylor series in mathlib, but it might just as easily be defined as the unique solution of a particular IVP, etc. Regardless of this, I have no idea why to you seemingly a definition via Taylor ser…

> Mathlib defines cos x = (exp(ix)+exp(-ix))/2 I am not understanding why you think this is at all relevant. Its like saying a^2 + b^2 = c^2 is not geometric because it doesnt make reference to triangles. But at the end of the day, everyone understands Pythagorean theorem to be an inherently geometric equation, because geomtry is just equations and numbers. I realize to some extent this is all subjective, but to me i…

You're analogy is flawed. Pythagoras' theorem is about right triangles. "a^2+b^2=c^2" isn't about triangles, it's not even a theorem, it's simply an equation (typically a diophantine one) that is satisfied by some numbers but not others. Something which typically belongs to number theory. Obviously the two things are closely related (which is the beauty of mathematics - there are a lot of connections between very different fields).

But really, I'll refer again to the part of my previous reply where I contextualise why I wrote what I wrote and how that answers the question of whether pi is somehow arbitrary due to the fact that we usually think of space as Euclidean: it's not.

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

#345

Post by jvanderbot »

randallsquared wrote: post #305
jvanderbot wrote: post #218

Earlier quoted context omitted.

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.

I respectfully disagree (without attempting to say you're wrong!) about the definition of coincidence and the requirement of being non causally related. If I'm riding on a bus and the light poles going past line up with my music, that's a coincidence even though they are cause soley by the bus motion BPM matching an essentially random choice of song BPM.

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

#346

Post by thayne »

jvanderbot wrote: post #345
randallsquared wrote: post #305

Earlier quoted context omitted.

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.

I respectfully disagree (without attempting to say you're wrong!) about the definition of coincidence and the requirement of being non causally related. If I'm riding on a bus and the light poles going past line up with my music, that's a coincidence even though they are cause soley by the bus motion BPM matching an essentially random choice of song BPM.

What you're describing is a coincidence.

What I'm describing is an artifact in your data that is caused by the motion of the earth.

To give a more concrete example, suppose you measuring the brightness of a trans-neptunian object, and observe that the brightness changes slightly with a period of about 1 year. You might think it has a non-uniform albedo, and a rotational period of one year, when in reality, it is just brighter when the earth is closer to it.

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

#347

Post by koolala »

theginger wrote: post #248
koolala wrote: post #214

Earlier quoted context omitted.

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.

I think I understand. Distance scales with Planet Size? That actually makes a lot of sense.

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

#348

Post by koolala »

ValentinA23 wrote: post #299
koolala wrote: post #211

Earlier quoted context omitted.

But does it work when you use the right Imperial technique?

If I come up with my own measuring unit, let's call it the sneezle (whatever the actual length I assign to it) I will be able to also define a duration unit (say, the snifflebeat) based on the time it takes for a pendulum one sneezle long to complete a full oscillation, and vice versa I can define the sneezle by adjusting the length of a pendulum so that it oscillates in two snifflebeats. Here are the maths: T = 2π√(…

Different planets, Different Sneezle length, Same Sneezle Beat

I'll try to see these units you linked thanks

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

#349

Post by tdiff »

jobigoud wrote: post #288
School-Cotton wrote: post #237

Earlier quoted context omitted.

The traditional definition of the second before modern timekeeping was 1/86400 of a day. I’m guessing that was precise enough for their purposes.

It must have been brutal to create the first clocks when your smallest external reference is a day long. You first make a huge hour glass or clepsydra and tweak it once per day until it's perfect. Very slow debugging loop.

Well, defining seconds via "caesium frequency" is perhaps more brutal.
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