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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

#331

Post by mistercow »

rvbissell wrote: post #320
mistercow wrote: post #184

Earlier quoted context omitted.

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.

Actually, no: that Tau-centric area formula you gave derives naturally from taking the integral. Your example actually fits the expectation you have from what you learned in Calculus I. You should _expect_ that 1/2 scaling to be there. If it seems awkward to you, it's only because of a lifetime of seeing it done in terms of pi.

You can argue that having an extra number to juggle around is somehow less awkward because, under specific and subjective criteria, it’s “expected”, but given that the whole hook for tau is “we keep having to put a multiplier of two everywhere”, I don’t find it very compelling.

I can also make arguments that pi/2 would have been a better constant from a teaching perspective. The pi/2 version of the Euler identity, for example, would give you all the tools you need to link complex multiplication to rotation.

But at the end of the day, the choice of multiple used for the constant is a convention. None is going to be ideal in every case, and no math fundamentally changes because of a particular choice. Trying to argue for a change in convention at this point is just silly.

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

#332

Post by amelius »

mistercow wrote: post #329
amelius wrote: post #323

Earlier quoted context omitted.

In case you didn't see this: https://tauday.com/tau-manifesto And this is the section that addresses your point: https://tauday.com/tau-manifesto#sec-circular_area

The entire tau manifesto is basically an exercise in how you can come up with rationalizations for just about any aesthetic preference, and the “area of a circle section” is a perfect example of how far you can go with the gymnastics.

I think that section shows that they tried to highlight arguments from both sides.

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

#333

Post by jrpt »

radiator wrote: post #14
BrandoElFollito wrote: post #7

As a physicist, this makes sense. Pi = 3, pi^2 = 10, which is g Not sure why everyone is surprised. Ah, and a year is pi*10e9 seconds (IIRC)

As a physicist? When we did physics at school, and we were solving problems, the answer was always a number together with its unit . pi² might be 10 because it is a pure number, but g can never be 10, because it is an acceleration, a physical quantity, so it must be 10 of some unit.

I mean it is pretty common to set c=1 though.

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

#334

Post by aydyn »

Tainnor wrote: post #324
aydyn wrote: post #315

Earlier quoted context omitted.

Any function can be defined as a taylor series. That is not at all an argument against its geometric nature.

> Any function can be defined as a taylor series. First of all, that's wrong. It's only true for analytic functions. Second of all, sin and cos appear in all sorts of contexts that are not primarily geometric (such as harmonic analysis). 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]

> 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 is so much material describing the geometric nature and interpretation of fourier/laplace transforms.

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

#335

Post by _a_a_a_ »

mr_mitm wrote: post #316
_a_a_a_ wrote: post #278

Earlier quoted context omitted.

Ur being rather snotty about this. I've just realised something important which is so obvious to you that you consider it trivial, but it's not. I realised something important today, you might just want to feel pleased for me, and a bit pleased that the world is a little less ignorant today...? Or not?

Sorry for coming off as snotty. It wasn't my intention, I thought my statements were rather matter of fact. It's possible that after having attended two lectures on differential geometry I have forgotten that some of these things like circumference ratios and sum of angles of triangles being different in a curved geometry are not obvious to every one. I'm glad you learned something! Edit: This picture should make it…

No probs, and yes I certainly learnt something important, thanks.

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

#336

Post by sfink »

rizzir wrote: post #330

Our mechanics professor in university told us that Pi squared is ten and 6xPi=20. He said "engineers do not care for the fourth digit after the dot. If you want some number very precisely calculated just hire a mathematician instead because they are cheaper per hour"

Multiple people have brought this up and I just don't get it. When would you ever use Pi squared for anything, and if you'd never use it, who cares what its value is?

The only thing I could come up with is marking out an area by rolling a wheel some number of times to measure each edge.

Long ago, I memorized the square root of pi precisely because it seemed like the least useful number I could think of. (I was frustrated by something. Never mind.) But pi squared seems like it's pretty much the same in terms of usefulness.

If you said pi is the square root of 10, then I could see the value. Maybe that is what is being implied?

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

#337

Post by dennis_jeeves2 »

kelnos wrote: post #274
koolala wrote: post #225

Earlier quoted context omitted.

Do miles connect to feet and feet vs. gravity connect to the Golden ratio? 6 feet is like 2 pi?

Not sure, but I am indeed often connected to the earth via gravity by my feet.

Me too.

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

#338

Post by muzika »

osigurdson wrote: post #191

Having pi^2 = g would annoy me a bit as g is fundamentally a measured value. Depending on the required accuracy of the calculation, you can't even use the idealized value.

We simply wouldn’t need “g” and would say that acceleration due to gravity on earth is sqrt(pi).

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

#339

Post by matt-attack »

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…

FWIW I enjoyed your original comment :)

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

#340

Post by Tainnor »

aydyn wrote: post #334
Tainnor wrote: post #324

Earlier quoted context omitted.

> Any function can be defined as a taylor series. First of all, that's wrong. It's only true for analytic functions. Second of all, sin and cos appear in all sorts of contexts that are not primarily geometric (such as harmonic analysis). 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]

> 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 series is not a "true" definition, you could probably crack open half a dozen (rigorous) real or complex analysis texts, they're likely to define sin and cos in some such way (or, alternatively, as the single set of functions satisfying certain axioms), because defining them via geometry and making this rigorous is much harder.

I can't argue whether cos has a "geometric nature" or not, because I don't know what that means. Undoubtedly cos is useful in geometry. It is, however, used in a wide range of other domains that make absolutely no reference to geometry. mathlib's definition of cos doesn't import a single geometry definition or theorem.

Remember that the starting point of this discussion was that somebody was claiming that "pi is different in non-Euclidean geometry", which, no, even in a completely different geometry, the trigonometric functions would be useful and pi is closely related to them.

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