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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

#311

Post by cameronh90 »

dueyfinster wrote: post #286
asimpletune wrote: post #283

If society were tasked one day with reinventing standards and units, it doesn't matter why, what do you think are some things they would change? For example, I think for human counting, base 12 is about as easy as base 10, but gives good ways to express division by 3, in addition to division by 2 or 4. It also fits better with how we count time, like how there are 60 seconds in a minute, 12 months in a year, etc... b…

To be honest I'm not a fan, time is cumbersome to do any sort of addition or subtraction to get exact days/hours/minutes (not to mention timezones etc). Compare to metric units, always base 10 and always easy to convert mm to cm to m and so on. Now that we live in a digital world - why do we consistently reinvent date/time libraries? To me that's proof enough the concepts are just hard to work with and over a long sp…

None of those issues with date and time are anything to do with the base, they're to do with date and time as defined by humans being inherently complicated concepts. Specifically, trying to have a single measurement "fit" for a load of different purposes.

If we had based our system around base 12, a base 12 version of the metric system would be just as easy to work with as metric is in decimal, with the added bonus that you can divide powers of the base (10, 100, 1000, etc.) into quarters, thirds and sixths without needing a decimal place, and thirds of 1 would be non-repeating.

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

#312

Post by cameronh90 »

asimpletune wrote: post #283

If society were tasked one day with reinventing standards and units, it doesn't matter why, what do you think are some things they would change? For example, I think for human counting, base 12 is about as easy as base 10, but gives good ways to express division by 3, in addition to division by 2 or 4. It also fits better with how we count time, like how there are 60 seconds in a minute, 12 months in a year, etc... b…

I think there's an argument for base 16 over 12. It's very slightly worse than base 12 for purely human use, since it's only divisible into halves, quarters and eighths. However, each hex digit maps to an exact number of binary digits, which I think outweighs the benefit of thirds and sixths in a digital world.

Of course, it's all theoretical, because there's no chance this would ever happen short of an apocalypse that takes us back to the stone age, and unless the radiation gives us 12 or 16 fingers, we'd probably just reinvent decimal.

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

#313

Post by kuschku »

mistercow wrote: post #240
kuschku wrote: post #239

Earlier quoted context omitted.

See, the issue with "foot" is that different people use different body parts to measure length. Germany used the "Elle", which is the distance between wrist and elbow, or roughly one foot. Other regions used the foot or the cubit instead. The primary advantage of the SI system is that it has only ONE length unit that you add prefixes to.

I’m saying that the single SI length unit could have been defined precisely as the light nanosecond, or “metric foot”, had people known that that length fit closely to an existing unit back around 1790. There would still be one unit with prefixes added, but that unit would have a really clean correspondence to physics rather than a hacky conversion factor. But you have to go back that far in time for it to work, beca…

Even if you made that kind of definition, it wouldn't have been that simple. Would you have used metric seconds or babylonian seconds?

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

#314

Post by School-Cotton »

School-Cotton wrote: post #306
rat_on_the_run wrote: post #258

Earlier quoted context omitted.

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.

Too late to edit, but I meant to say “consider this to be an article about history, not physics”.

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

#315

Post by aydyn »

Tainnor wrote: post #294
aydyn wrote: post #280

Earlier quoted context omitted.

What do you mean? Cos is an inherently geometric function.

To quote myself: > cos can be defined via its power series or via the exp function (if you use complex numbers).

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

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

#316

Post by mr_mitm »

_a_a_a_ wrote: post #278
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…

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 pretty clear for anyone who is new to this concept: https://en.wikipedia.org/wiki/Non-Euclidean_geometry#/media/...

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

#317

Post by kazinator »

mistercow wrote: post #259
kazinator wrote: post #257

Earlier quoted context omitted.

It is not unusual case. The heuristic you want is working. It's nothing more than a coincidence.

What are you disputing about the explanation given in the post? As far as I can tell, it’s basically accurate (although the pendulum unit was called the toise, and the meter seems to have targeted half a toise). If you accept that account, it’s not a coincidence.

I initially had an objection due to a misconception I was carrying.

I see now that the pendulum formula is a pure relation between time and distance/length. It will apply regardless of the units used. For example if we measure time in fortnite and length and furlongs the formula will be the same. The gravitational acceleration of course will be in those units: furlongs per fortnights squared. Needless to say that will not be 9.81.

Now the meter unit was chosen in relation to the second unit by the length of a pendulum that produced an integer period. So that choice/relation caused the gravitational acceleration g to take on such a value that its square root cancels out the π on the outside of the root.

I was confused for a moment thinking that the definition of the kilogram would somehow be mixed up in this but of course it isn't. g doesn't incorporate mass; and of course pendulum swings are dependent only on length and not mass.

There are all sorts of situations in which certain units either give us a nice constant inside the formula or eliminate it is entirely.

For instance Ohm's law, V = IR. It's no coincidence that the constant there is 1. If we change resistance to some other unit without changing how we measure voltage and current we get V = cIR.

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

#318

Post by xapata »

JadeNB wrote: post #246
jvanderbot wrote: post #232

Earlier quoted context omitted.

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.

You're right. Thanks for being that guy!

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

#319

Post by amelius »

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.)

Any examples of textbooks or papers where the author used tau instead of pi (and the topic was not tau or pi)?

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

#320

Post by rvbissell »

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.

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.

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