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Happy Tau Day

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Re: Happy Tau Day

#51

Post by jeffwass »

MrManatee wrote: post #42

I have found the idea of tau useful even though I have never used it in writing. One argument in favor of tau is that in many formulas pi often has the multiplier 2 in front of it. If these formulas are written in terms of tau, they may become slightly easier to memorize and manipulate. Perhaps so, but I don't really care about this. It’s not a big difference. Besides, there are also lots of formulas that are easier…

Your Cauchy argument is actually misleading and runs right into my joking example elsewhere in this thread about needing to define a new constant Sigma=4pi to work better with steradians.

Pi doesn't always represent only a pure circle. Eg, in solid angles there are 4pi steradians over a sphere. Or 2tau. Or what I define as Sigma.

This extra factor of two when using Tau should be just as disconcerting to tau enthusiasts for steradians as pi is for radians.

Your example is talking about shooting particles in all directions over 3D space. This calls for a solid angle approach. Which means you should be integrating over steradians just as you'd use radians for an angular system. There are 4pi steradians over a sphere's full solid angle, the wall only covers 2pi steradians. Meanwhile the extra factor of two comes from the integration of decreasing infinitesimal wall cross sections over the azimuthal angle.

The fact that it comes out to 1/2 tau is merely happy coincidence. Ie, the geometry introduced an extra factor of two because it's just as easily 1/4 sigma, and for a solid angle system (shooting particles in all directions) you should be using steradians. Hence sigma.

Re: Happy Tau Day

#52

Post by agumonkey »

BenFrantzDale wrote: post #23
nabla9 wrote: post #15

While choice between tau and pi is arbitrary from mathematical point, things like pedagogy, notational simplicity and aesthetics matter. The reason why I think π is probably better choice is because small multiples of constant are cognitively easier to process than fractions. 2π is easier to write and see as single object than τ/2. All we need to do is to make slight cognitive adjustment and think and teach 2π as a n…

My favorite example of why τ is the true circle constant is actually the equation A = 1/2 τ r^2. It fits the usual quadratic form and shows how circumference and area relate: c = dA/dr = τ r and so A = ∫ c dr = ∫ τ r dr = 1/2 τ r^2. At first glance those 1/2s seem superfluous and like it would be nicer to just have a 2 in the other terms, but the 1/2 shows that differentiation and integration with respect to r are ha…

Reminds me of Taylor series. The factorial terms eluded me for so long until I realized it was a cancelling factor for accumulated derivation of polynomials. Now there's this link in my mind between powers / derivation / factorials.

Re: Happy Tau Day

#53

Post by agumonkey »

jeffwass wrote: post #32
jobigoud wrote: post #22

Earlier quoted context omitted.

> 2π is easier to write and see as single object than τ/2 The point is that you shouldn't have to write 2π in the first place. So notational simplicity would mandate the use of τ, not π. Noting that you will have to write τ/2 for a half turn is moot, in your case you'll also have to write π/2 for a quarter turn anyway.

It gets worse! When using solid angles over a sphere, you're still going to need to use 2Tau steradians to cover a sphere. This factor of 2 will continue to cause confusion for the Tau fans. Therefore we must also define a new constant Sigma = 4pi so we can cleanly and easily deal with steradians. Anybody up for writing the Sigma Manifesto?

Maybe we could subscript tau with dimension n.

Re: Happy Tau Day

#55

Post by tgb »

jeffwass wrote: post #51
MrManatee wrote: post #42

I have found the idea of tau useful even though I have never used it in writing. One argument in favor of tau is that in many formulas pi often has the multiplier 2 in front of it. If these formulas are written in terms of tau, they may become slightly easier to memorize and manipulate. Perhaps so, but I don't really care about this. It’s not a big difference. Besides, there are also lots of formulas that are easier…

Your Cauchy argument is actually misleading and runs right into my joking example elsewhere in this thread about needing to define a new constant Sigma=4pi to work better with steradians. Pi doesn't always represent only a pure circle. Eg, in solid angles there are 4pi steradians over a sphere. Or 2tau. Or what I define as Sigma. This extra factor of two when using Tau should be just as disconcerting to tau enthusias…

Surely the Cauchy distribution you're replying to doesn't have any steradians going on, right? It's a 1D distribution and the gun described would point towards a random point on a circle, not on a sphere.

Re: Happy Tau Day

#56

Post by jeffwass »

adwf wrote: post #46
jeffwass wrote: post #27

Earlier quoted context omitted.

So to save confusion over a mere factor of two, you're proposing devoting an entire new Greek letter to just double a constant, and requiring students to understand pi anyway to make sense of hundreds of years of mathematical and scientific corpus? It's just like that xkcd comic about introducing a new standard and now having N+1 standards. Except in this case the new standard offers only a factor of two.

Relax, the whole thing is a joke, including my comment above.

Ha, yeah I know. On rereading my comment, it does come across as somewhat defensive but I definitely wasn't meaning it to be that way. Actually I was smiling as I wrote it.

See my other comments on this thread about sigma, for instance.

Re: Happy Tau Day

#57

Post by peterbonney »

greydius wrote: post #8
thecopy wrote: post #2

The choice between Pi vs. Tau is completely arbitrary and should be based on minimizing the friction of use. Everyone knows Pi, few knows about Tau. I see no point in spending any more energy than that on this kind of nonsense decisions.

"Because that's the way we've always done is" is never a valid argument.

Of course it's a valid argument - you have to consider the costs and benefits of switching whenever you propose a change in a broadly accepted convention.

For example, one could make an argument that oral and written communication in the United States would be more efficient if everyone was forced to communicate in Klingon. But the switching cost would be so astronomically high that any possible gains pale in comparison.

(I'm not saying the pi/tau cost/benefit is comparable to English/Klingon, I'm just pointing out that "that's the way we've always done it" is in fact a perfectly legitimate argument. The onus is on the person proposing a change to show that the change is worth it.)

Re: Happy Tau Day

#58

Post by phkahler »

greydius wrote: post #8
thecopy wrote: post #2

The choice between Pi vs. Tau is completely arbitrary and should be based on minimizing the friction of use. Everyone knows Pi, few knows about Tau. I see no point in spending any more energy than that on this kind of nonsense decisions.

"Because that's the way we've always done is" is never a valid argument.

>> "Because that's the way we've always done is" is never a valid argument.

But it is. For the pi/tau thing, it's better if the whole world uses one set of formulas. For the guy who pointed to which side of the road we drive on, it's best if we all agree to drive on the same side. And for units, it's best if we all agree to measure things using the same units - otherwise probes crash into mars and such. All of those are arbitrary decisions and we're better of just sticking with them - except the units one, by using SI units a lot of formulas simplify and arbitrary constants go away. I'm not so sure tau/pi have a similar advantage. If you do change, you'll put the world in a weird transition phase until the change is widespread and that causes problems until it's resolve. Also, with mathematics you have existing and historical papers that use pi, so what to do about those? The legacy of pi would be around a long long time.

Re: Happy Tau Day

#59

Post by gravypod »

jeffwass wrote: post #41
gravypod wrote: post #38

Earlier quoted context omitted.

This isn't to save time multiplying by two. This is to uniform a hole slew of previously unrelated equations that can now be represented similarly.

I honestly can't tell if you're being sarcastic or not.

I don't understand your comment. Have you watched the video linked within this post?

This video is very good at pointing out many of the benefits of tao. [0]

[0] - https://www.youtube.com/watch?v=H69YH5TnNX

Re: Happy Tau Day

#60

Post by jeffwass »

tgb wrote: post #55
jeffwass wrote: post #51

Earlier quoted context omitted.

Your Cauchy argument is actually misleading and runs right into my joking example elsewhere in this thread about needing to define a new constant Sigma=4pi to work better with steradians. Pi doesn't always represent only a pure circle. Eg, in solid angles there are 4pi steradians over a sphere. Or 2tau. Or what I define as Sigma. This extra factor of two when using Tau should be just as disconcerting to tau enthusias…

Surely the Cauchy distribution you're replying to doesn't have any steradians going on, right? It's a 1D distribution and the gun described would point towards a random point on a circle, not on a sphere.

Argh, you may be right.

I was actually thinking of a separate example from physics (years ago) calculating the classical flow of particles through an aperture. Where you assume particles have maxwell-Boltzmann distribution of speeds and consider them travelling in all directions over 3D and also the effective size of the aperture vs the azimuthal angle of travel. It's quite similar to this example actually but given the speeds you also must account for which velocity range [v,v+dv] as a function of angle will put a particle through the hole in time dt.

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