What is it?

An enlightening discussion about pi and tau.

What is it?

by SpikedMath » Mon Jul 04, 2011 9:17 pm

The Pi Manifesto is dedicated to defend one of the most important numbers in mathematics: [tex]\pi[/tex]. Quite recently, a phenomenon known as the Tau Movement has steadily grown and is gaining more and more followers (called Tauists) by the day. This is largely due to three driving forces:

The original article [tex]\pi[/tex] is wrong written by Bob Palais (published in 2000/2001).
The Tau Manifesto written by Michael Hartl (launched on June 28th, 2010).
The video Pi is (still) wrong by Vi Hart (uploaded on March 14th, 2011).

We encourage you to first check out these links in detail to see the possible benefits of defining the constant [tex]\tau=2\pi\approx 6.283185\ldots[/tex].

The full version of the Pi Manifesto can be read here:
http://www.thepimanifesto.com/

This forum is meant for discussion about tau and pi, including arguments for and against pi.
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Re: What is it?

by xander » Mon Jul 04, 2011 10:54 pm

One of the things that really gets to me about this whole debate is insistence on both sides (however jocular it may be) that they are right and the other side is wrong. It is about context. There are times when [tex]\tau[/tex] is clearly superior, and other times when [tex]\pi[/tex] is better. There are convincing arguments in favor of both---your argument regarding the area of a unit circle is pretty good.

On the other hand, I find your argument regarding squaring the circle to be a draw, at best (one that I honestly had not thought about before). Squaring the circle cannot be done with compass and straightedge. Using [tex]\tau[/tex], terms involving [tex]\sqrt{2}[/tex] pop out. That should raise eyebrows regarding the impossibility of the construction. The [tex]\sqrt{\pi}[/tex] terms are interesting, and should also raise questions, but they don't really have the same pizazz.

I also find the argument that [tex]\tau[/tex] is overloaded to be a very weak argument. [tex]\pi[/tex] is also overloaded. Big [tex]\Pi[/tex] notation for products could be confusing, but more importantly, lowercase [tex]\pi[/tex] is used for osmotic pressure, and I've seen it used to express long term probabilities for Markov processes. I'm pretty sure that I have also seen it in economics. People make the argument that Hartl and Palais are showing their true colors as pure mathematicians, but it seems to me that the anti-[tex]\tau[/tex] side has lost track of the various uses of [tex]\pi[/tex]. Both letters are a bit overloaded---which is why I kind of like Palais' three-legged [tex]\pi[/tex].

Of course, at the end of the day, as I said, both sides have something going for them. The real question we should be asking is not "Which is better?" but "In what contexts is which better?" To someone immersed in the study of mathematics or its applications, it doesn't really matter which is used---we are sophisticated enough to use the symbol that best expresses the idea that we wish to express. On the other hand, what is a boy or girl to do when encountering trigonometry for the first time? In the US, the better part of a semester of high school is spent learning trigonometry in the context of the unit circle. I've had much better luck using [tex]\tau[/tex] than I ever had with [tex]\pi[/tex] (though, admittedly, my samples of each are small).

tl;dr version: Both symbols have pros and cons. [tex]\tau[/tex] is easier for beginning students. Advanced students and professionals can fend for themselves.

xander

EDIT: I should also note that I have started to add "Let [tex]\tau\equiv 2\pi[/tex]," to the beginning of most of my papers, despite the fact that I have this on my arm:

Image

Clearly, I like both. ;)

FURTHER EDIT: The Russian letter т may be a good substitute for [tex]\tau[/tex], as well. We already use Hebrew, why not Russian as well? The cursive version looks like this:
Image
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Re: What is it?

by bmonk » Wed Oct 05, 2011 9:55 am

The worst part about using tau?

I'd have to re-memorize it to a useless number of decimal places.
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Re: What is it?

by bmonk » Sat Dec 24, 2011 1:48 pm

I came across another reason for keeping pi, in a magazine recently:

If you have a cylinder of Mozzarella Cheese, height a and radius z, the volume will be pi z z a :roll: !
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3 is WRONG!

by japanada » Wed Jun 06, 2012 3:42 pm

I honestly don't see what people mean by "pi is wrong" or "tau is wrong." They're both real numbers; neither one "exists" any more than the other does. I mean, you never hear anyone go around yelling "3 is wrong! 3 is just 6/2! Let's use 6 instead!" And that's pretty much exactly what I'm hearing whenever I hear debates about tau and pi.

In the end, if you're debating which one is a more natural circle constant, your argument is cultural, linguistic even, not mathematical. Which is fine, I love culture as much as the next guy, but unless you can come up with a naturally defined "rightness function" that is maximized at one or the other, please don't think that you're doing mathematics by arguing over which one is "right."

The one argument for tau that I do like is that it makes it easier to learn arc length and radian measure. And I totally support the idea of introducing students to the number 2pi, even calling it tau, to make those calculations easier. Just as it makes more sense to use the symbol 6 instead of the symbol 2x3 sometimes. But that doesn't mean 3 is "wrong," and that you should never use a 3 when you're doing calculations and always replace it with 6/2. And as noted on the pi manifesto, pi now has so many uses beyond just circles, many of which don't incorporate the 2 in there, that it makes no sense to try to convert everything over even if tau is more natural as a circle constant.

Just my two cents. But before I make myself sound too snobbish, I do want to say that I love the discussions. Though I don't think the question itself is really mathematical, the arguments in favor and against do raise some awesome mathematical ideas that I love to see. So by all means, keep the discussion going! It's just that I don't see it as a "which one is right" discussion, so much as a "what are some cool facts about pi and tau that you know of" discussion, which is far more awesome in my opinion. This is one of my personal favorites so far (in particular, giving a pretty solid reason for why diameter might actually be a more natural measurement than radius: the diameter can be generalized to any shape, allowing the circle to be compared to them more easily, while radius is a very isolated concept that can't be generalized, at least not in a way that's as natural).
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Re: 3 is WRONG!

by xander » Wed Jun 06, 2012 6:23 pm

japanada wrote:I honestly don't see what people mean by "pi is wrong" or "tau is wrong." They're both real numbers; neither one "exists" any more than the other does. I mean, you never hear anyone go around yelling "3 is wrong! 3 is just 6/2! Let's use 6 instead!" And that's pretty much exactly what I'm hearing whenever I hear debates about tau and pi.

I don't think that anyone is seriously saying that [tex]\pi[/tex] or [tex]\tau[/tex] are objectively right or wrong. Rather, they are arguing that either [tex]\pi[/tex] or [tex]\tau[/tex] is the better constant to use in some particular situations (with some people then going on to say that, for instance, [tex]\tau[/tex] is the better constant in more situations, hence is more right than [tex]\pi[/tex]). For instance, I would argue that [tex]\tau[/tex] is the better constant to use when introducing trigonometry to middle school students, since [tex]\tau[/tex] radians is equal to a full circle.

japanada wrote:In the end, if you're debating which one is a more natural circle constant, your argument is cultural, linguistic even, not mathematical.

That is exactly the point. The choice of [tex]\pi[/tex] or [tex]\tau[/tex] is a choice of "spelling" or "word choice." It is entirely a cultural debate (or, perhaps, a pedagogical debate).

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Re: What is it?

by japanada » Thu Jun 07, 2012 1:30 am

Thanks for the response! And I'm sure that a lot of people involved in the debate do take your attitude - that it's just a matter of convenience of notation. Unfortunately, I've seen a lot of people take it way too literally, even going as far as saying "pi should be banned." I guess that's more of what I was reacting against.

But then again, I guess that's a problem with any discussion that spreads on the internet - you'll end up with 95% of the responses by people who never cared about the issue until it shows up on their Facebook or Twitter feed, at which point they suddenly become "experts" on the subject, supporting the most radical position they can :roll: . Though these forums definitely have far less of that than many other places on the internet.

Even so, it makes me a little uncomfortable to talk about "which is better to use in specific situations," because it makes it seem like you can always choose whatever number you want, just whatever makes life easiest for you. Again, I don't think most people (on this forum at least) will make that mistake, since they'll understand it's just a notational change - that you're not actually "using tau instead of pi," you're just "writing tau to stand for 2pi" - but I've definitely seen people take it the wrong way. "Any publicity is good publicity?" Not when it can convince students that what they've been learning is "actually wrong" and makes them more confused about math than they already were.

Again, tau vs. pi is a great discussion... I only wish it had been labelled and publicized differently.
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Re: What is it?

by rdococ » Fri Oct 18, 2013 2:19 am

That is it. I think [tex]\pi[/tex] is worse than ever. I have been able to find a counter-argument to nearly all of what shows up on this website. Obviously [tex]n\pi[/tex] IS special when [tex]n = 2[/tex].
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Re: What is it?

by bmonk » Tue Oct 22, 2013 4:42 pm

There is one argument for π over τ--I like to eat pie, but have yet to eat a tau. We need to come up with a good food named tau, preferably one served in a circle.
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Re: 3 is WRONG!

by 1=0 » Sun Mar 19, 2017 2:07 pm

xander wrote:
japanada wrote:I honestly don't see what people mean by "pi is wrong" or "tau is wrong." They're both real numbers; neither one "exists" any more than the other does. I mean, you never hear anyone go around yelling "3 is wrong! 3 is just 6/2! Let's use 6 instead!" And that's pretty much exactly what I'm hearing whenever I hear debates about tau and pi.

I don't think that anyone is seriously saying that [tex]\pi[/tex] or [tex]\tau[/tex] are objectively right or wrong. Rather, they are arguing that either [tex]\pi[/tex] or [tex]\tau[/tex] is the better constant to use in some particular situations (with some people then going on to say that, for instance, [tex]\tau[/tex] is the better constant in more situations, hence is more right than [tex]\pi[/tex]). For instance, I would argue that [tex]\tau[/tex] is the better constant to use when introducing trigonometry to middle school students, since [tex]\tau[/tex] radians is equal to a full circle.


"Right" and "wrong" aren't exactly the right words to use, but I do argue that tau is objectively the circle constant, and pi is objectively the semicircle constant. This of course means tau is more convenient when looking at full circles and pi is convenient when looking at half circles. However since a semicircle is only defined as a bisected circle, and I have yet to find any way to define a semicircle without first defining a circle, this shows that pi is really just half of tau and that defining pi as the semicircle constant is the same thing as defining it as half of tau, which means that pi really does only work because it is half of tau, and pi comes from tau. So I would say tau is objectively more natural, and that's what people mean when they say "Pi is wrong." Obviously the number isn't literally "wrong," but it catches your attention more than saying "Pi is unnatural" would. If the only reason pi is important to mathematics is because it's half of some other number, why not use that other number instead? It doesn't make any sense not to. It would be like using e/2 instead of e, and I think we can all agree that e is more natural than e/2, so why isn't it possible for one choice of the circle constant to be more natural?

The main thing that shows that tau is the true circle constant is that the standard definition of pi, C/D, is inconsistent with the rest of mathematics, which uses radius. Obviously, then, we should use C/r, but the πists will object to this by trying to change the definition to A/r^2, another definition that is inconsistent with the rest of mathematics because circumference is used everywhere, while area of a circle is used almost nowhere. The only definition of pi that is consistent with the definition of radians and with the rest of mathematics is that pi is half the circumference of a unit circle. However, this definition has a factor of 1/2 thrown in for no reason except to make it equal to pi rather than tau. If we remove the arbitrary factor of 1/2, we are left with the simpler, more natural definition of tau as the circumference of a unit circle.
Last edited by 1=0 on Mon Mar 27, 2017 11:11 pm, edited 1 time in total.
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Re: What is it?

by 1=0 » Sun Mar 19, 2017 7:52 pm

bmonk wrote:I came across another reason for keeping pi, in a magazine recently:

If you have a cylinder of Mozzarella Cheese, height a and radius z, the volume will be pi z z a :roll: !


But you'll have to go downTAUn to buy that pizza. And don't worry, there are still puns with tau; for example e^(iτ)=1 tells us "Be one with the tau."

I have recently come across another reason to switch to tau. Infinity factorial is obviously infinity, right? Nope, it's actually  √τ! (That's an exclamation mark, not a factorial). To add to the awesomeness, infinity primorial is τ^2. That's right, ∞!=√τ and ∞#=τ^2. Those have to be the most awesome equations I've ever seen! Tau has just surpassed -1/12 as my favorite number.
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