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MFT - Tau - March 13, 2012
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Spiked Math Comic - MFT - Tau

It's Math Fact Tuesday, yay! If anyone celebrating "Pi Day" wants to spread Tau Propaganda, I've prepared a black & white summary sheet highlighting key tau facts with relevant sources for the curious. Make sure to print off a bunch of copies and distribute to your liking!!

Relevant Links:
  • Joseph Lindenberg's "Tau before it was cool" - Awesome images and detailed descriptions that will convince any reader of the power of Tau
  • Robert Palais' "Pi is wrong!" - The 2001 article in The Mathematical Intelligencer
  • Michael Hartl's "The Tau Manifesto" - Detailed article on why Tau is the correct circle constant
  • Wikipedia entry on tau (2pi) 
  • Peter Harremoës' page on "Al-Kashi's constant τ" with lots of links to articles and resources
And make sure you follow Joseph's advice and celebrate Tau Hour!

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Any ideas for improvements/modifications are more than welcome! Note that the amount of space is limited for a one sided 5.5" x 8.5" summary sheet (the dimensions in the first file I used).

Isn't it easier to measure the diameter than the radius?

Not if you're standing at the radius.

You mean center. And even then, it's pretty hard to find the exact center of a circle without bisecting the diameter.

That doesn't have anything to do with naturality.
Maybe you could define the circle constant in terms of the smallest possible square that fits a circle inside it, because that is easier to measure. So, circle constant = d^2/circular area = crazy!

By the way, d^2/(circular area) = 4/pi

d^2/(circular area)
Circular area = pi * r^2
d = 2 r
Expand: (2r)^2=(2r)(2r)=(2*2)(r*r)=4*r^2

Surprised to see tau "propaganda" here. Wasn't it only last year that you posted the pi manifesto alleviating many of the tauists' arguments? ;)


Treason, I say! Treason!

Tis true, I'm still not convinced of this whole "tau thing", but I do enjoy causing tau-rouble.

Admit it – it was Tau Hour that won you over, wasn't it? That's OK. Don't worry – we won't ask what's actually in your stein tomorrow.

This is a good one, too. Vi Hart rocks.

I agree that Vi Hart rocks. I met her recently. :) But I disagree with that video, because I have never measured how much dessert I have by its perimeter. If I did that, I may as well just eat Koch cakes. Area is more important. I am somewhat amused by this pi-rotation Mike seems to have made on the issue after writing the Pi Manifesto.

Actually I don't really care either way; you either multiply by 2 in some cases or divide by 2 in others. May as well stick with the status quo.

Disagree about Euler's identity being made more elegant. exp(i*tau) = 1 ? Big deal, exp(0) = 1 too. The equation with tau is thus far more liable to calls of "so what?", whereas exp(i*pi) = -1 immediately looks different to the relatively uninitiated, as all real values of exp come out positive.

In any case, Euler with pi is more elegant if you write exp(i*pi) + 1 = 0, thus making it a clear relation between the five most important quantities in mathematics.

That "five most important quantities in mathematics" thing is rather contrived (why isn't "infinity" an important enough quantity in their minds, or the golden ratio, or any number of other mathematical constants), and only comes from people arbitrarily choosing rearrange the equation to get rid of the minus sign. But if you insist on including the unnecessary zero just to get your "five quantities", "exp(i*tau) = 1 + 0" is a tau-version of the euler identity, _and_ follows directly from the exp(i*theta) = cos(theta) + i sin(theta) euler equation.

There's another option for Euler's Identity. If my website can convince you that 2 is also an "important number", then consider the following identity. It's just the complex conjugate of the original Euler's Identity after swapping sides:
0 = 1 + e^{-i*(τ/2)}
You'll have to imagine it properly formatted, but consider that it has:
+,-,*,/ All four basic arithmetic operations IN STANDARD ORDER
0,1,2 in numeric order
e,i,τ in alphabetical order
As well as exponentiation, and everything occurs EXACTLY ONCE.

I heard about tau, it sounded interesting (as in "it sounded right"). Then I heard about your pi counterargument, and it sounded even more "right".
Then I talked to my nephew who hated the fact that "our school math only gives you formulas and never explains them" and I tried to explain pi to him in a simple way. I ended up with a comparison of circles to squares that ended with a "pi is the number that is not 4" explanation. And that assured me that pi does make sense.

And yeah, you writing the pi manifesto first and then this doesn't make sense. I expect you to add a "tau is wrong" summary sheet below this one to balance things out.

Do not listen to this tau propaganda! How can we forget about PI(e)?

And Euler's identity is waaaay prettier with pi. I remember when I was in high school and I was fascinated that e^(sth sth) can give a negative number. Probably one of main reasons why I like complex analysis now.

I can see that tau tends to make stuff look prettier, but I am getting very angry, when people write "pi is wrong". Dammit, we talking about maths here and people use "wrong" like sociologists. At best, he could say that tau is more elegant than pi, though I agree with the above comments that the Euler identity with pi is more exciting than the one with tau. Cheers.

I was holding onto pi for the A=pi*r^2, but your comparison of 1/2*tau*r^2 with the different shapes and dimensions of circular geometry, really sealed the deal. I am tau-converted

Uh oh... we lost one. We'll just have to convert you back tomorrow!

I think I'll celebrate π day tomorrow (3.14) and τ day in a few months (6.28). At least that will give me some time to think of appropriate foods for tau.

The Euler identity with, as here stated, is not equivalent to the classical Euler identity. If you tell me that e^{i \tau} = 1, I can't decide whether is e^{i \pi} = 1 or e^{i \pi} = -1. You should write e^{i \tau/2} = -1

OK, then I can say: "If you tell me that e^{i\pi} = -1, I can't decide whether e^{i\pi/2} = i or e^{i\pi/2} = -i." And that does not make sense either. Euler's identity is elegant because it states so much in so little number of terms, and because those terms are some of the most important numbers known to mathematics. It is simple and beautiful. Tau version of the identity has the same properties, and it is a fundamentally(not arbitrarily) simpler statement, because it states what happens when you make one FULL turn in complex plane.

@ ers? Um.... fundamentally more simple? 1 turn of the circle in the complex plane? Eulers identity is important because it identifies that e^(i*theta) represents ANY turn in the complex plane. Period. One turn or half a turn, none is 'more fundamental' than the other.
Now, if perhaps you made some argument about residues on poles in the complex plan I'd say that integrating for one 'full' turn yields more fundamental insight, but even that is somewhat contrived and has nothing to do with pi or tau, really.

Maybe I am choosing wrong words ("fundamentally"???).(English is not my first language.) I know both e^(i*pi) and e^(i*tau) are special cases of Euler's formula, and I know the formula is the real deal, not its special cases. But as I see it, the argument was about *ruining* or *perfecting* the *beauty* (So much subjective words, not sure if I am still talking about mathematics) of the most famous special case of the formula: "Euler's identity" e^(i*pi)+1=0. Everyone agrees it is one of the most beautiful identities in mathematics because it has e, pi, i, 1 and 0 in it and because it is simple. And my point is: Doing a full turn is more basic and simpler than doing a half turn. Doing a half turn and calling it more beautiful than a full turn is like teaching a child 1/2 before teaching them counting OR before defining a circle, defining a half-circle and calling it and deriving the circle from it and calling it double-. Whole point of choosing tau over pi is about that tau represents a more basic concept. And mathematics is all about choosing the most basic concepts and defining more complex things using them.

Correction: "....OR before defining a circle, defining a half-circle and calling it {something} and deriving the circle from it and calling it double-{something}....."

Sorry, I mistakenly used <something> and it is invisible because of HTML-styled comment system.

tau is really a misnomer... since τ = 2π, either tau and pi be switched, or call the new variable ππ.........

@Iva,n I've seen the argument that you can look at them as fractions (turn/I and turn/II), and I'm satisfied by that, since "legs" argument is equally arbitrary.

When I was in a multiple choice, there was a no calculator question of converting degrees into radians. (by finding the closest choice), and I made a mistake: I confused a half circle with a complete circle (pi vs. 2pi).

I'm pretty neutral at the tau issue, but I want to focus on the standard unit for angles instead (degrees. vs. radians vs. gradians vs. fraction of a circle vs. tau)

But the tau reform is much better than these "Watch a YouTube video" social reforms, I can start using tau by using it in my unmarked works.

One interesting thing tau allows is you can take the percentage labels on any PIE chart and instantly have the associated angles by just writing "τ radians" after them. So a slice labeled "35%" has an angle of "35%τ radians", or ".35τ radians" if you get rid of the % sign. I imagine the idea of using % signs to label angles would make some purists a little queasy, but it's an interesting option considering all the PIE charts people see.

Which is larger e^tau or tau^e ?

Just doesn't have the same feel to it as the traditional version:

Which is larger e^pi or pi^e ?

Also in favour of tau is that tau day usually has warmer weather than pi day.

Not here it isn't. Tau day is freezing in comparison.

Even in the Northern hemisphere, going by the average monthly temperatures for two random places I just looked at, the more accurate pi day (22/7) would probably be warmer than tau day. I wouldn't necessarily call that a good thing, though.

it's a number, it's like saying 4 is right and 2 is wrong...
pi exists and tau exists. I won't use tau, because I don't want to.

I won't use tau, because I don't want tau.


Try deriving tau with respect to x. Try doing it again. And again. Circumference, circle area, cone volume. Try doing it with pi.

I miswrote, I meant integrating.


as you said, if pi is wrong, then tau is twice as wrong...

If tau is so perfect and pi is so wrong, how come I never heard of tau in the sense of 2pi until 2011? I earned a BS in Math in 1984, and there was never any mention in any of my classes of a single constant being named that was equal to 2pi. Is this a recent "discovery"?

If the mathematicians introduce tau, then physicists should change the electron charge to positive, and ban the CGS-system, and stop writing "dx" right after the integral! So many annoyings to be fixed, and none of this is probably ever going to happen

While we're at it, we might aswell reverse the notations on electrical charges so they make sense. Even if it's less than optimal, pi is not 'wrong', just a tiny bit less practical, not really a ground to change a rooted convention...

Ada says: Blasphemy x 3 and a little bit more! Support Pi at www.cow-pi.com.

I proposed this in 1995, though I didn't call it "tau", and I didn't have the marketing insight to frame it as a resistance to "pi day" (though I don't believe anyone celebrated pi day back then). I coupled the halving mistake with the mistake of the 12 hour clock. As others have mentioned, though, if we actually lived in a sane world there'd be a LOT of other changes made before this one... http://hostilefork.com/2010/07/18/clocks-that-run-backwards/

Just seems a little awkward. But then all change does at first. Still, elegant as it seems, I'm not convinced. Plus, I don't want to have to buy another new scientific calculator for my engineering approximations. Gotta go find a slice of Pi(e) now...

Not sure if this has been mentioned, but the fact that e^(pi i) = -1 is more fundamental than e^(2 pi i) = 1, seeing as the latter is immediately implied by the former without any knowledge of complex arguments in an exponent. Tau is just silly, and no matter what constant you use there will always be areas in which it is nicer and areas in which it is worse. Much better things to worry about changing.

I came here on Pi Day for this?
Bah, humbug!

Why can't we use BOTH pi and tau?
And seriously, why do you want to get rid of pi when it takes a part in what happened before in the past?

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