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The Pi Manifesto - July 4, 2011
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No comic today, but something you might find interesting.

Recently, pi has been under "attack" by a new constant called tau. You can read about it here:

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

I find tau to be quite interesting, but alas, someone has to defend pi. Check out the Pi Manifesto on why pi really is the true circle constant!

I've also set up a subforum for more discussion on this topic. For the record, I'm neither a tauist nor piist for the time being.

Is not \tau=2\pi?

What is the big deal?

This is a common response to the introduction of Tau. But if it truly has more benefits in being the circle constant over pi, then surely it should be discussed by mathematicians rather than ignoring it as a triviality.

but if we use it in equations, sin'(x) = 2cos(x)
this is terrible!

>What is the big deal?

people like to waste time arguing over trivial, pointless stuff. (thus, the internet.)

If students were taught Tau instead of Pi at first, they would find trigonometry incredibly easily rather than insufferably confusing.

Come on! It is a simple conversion. It is a factor of two. Think of it as (2pi) * fraction. Don't simplify the fraction until the end. Pi has it's uses like πr²

Well written I thought.

Vi Hart is hot, therefore I'll take her side. :D

Vi Hart is incredibly snotty, so I'll vote against her.

VI HART SHALL BE SLAUGHTERED

(I was there. I disagree with her about tau, but she is not snotty and she makes people love maths, so I didn't help with the slaughter.)

tau and pi are both wrong; pi/2 is right

haha!! :)

I still think tau is more important, but being the radian angle measure for a quadrant, pi/2 is tau's main competitor.

pi is RIGHT tau is WRONG for the simple reason that the volume of a pizza with radius z and thickness a is pi*z*z*a

If you insist on a battle of puns,
"A famous tauitology states that 'tau is the way.'"

it's a question about we feel about something. ergo, no question I am interested in.

Dude, the letter tau seems like HALF the letter pi instead of TWICE it. That's confusing...

holy crap, good point...

The DENOMINATOR is half, therefore, tau = 2 pi.

e^{\pi i} + 1 = 0 is reason enough to consider pi more fundamental than tau.

If you took the time to watch vihart's video, you would know that e^(i{tau})=1!

The former implies the latter while the latter does not imply the former without use of complex analysis, thus the version with pi is more fundamental.

Also, the idea relating the "new" area equation to stuff in Physics is totally bullshit since they aren't related at all, making it a terrible argument.

They are completely related. Both reflect the integration of a simple linear function over its parameter. Int[c x,x ] = 1/2 c x^2 . Using c = 2 pi hides the 1/2 .

Euler's identity connects 5 very important concepts in math:
e, i, pi, 1, and 0.

If you take the tau version, it's just not the same. Where'd 0 go?

If you insist the zero must be there:

e^(i*tau) = 1 + 0

Which is truly the fundamental form, because it follows directly from the identity e^(i*theta) = cos(theta)+ i*sin(theta)

Still not convinced. It's like you just forced the 0 to be there.

e^(i*tau) - 1 = 0, then?

e^(i theta) = cos(theta) + i sin(theta)
e^(i pi) = cos(pi) + i sin(pi) = -1 + 0i = -1
e^(i pi) + 1 = 0 (rearrangement!)
e^(i tau) = cos(tau) + i sin(tau) = 1 + 0i (no rearrangement!)(win for tau!)

It would make me happier if we could find a way to fix the electrical sign convention so that the charge on electrons is called the positive charge, and current-flow diagrams matched electron-flow diagrams.

Exactly! Until physicists change that, they have no right to mess with pi!

EEs use "j" for the imaginary root. I fear getting the charge convention changed would be an uphill battle.

In terms of measuring, "circumference" is a more basic feature of a circle than "area". If circumference is measured in some units then area is measured in some units "squared". So the point you make when you say "area of unit circle is pi" is not as strong as "one turn is one tau" point is.

Sum of interior angles in any polygon is not one pi, it's just for triangles. And (sum of interior angles)=(k-2)*pi formula derived from putting k-2 triangles side to side in order to get a k-sided polygon cannot be as elegant as the fact that sum of exterior angles is always 360 degrees (which is one tau) no matter how many sides a polygon has.

Length is not a more basic feature than area. Consider the Koch Snowflake or any number of other 2 dimensional fractions. The perimeter goes to infinity while the area remains finite. And personally, I can think of a much simpler way of constructing the limit for the area of a circle (cover the unit circle with n^2 almost disjoint squares and add the areas of the squares that contain any part of the circle. as n goes to infinity, the area will go to the area of the circle). Give me a construction for the circumfrance? I thought not...

The Koch snowflake argument is weak since an analogous solid can be created in three dimensional space = infinite surface area with finite volume.

Or just consider the negative space of the circle.

It is just as easy to find many ways to integrate infinitesimally short line segments over the circumference of a circle as it is to find many ways to integrate infinitesimally small rectangles over the area. The circumference requires one integration, whereas the area requires two if the rectangle is used.

Which just shows that the surface area of a 3d object isn't more basic than the the volume. How does that say ANYTHING at all about CIRCUMFERENCE/PERIMETER and area of a 2D object?

And for the sake of Mathiness, I'm taking "X is a more basic property than Y" to have "if Y is finite than X is finite" as a necessary (albeit not a sufficient) condition. Basically, I'm taking "circumference is a more basic property than area" to imply "a 2D shape will have a finite area if it has a finite circumference/perimeter"

Oh, and since I'm basically only assuming I have a 2 dimensional shape in a R^2, yes. I'm making a weak argument (a weak arguments is one that doesn't require very strong assumptions).

THANK you! And what about the "inside" angles plus the "outside" angles? k * tau!

Count me in the Pi camp. Dudes like me spent a lot of time memorizing the decimals of Pi. Our efforts shall not be voided!!

The frequency of occurence depends on what you are doing, I guess. But our typesetting conventions favor multiplication over division, so I prefer $2\pi$ to $\frac{\tau}2$, because the latter looks bad in-line.

Anyway. This ship sailed, and the train left the station a long time ago. Pi it is.

http://www.puremaththeory.com/
Okay, forget about tau for the moment, read THIS crazy website. It attempts to explain that pi is rational.

Oh, and back on topic, I am neutral. I love pi and tau equally, and use them both on my math tests. ^^

Oh god... someone please shoot whoever made that website in the face.
No one should be allowed to be that full of shit...

They also attempt to prove that the square root of 2 is rational. Any college math student can give you a proof that is it, in fact, irrational.
"Fermats Last Theory" seriously, if you don't know the difference between a theorem and a theory, I won't take you seriously.

The site is currently (check my post time) blocked due to excessive bandwidth usage.

It's up now. Anyone know where I can find 5 monkeys with typewriters.

Classic. I've just read paper that disproves current Theory of Evolution using (in some other paper resolved) Riemann hypothesis and little bit of "Quantum fractions".
Lets enjoy the bright light of master's wisdom: "The solution to the “Riemann Hypothesis” along with the Geometry Solution of the “Poincare Conjecture” gives us a clear understanding of time, space, mankind’s biology, and the structure of our universe as we will show."

Count me in the I don't care as long as you don't mix up the symbols camp.

No matter what other people use you can still use pi, you can also define any symbol to mean pi/17 at the start of your paper if you want. I really don't see the big deal (either way). I agree that perhaps in high schools it would be easier for people to learn the trig formulas and stuff but other than that...

as long as I get my pie on pi day
tau is 2pi, (or 2/pi, idk) so what is the big deal having another math symbol?

$\pi$ is how to get a pi symbol in LaTeX, 2\pi means two times pie, not two over pie

kk thanx 4 clearing up ;)

You could always get two pies on tau day. How's that for an argument?

That article is bloody confusing. What the hell is up with the extensive use of TeX code and (wrong) assumption that everyone understands it.

I have nothing to add, just that piist looks quite a lot like pissed :P

Let's find an alien who knows math and ask for their circle constant

If this has been around for a while, then it probably still has yet to catch on. However, I do think that it won't take over pi, just exist alongside. I do support tau, I think it's a great idea. tau/4 is a quarter circle. This will make trig applications in higher mathematics a lot simpler.

tau is the ratio between circumference and radius. when you integrate over the rings of a circle to get the area you get 1/2 * tau * r^2, as for all integrations of this kind. for simplicity when dealing with areas the coefficient 1/2 * tau may be called pi. i see no problem here.

I'm going to be pious and say that the taoist's are wrong

"tao" is pronounced "dao" not "tau".

Furthermore Euler's Identity should be written as e^(i*1/2*tau)=-1 because that means: starting at real 1 you need to make half of a turn through the complex plane to get real again at -1. this makes much more sense thinking of pi which is related to areas for me. pi happens to be a half turn because integrating a linear function (such as the circumference over radius) yields a 1/2 -- however that 1/2 has nothing to do with the half turn in euler's identity.

So let's introduce
~
(cutting the symbol in half again) for the volume of a unit sphere.

I should have also spelled that out "tilde".

As a physicist, my opinion is that no matter what convention mathematicians choose, I absolutely refuse to memorize loop factors as anything other than 1/(4pi)^2.

Although pi has the advantage when referring to equation and formulas, tau has more sense since the defining part of a circle is the radius not the diameter, being that the case, the true circle constant is tau, but we don't have to limit ourselves to just 1 of them, we can use both to whenever it's convenient. Personally I liker tau, I support tau but i don't think we should replace pi by it, we should embrace both

This is silly, "if it's not broken, why fix it?" - everyone use pi, school kids use pi, since ancient times the circle constant (no one called him that, but fine) was known as three, in some cultures even something closer to pi itself. there is a reason it was chosen - it is easier.

"if it's not broken, why fix it?" Because it _is_ broken, at least pedagogically speaking. Teaching radians using pi is a nightmare - why is half of pi equal to a quarter of a circle, instead of half? Why does a sixth of a circle get a third of pi, instead of a sixth? With tau it's easy: half a circle, half a turn, half a tau.

Before the tau manifesto, I was thinking that, since the functions sine and cosine are so related, we should probably just consider only from 0 to pi/4 as the primary circle constant. Or pi/2. However, once you work with cyclical functions, you really do want the modulo 2pi to work well. The -pi to +pi symmetry works very well, but does not look so well when we get slightly bigger errors to find out if we are outside the primary region (more computational overhead too).

Nowadays, there is an idea that maybe nature itself wants the circumference to be 1 (angular momenta is quantised, not linear momenta). That suggests that we also try to work with 1/2pi for linear lengths. Of course, this is just my musing for now.

It may seem that the halves may be annoying elsewhere, but it can prove to be important. For example, nobody really wants to keep the half inside the k in half k x^2. If you actually tried it out, the version with the half in there actually is much more sensible especially from the viewpoint of the expansions. A nice factorial is easier to work with than having the stuff clog it up later. Bringing the circle constant in line may also make sense later that way.

Finally, I actually propose a different approach. Instead of using tau everywhere at once, we should just do (2pi) everywhere first, so as to see the impacts for ourselves. That observational view is quite important, and allows us to transition to tau properly later.

Sorry, I forgot to mention that, because the exponential function is usually used with the energy, the half really does belong there too.

Just thought I'd say that Pi is dangerous, and should be avoided:

I'll never understand why people so readily mix up pi and pie. If you can so easily ignore a factor of 2.718281828459… then this whole pi vs. tau debate is meaningless, since by a similar approximation pi=tau.

It makes lots more sense to have a constant that is simply defined as the ratio between the diameter of a circle and it's circumference, which is also easy for anyone to understand. Everything about sines and cosines and angles came in lots later, and it'd be messy to use 2 different constants, if you change something, you should only do it if it's worth the hassle.

Tau is defined as the circumfrence/radius of a circle. Sure, pi is easy to measure, but that doesn't make it more important.

mmm ... no. All tau does is allow you to remove a constant or two from various expressions. The people who think trig is difficult probably shouldn't be doing anything that requires a math background in the first place.

Should use Russian Ш (sh), then

Why not take 2\pi\i as a constant, as long as we are changing everything?

Hey, this was mentioned in the '60 Seconds' section of New Scientist magazine! Printed in ink on actual dead tree!

Here it is on the actual internets: http://www.newscientist.com/article/mg21128203.600-60-seconds.html

Sweet. Good to know some people are quite passionate about pi :D

I just found out that tau supporter Vi Hart is going to be on a cruise ship with me. I'd better pick a side and memorise some arguments. :D

As pointed out previously,
e^(pi * i) = -1, and e^(tau * i) = 1.

So let's end the argument by noticing that e is more basic then pi or tau (since it's the base here) and call it the winner.

Here's one claim for e-ism in an e manifesto:
http://timwit.wordpress.com/2011/07/22/the-e-manifesto-e-pi-tau/

I say 1 is more basic than any of them, and what's more, I know it to infinite decimal places, in all bases (even irrational ones.)

e^(i theta) = cos(theta) + i sin(theta)
e^(i pi) = cos(pi) + i sin(pi) = -1 + 0i = -1(rerarrangement!)
e^(i pi) + 1 = 0 (rearrangement!)
e^(i tau) = cos(tau) + i sin(tau) = 1 + 0i (no rearrangement!)(win for tau!)

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