# "Pi is wrong" - the Tau Manifesto



## some1rational (Dec 31, 2010)

This may be old, but I couldn't find any posts about it (and for me, speedsolving is just as good an outlet for mathematics haha).

Read:

http://tauday.com/

or for pdf version:

http://tauday.com/tau-manifesto.pdf

I have to say, as a physics and mathematics major, I've always had an inkling that something felt 'wrong' about the usage of Pi, but it wasn't until I read this article that I could pinpoint exactly what was the problem (credit goes entirely to the author, Michael Hartl).

[for the record, I wholeheartedly agree with the author, that the usage of Pi is a "pedagogical disastor", and not that Pi the number is in some way wrong...obviously, but just in case]

So for those of you who enjoy mathematics, I'm just curious, what are your thoughts/opinions?



EDIT: The author, Michael Hartl, is a Harvard Physics PhD graduate and was a professor of theoretical/computational physics at Caltech. Just to give the article some more weight (though it doens't really need it imho).


----------



## y3k9 (Dec 31, 2010)

That's an interesting read. And btw there's a awesome math forum on aops.


----------



## BigSams (Dec 31, 2010)

y3k9 said:


> That's an interesting read. And btw there's a awesome math forum on aops.


Are you honestly saying that you read and understood this in 2 minutes?
btw I'm on aops, almost done the intermediate series


----------



## y3k9 (Dec 31, 2010)

BigSams said:


> Are you honestly saying that you read and understood this in 2 minutes?
> btw I'm on aops, almost done the intermediate series


No I read a bit posted on here, and went back to reading.


----------



## some1rational (Dec 31, 2010)

The coupe de grace was really the final straw for me. I have to commend Michael on a job well done in this manifesto. So what are your thoughts? 

I for one plan to start substituting Tau for 2*Pi from now on (though it might be awhile before I will use it again lol, I just graduated and am in the midst of applying for grad school)


----------



## aronpm (Dec 31, 2010)

**** GUYS, CALL THE MATH POLICE

SOME FORMULAS MULTIPLY PI BY 2


----------



## some1rational (Dec 31, 2010)

@aronpm: I don't know if your being sarcastic or not, but the point is not that the factor of 2 is confusing for anyone who is even relatively acquainted with mathematics (trigonometry, complex analysis, etc.).

The point is to look at it through "the eyes of a child". A quarter turn is 1/4 of Tau, but a quarter turn is 1/2 of Pi. That, plus all the other arguments given, makes me more than convinced that Tau should be the 'natural' choice. Anyway, more discussion is always appreciated. If I can convince just one other person on this forum, I feel like I've done something lol.


----------



## Keroma12 (Dec 31, 2010)

This is very convincing, and does make many good points. It would be interesting to see somebody defend pi.


----------



## cmhardw (Dec 31, 2010)

Wow... I'm convinced. I like the significance of Tau. The rewritten Euler's identity was what sealed it for me. Pi does feel somewhat like my S-slope after reading that.


----------



## some1rational (Dec 31, 2010)

Yay! lol thank you for responding Chris. If this thread was a competition, convincing you is like first place prize for me haha.


----------



## maggot (Dec 31, 2010)

i found this to be really stupid. 
lets declare another variable (as if there wasnt enough already) thats completely insignificant!!!
2pi = tau. 
retarded. 
and it seems as though his motive is because he had a hard time grasping radian measure.
pi has its coup de grace in pi r^2... actually pi has its coup de grace because its one of the most significant numbers used in math, regardless of wether its multiplied or divided to make it look "ugly". there are a lot of ugly formula out there that need some help... PMCC's, standard deviations... just about any statistics formula lol... they're all horrible imho.


----------



## einstein00 (Dec 31, 2010)

lololololol.

In section 3, Hartl makes a big deal about the circle area formula being A = pi*r^2, when several physics formulas take the form y = 1/2*k*x^2. He then proposes that we change the formula to A = 1/2*tau*r^2. This is completely unnecessary: C = 2*pi*r, so A = 1/2*2*pi r^2 = pi*r^2. But because C = pi*D, A = 1/2*pi*D^2, which follows the general form y = 1/2*k*x^2. He's just playing with variables and numbers to fit his persuasive needs, when his argument is actually invalid.


----------



## y3k9 (Dec 31, 2010)

The quarter turn did it for me. But what I still don't understand what pi and tau are fighting for? Being called the most important constant relating to circles?


----------



## some1rational (Dec 31, 2010)

einstein00: actually A = 1/4 * Pi * D^2, you did your calculation wrong

A = Pi * R^2 = Pi * (D/2)^2 = (Pi * D^2)/4 which deviates from the standard of quadratic forms.


maggot: Fair enough, everyone is entitled to their own opinions and I merely posted this to see what people would say. But in (my?) defense, if your a seasoned math/physics/engineering veteran, defining variables is half your job. The usage of Tau, imo, simplifies this process. Also, Michael Hartl was a professor of theoretical/computational physics at Caltech, I'm pretty sure grasping radian measures is the least of his worries.

y3k9: What Hartl is arguing for (the fighting you refer to), is that Tau is the more natural choice and is more readily understandable by a person who has little to no experience with mathematics (e.g. a child). The point being that, perhaps more people would be able to grasp the idea, or to take up mathematics as a hobby/profession/etc if the change were to take place.


----------



## blakedacuber (Dec 31, 2010)

i had a thought that it was wrong

then i asked my maths tweacher why pi was 3.17 and not 3.12 etc and she couldnt answer


----------



## keemy (Dec 31, 2010)

Yo I have actually heard this augment before but it didn't really convince me then and not now either, sorry. My problem is there are really no practical differences (just aesthetics) which doesn't warrant any actual change imo.


----------



## y3k9 (Dec 31, 2010)

blakedacuber said:


> i had a thought that it was wrong
> 
> then i asked my maths tweacher why pi was 3.17 and not 3.12 etc and she couldnt answer


Weird, I always thought pi was irrational and started with 3.1415... but if you say it's 3.17 I believe you because you obviously are some genius here.


----------



## some1rational (Dec 31, 2010)

keemy: Your right, it's purely aesthetics. But the aesthetics has the advantage of being simpler (a las it's still just an opinion) from the vantage point of the unacquianted. Also it allows for an understanding of analogous ideas (quadratic forms). Anyway, if reading it is not convincing, then there is nothing I can do. I still love Pi, but I feel Tau is the better choice.


----------



## BigSams (Dec 31, 2010)

keemy said:


> Yo I have actually heard this augment before but it didn't really convince me then and not now either, sorry. My problem is there are really no practical differences (just aesthetics) which doesn't warrant any actual change imo.


 
Dude. Yes, aesthetics plays a big factor; we're human, we like pretty stuff. But on the more practical end, if the ratio of simplified to complicated formulas as a result of switching to tau is more than 1, this could make a lot of computer programs more efficient.


----------



## einstein00 (Dec 31, 2010)

ooh, we like pretty stuff! e^(i*pi) + 1 = 0 vs. e^(i*tau) = 1 + 0. Which is prettier? Does Hartl really think that adding a "+0" makes the equation more meaningful? If so, then (e + i + pi)^0 = 1! I am a genius!


----------



## cmhardw (Dec 31, 2010)

keemy said:


> Yo I have actually heard this augment before but it didn't really convince me then and not now either, sorry. My problem is there are really no practical differences *(just aesthetics)* which doesn't warrant any actual change imo.


 
Normally mathematicians are obsessed with elegancy in math. A proof must be short and _elegant_. A concept must be able to be summarized in a clear and concise way that is easy to understand. Also, things in seemingly unrelated topics generally end up being consistent with each other through simplifying them this way.

This manifesto presents a _really_ strong case that \( \tau \) is the _more elegant choice_ compared to \( \pi \), and I would tend to agree based on what I've seen so far.


----------



## vcuber13 (Dec 31, 2010)

blakedacuber said:


> i had a thought that it was wrong
> 
> then i asked my maths tweacher why pi was 3.17 and not 3.12 etc and she couldnt answer


 
are you sure she couldn't answer or that she wouldn't dignify that with a response?


----------



## keemy (Dec 31, 2010)

some1rational said:


> keemy: Your right, it's purely aesthetics. But the aesthetics has the advantage of being simpler (a las it's still just an opinion) from the vantage point of the unacquianted. Also it allows for an understanding of analogous ideas (quadratic forms). Anyway, if reading it is not convincing, then there is nothing I can do. I still love Pi, but I feel Tau is the better choice.


Don't get me wrong I'd have nothing against seeing someone use tau (though I would except them to define it first because most people wouldn't know what they mean). But you can't really expect someone to change what they use over something trivial/subjective.




BigSams said:


> Dude. Yes, aesthetics plays a big factor; we're human, we like pretty stuff. But on the more practical end, if the ratio of simplified to complicated formulas as a result of switching to tau is more than 1, this could make a lot of computer programs more efficient.


Not really considering multiplying or dividing by 2 is just a digit shift in binary.



cmhardw said:


> Normally mathematicians are obsessed with elegancy in math. A proof must be short and _elegant_. A concept must be able to be summarized in a clear and concise way that is easy to understand. Also, things in seemingly unrelated topics generally end up being consistent with each other through simplifying them this way.


Yes I agree 'elegance' does seem to be important to many mathematicians and there is a good reason for it to (easier to read and verify proofs ect.) but standardization is also important and pi is the standard and I just don't think the reasons why tau 'should' be justify a change.


----------



## maggot (Dec 31, 2010)

cmhardw said:


> Normally mathematicians are obsessed with elegancy in math. A proof must be short and _elegant_. A concept must be able to be summarized in a clear and concise way that is easy to understand. Also, things in seemingly unrelated topics generally end up being consistent with each other through simplifying them this way.
> 
> This manifesto presents a _really_ strong case that \( \tau \) is the _more elegant choice_ compared to \( \pi \), and I would tend to agree based on what I've seen so far.


 

I also agree that tau is definately more elegant in a lot of applications, however, pi in itself is elegant. Simply reintroducing 'something new' can be more misleading. I like elegant proofs where systems are integrated for simplicity, but this is just taking the number 2 out of the equation... for me, this is unreasonable. It would be more efficient to declare things like standard deviation variables, so that proofs are more pleasant to the eye.


----------



## qqwref (Dec 31, 2010)

cmhardw said:


> This manifesto presents a _really_ strong case that \( \tau \) is the _more elegant choice_ compared to \( \pi \), and I would tend to agree based on what I've seen so far.


I agree too. \( \tau \) does seem more elegant. Seems like \( \pi \) is much more likely to appear next to a 2 than not. I don't think it is elegant to suddenly replace everything with a \( \pi \) in it, but I think math would be prettier if we had been using \( \tau \) from the start.

And sorry y'all, the \( e^{i \pi} + 1 = 0 \) identity is nowhere near as elegant or amazing as people will tell you.



BigSams said:


> this could make a lot of computer programs more efficient.


No, not really, since it's just a change in notation. And any really efficient math computation program should have 2pi as a hardcoded constant anyway.


----------



## einstein00 (Dec 31, 2010)

qqwref said:


> Seems like \( \pi \) is much more likely to appear next to a 2 than not.


 
volume of sphere: \( 4/3\pi r^3 = 2/3\tau r^3 \). Neither is more "elegant" than the other.

surface area of sphere: \( 4\pi r^2 = 2\tau r^2 \). Again, neither is more "elegant" than the other.

Volume of circular cone: \( 1/3\pi r^2 h = 1/6 \tau r^2 h \). Ditto.

SA of cone: \( \pi r^2 + \pi r l = 1/2\tau r^2 + 1/2\tau r l \). "annoying" fractions "get in the way".


----------



## qqwref (Dec 31, 2010)

einstein00 said:


> volume of sphere: \( 4/3\pi r^3 = 2/3\tau r^3 \). Neither is more "elegant" than the other.
> 
> surface area of sphere: \( 4\pi r^2 = 2\tau r^2 \). Again, neither is more "elegant" than the other.
> 
> ...


You're right, these aren't really any more elegant either way. In these cases you pretty much have to admit that it doesn't matter which one you choose.

But if you do more complex stuff with trigonometry, calculus, Fourier series, differential equations, statistics, etc. then you will see \( 2\pi \) crop up by itself a LOT.


----------



## cmhardw (Dec 31, 2010)

lol I just thought of a good logo for \( \tau \) supporters:

\( \tau > \pi \)

It's true mathematically, but it also has the psychological implication too.


----------



## aronpm (Dec 31, 2010)

einstein00 said:


> volume of sphere: \( 4/3\pi r^3 = 2/3\tau r^3 \)[/math]



Hartl must write another manifesto explaining why we need a new symbol to represent \( 2\tau \)


----------



## Keroma12 (Dec 31, 2010)

cmhardw said:


> lol I just thought of a good logo for \( \tau \) supporters:
> 
> \( \tau > \pi \)
> 
> It's true mathematically, but it also has the psychological implication too.


 
Haha, I love it.

Also, imagine if we had begun by using \( \tau \), instead \( \pi \). Many people would find it silly if someone wanted to use \( \pi \), because they would be used to a circle being 1\( \tau \). I agree that it doesn't matter _that_ much, but I still like \( \tau \) over \( \pi \)... for now at least.


----------



## y3k9 (Dec 31, 2010)

Keroma12 said:


> Haha, I love it.
> 
> Also, imagine if we had begun by using \( \tau \), instead \( \pi \). Many people would find it silly if someone wanted to use \( \pi \), because they would be used to a circle being 1\( \tau \). I agree that it doesn't matter _that_ much, but I still like \( \tau \) over \( \pi \)... for now at least.


Instead of pi, write \( \tau/2 \)


----------



## Keroma12 (Dec 31, 2010)

y3k9 said:


> Instead of pi, write \( \tau/2 \)


 
Good idea :tu


----------



## kinch2002 (Dec 31, 2010)

I now see why we should have used Tau from the start. But I fear it's too late 
It certainly would be more beautiful for Mathematics if it were switched, but Maths is already beautiful enough for me


----------



## da25centz (Dec 31, 2010)

brilliant. purely brilliant. I will tell my calc teacher about this right away


----------



## y3k9 (Dec 31, 2010)

kinch2002 said:


> I now see why we should have used Tau from the start. But I fear it's too late
> It certainly would be more beautiful for Mathematics if it were switched, but Maths is already beautiful enough for me


Switched? No. But added, yes if we try.


----------



## maggot (Dec 31, 2010)

cmhardw said:


> lol I just thought of a good logo for \( \tau \) supporters:
> 
> \( \tau > \pi \)
> 
> It's true mathematically, but it also has the psychological implication too.



i lol'd

im too lazy to look up 2pi applications... but we should compile a list of formulae that use the 2pi.

pendulum T=2pi (L/g)
normal distribution . . . yeah. . 
fourier transforms. . again . . yeah. . . can someone latex these with tau? 
stirling's, spouge's, and lanczos approximations. . .


----------



## Lucas Garron (Dec 31, 2010)

BigSams said:


> But on the more practical end, if the ratio of simplified to complicated formulas as a result of switching to tau is more than 1, *this could make a lot of computer programs more efficient*.


Huh?


----------



## BigSams (Dec 31, 2010)

keemy said:


> But you can't really expect someone to change what they use over something trivial/subjective.


It's because of people you that we're stuck with the current calendar system -_- 



keemy said:


> Not really considering multiplying or dividing by 2 is just a digit shift in binary.


Hmmm? I was referring to cases where there is a factor of 2 like \( C=2\pi r=2\frac{ \tau}{2} r= \tau r \). Saves an operation. I'm not qualified to say that the simplification/no change to more complication ratio is more than 1, though.

EDIT: @ LG, like if you had to find the circumference of a million circles, you just saved a million operations.


----------



## cmhardw (Dec 31, 2010)

y3k9 said:


> Switched? No. *But added*, yes if we try.


 
I agree with this. I've been talking to one of my good friends from high school, who is finishing up a PhD in graph theory, about this. His opinion is that there are certain instances where using \( \pi \) is more elegant. In Euclidean geometry the sum of the angles in a triangle is \( \frac{\tau}{2} \) which is not as elegant as just saying \( \pi \). Also the sum of the angles in a 2-dimensional k-gon is \( \frac{\tau}{2}(k-2) \) which is not as elegant as \( \pi(k-2) \)

His opinion was that \( \tau \) will likely be the mathematical equivalent of the dvorak keyboard layout - embraced by some but not all. I guess it's neat to have both, since each one can be more elegant in certain situations. I will certainly include \( \tau = 2\pi \) whenever using \( \tau \) in any work that I write, and I will not use this in the classroom as it would probably confuse students. Still, I personally find \( \tau \) to be very useful. I mean look at the labeling of the unit circle using \( \tau \) - so much clearer to understand in my opinion.


----------



## Kapusta (Dec 31, 2010)

Very cool, it makes pi look much less interesting than what it is made out to be. Perhaps over the next generation such a number will be adopted.


----------



## y3k9 (Dec 31, 2010)

I got a perfect idea, let's just divide \( \tau \) by 2 and you get \( \pi \), and so we can use \( \tau \) to find the digits of \( \pi \)! jkjk.


----------



## Keroma12 (Dec 31, 2010)

cmhardw said:


> I mean look at the labeling of the unit circle using \( \tau \) - so much clearer to understand in my opinion.


 
This was the point that convinced me. I'd always thought it was odd to have 2\( \pi \) in a circle. It makes so much more sense to a beginner to use 1\( \tau \) for a circle.


----------



## Ranzha (Dec 31, 2010)

Tau = 2 * pi. Cool story.
Sure, it's significant, but I don't think this will "obtain superiority" over pi.


----------



## y3k9 (Dec 31, 2010)

Ranzha V. Emodrach said:


> Tau = 2 * pi. Cool story.
> Sure, it's significant, but I don't think this will "obtain superiority" over pi.


Did you read the article???


----------



## maggot (Dec 31, 2010)

i agree with chris's friend from high school. this is why i say tau is elegant in some cases, however, pi in itself is also elegant. there are just as many elegant formulae with pi, as i could see with tau... i do agree with the computational ease with tau, seeing as 2pi comes up a lot in computations in elementary maths (trig calc DE's PDE's statistics etc)


----------



## y3k9 (Dec 31, 2010)

maggot said:


> i agree with chris's friend from high school. this is why i say tau is elegant in some cases, however, pi in itself is also elegant. there are just as many elegant formulae with pi, as i could see with tau... i do agree with the computational ease with tau, seeing as 2pi comes up a lot in computations in elementary maths (trig calc DE's PDE's statistics etc)


Yeah, I suggest tau be taught *in addition to* pi.


----------



## maggot (Dec 31, 2010)

then it becomes really interesting when you start looking at things like dedekind eta function. . . 
http://en.wikipedia.org/wiki/Dedekind_eta_function
i might use it for myself. but when presented to a public audience, i would take it out again... unless it became extreeeeemly elegant like eulers identity.


----------



## gpt_kibutz (Dec 31, 2010)

That is actually a very interesting article. I think it would be fun to use tau instead of pi in some equations/demonstrations and see what other people think xD


----------



## aronpm (Dec 31, 2010)

maggot said:


> then it becomes really interesting when you start looking at things like dedekind eta function. . .
> http://en.wikipedia.org/wiki/Dedekind_eta_function
> i might use it for myself. but when presented to a public audience, i would take it out again... unless it became extreeeeemly elegant like eulers identity.


 
\( q = e^{\tau i \tau} \)

Mathematics is now 18+++


----------



## some1rational (Dec 31, 2010)

Yea, for the record, I'm not saying we must all switch now. I just think it makes more sense from a pedagogical point of view. It can happen incrementally, as Hartl put it.

As a side note, I like Chris's point about how in Euclidean geometry the interior angles of a triangle must sum up to Pi. From my point of view, there seems to be almost something metamathematical about how Pi (half a turn, half of Tau) is intimately connected with the parallel postulate in this sense (equivalent to interior angles of a triangle sum up to Pi); whereas going higher or lower gives one spherical and hyperbolic geometries, respectively.


----------



## EnterPseudonym (Dec 31, 2010)

The unit circle sealed it for me. I wasnt able to understand it all though. High school calc doesnt really help. I also agree it should be taught along side pi.


----------



## qqwref (Dec 31, 2010)

Looking at the triangle angles from a tau perspective:
- Imagine a line segment on one of the sides, the length is irrelevant (make it longer than any of the sides, say) - just pay attention to the slope.
- Rotate the line segment around each corner by that angle, so that it goes to the second side, then the third side, then back to the first.
- In Euclidian geometry, the line has clearly now been rotated by the sum of those angles. And it is pretty clear in an intuitive sense (if you know the line is parallel to the first side again) that the line has been flipped by half a turn.
So the sum of the angles of a triangle is \( \tau/2 \).


----------



## some1rational (Dec 31, 2010)

o0o nice one qqwref, that's a very intuitive way to look at it.


----------



## Stefan (Dec 31, 2010)

some1rational said:


> The coupe de grace was really the final straw for me. I have to commend Michael on a job well done in this manifesto. So what are your thoughts?



_"Look I found some ugly formulas, let's make the circle area formula just as ugly and pretend that's better."_
:fp

Besides, \( \tau r \) just looks terrible, \( \tau \) and \( r \) look too similar.



aronpm said:


> **** GUYS, CALL THE MATH POLICE
> SOME FORMULAS MULTIPLY PI BY 2



Yeah, reminds me of the 9/11 conspiracy theorists who see 11 everywhere.



BigSams said:


> like if you had to find the circumference of a million circles, you just saved a million operations.


 
Why would the compiler/interpreter not precompute that, multiply the two constants right away only once and use their product afterwards?


----------



## some1rational (Dec 31, 2010)

Stefan said:


> _"Look I found some ugly formulas, let's make the circle area formula just as ugly and pretend that's better."_
> :fp


 
lol, maybe it's just me having used those quadratic forms (mass/spring, Hooke's approximation) so much that they've grown on me. I guess beauty is in the eye of the beholder on this subject.

I have to clarify that I don't know whether or not tau is the best choice as a variable to use. But I am convinced that 2*Pi deserves it's own character and that it really is the more 'natural' circle constant.


----------



## Kapusta (Dec 31, 2010)

Stefan said:


> Yeah, reminds me of the 9/11 conspiracy theorists who see 11 everywhere.



I actually just skimmed that, and passed off as junk, when I looked at my clock... and it was 9:11! :O


----------



## Kynit (Dec 31, 2010)

Very cool! I would be very interested in seeing where this goes. I do agree that the tau and r look a little too similar, but I love the points that were made for the numerical value.


----------



## Quadrescence (Jan 1, 2011)

wow whoever discovered tau is a genius

I invented *iota* which represents 2*pi*_i_, check it out

_e_^*iota* = 1

cauchy's integral formula: _f_(_a_) = 1/*iota* integral _f_(_z_)/(_z_-_a_) d_z_

the nome _q_ = _e_^(*iota* tau)

what do you guys think??

I always thought pi was wrong, go *tau & iota*!!!


----------



## y3k9 (Jan 1, 2011)

Quadrescence said:


> wow whoever discovered tau is a genius
> 
> I invented *iota* which represents 2*pi*_i_, check it out
> 
> ...


How would iota be useful?


----------



## Quadrescence (Jan 1, 2011)

y3k9 said:


> How would iota be useful?


 
iota representa one full complex turn

also note that three iotas together look like pi

it would vastly simplify complex analysis for example

also you can't deny the beauty of e^iota = 1


----------

