No, really, pi is wrong: The Tau Manifesto

by llambda • 14 years ago

186 110 14 years ago

tzaman• 14 years ago

Sometimes I just feel stupid. Reading this article is one of those times :)

Kudos to author though, I think most of mathematicians out there just take math as it is. Lack of entrepreneurial spirit I'd say.

archgoon2• 14 years ago

> I think most of mathematicians out there just take math as it is.

Really? People whose job is continually creating new math daily are just taking math as it is?

serge2k• 14 years ago

unless there is a reason not to.

math builds on earlir work

huxley• 14 years ago

Sure, creating new stuff doesn't necessarily imply throwing everything out each time you make an addition or change.

It would be like a startup guy re-inventing alarm clocks and breakfast every morning, followed by re-inventing showering, shaving, dressing, opening the door, entering the car, starting the car, what lane you drive in, etc.

You'd never get to your workplace to do whatever it is that your startup is creating ("We Are Re-inventing Innovation!" -- you might think I'm joking but that's a common tagline even PARC used it).

Some stuff you take as it is (a baseline) and you create around it.

Maybe tau is something to consider more seriously but pi has done pretty well for itself and many don't consider changing it to be a big priority.

ianterrell3• 14 years ago

Email me your address for a free* patch if you're the type to wear one.

http://www.kickstarter.com/projects/ianterrell/say-no-to-pi

*Free if I can get it to you with a stamp in the US.

jff• 14 years ago

Why not make another one, "say no to tau", and see which gets the most buyers? :)

I'd buy a "say no to tau" patch.

ctdonath• 14 years ago

[ack, fat-fingered the downvote icon. sorry. someone please upvote!]

andylei• 14 years ago

http://www.thepimanifesto.com/

ddlatham• 14 years ago

Be sure to check out the rebuttal.

http://tauday.com/tau-manifesto#sec:the_pi_manifesto_a_rebut...

onemoreact• 14 years ago

That's hardly a meaningful rebuttal.

  C = 2 * Pi * r = Tau * r
  A = Pi * r ^2 = Tau / 2 * r

PS: When dealing with radians Tau is fine, but I like showing +Pi and -Pi both point away from 0.

mhartl4• 14 years ago

The awesome thing is, today he posted his own rebuttal:

http://spikedmath.com/fact-005.html

Dove3• 14 years ago

‎#define tau 6.2831853

If you haven't gotten around to it, today's a good day.

hdevalence2• 14 years ago

Wouldn't it be better to do

    #define M_TAU = (2.0*M_PI)

instead, so that it's clear that it's a math constant instead of a local variable? Also, I suspect that the math.h pi constant is accurate to more than 8 decimal places, so this way you don't lose precision.

Dove3• 14 years ago

Whatever is appropriate in your context. :)

What I really mean is this: use it. If you're a tauist, but tau isn't in your header file / constant library / whatever you use, today might be a good day to put it there.

bascule• 14 years ago

The best part of this is the area of a circle, which is the integral with respect to r of rtau, or taur^2/2.

Sure, pi*r^2 is nice and easy to remember, but also remember even the mnemonic is wrong... pies are round!

DrHankPym2• 14 years ago

It's also easier to remember that the area of a unit circle is pi.

Also, e ^ (-i * pi) + 1 = 0

RuggeroAltair5• 14 years ago

What's the kinetic energy of a unit speed, unit mass?

What's the potential energy of a unit distance unit spring constant?

What's the energy of a unit mole, unit temperature in Boltzmann units?

:-)

4ad2• 14 years ago

π is not "wrong", that doesn't make any sense. However, I agree with the author that expressing things by using π is more complicated and not as elegant as expressing things with τ.

FaceKicker• 14 years ago

Fourth paragraph:

> It should be obvious that π is not “wrong” in the sense of being factually incorrect; the number π is perfectly well-defined, and it has all the properties normally ascribed to it by mathematicians. When we say that “π is wrong”, we mean that π is a confusing and unnatural choice for the circle constant.

4ad2• 14 years ago

Yes, the article is correct and I agree with it, I just feel the wording is unfortunate.

nandemo2• 14 years ago

I don't give a damn about tau. On the other hand, the way π　iｓ rendered on Firefox on Windows is so wrong... it looks just like a small version of Π (capital pi), or the Cyrillic small Pe:

http://en.wikipedia.org/wiki/Pe_%28Cyrillic%29

gbaygon• 14 years ago

TL; DR: “Let τ=2π”

It's a simple substitution, and it should be used "when it's convenient to".

I really don't see any innovation in the article as everything in it is directly derived from the above mentioned substitution.

sophacles2• 14 years ago

It isn't innovation from a math sense, it's innovation in a pedagogical sense. Once you've internalized pi (and math in general) it is pretty hard to see what is so difficult about it (a well studied issue with expert knowledge and teaching things to novices). However a simplification from the point of view of the novice who doesn't "just get it" is a good thing.

tokenadult• 14 years ago

However a simplification from the point of view of the novice who doesn't "just get it" is a good thing.

Questions about mathematics pedagogy are inherently empirical, and should be answered by observation of actual learners. So where is the evidence that learners who don't get how to use π will be better able to learn mathematics if they use τ to tackle the same problems?

jessriedel• 14 years ago

Using τ is better than using π for the same reason (albeit not as strong a reason) that using π is better than using g=17π. It has little to do with convenience, and more to do with conceptual clarity. Actually, the difference between g and π might be less bad, because at least when mysterious factors of 17 appear you would know where they came from. On the other hand, the factor of 2 that rides along when you choose to use π over τ could come be confused with a million other factors of 2.

polemic3• 14 years ago

pi is "wrong" in the same way as imperial measurements are "wrong".

philwelch2• 14 years ago

pi is "wrong" in the same way as non-Planck units are "wrong". SI is almost as arbitary as imperial.

polemic3• 14 years ago

As in - because pi is a unit-less ratio? Plank units are still units (albeit a special case), so I don't agree with your syllogism.

However, what I meant was the human processing overhead in converting between units. I mean, pi is correct, for what it is. Tau would simply be easier for humans to use, much like SI units.

angersock5• 14 years ago

As a mechanical engineer by training, I will be a sad panda if you use tau for something other than torque.

EDIT: And shear stress. Derp. That's why I write software now. :)

aqme28• 14 years ago

Tau is used for tons of other things already. http://en.wikipedia.org/wiki/Tau#Modern_usage

I wish each letter could have only one use, but sadly that isn't possible (like some notorious equation from my plasma physics days that forced us to use different versions of P and Rho for density, pressure, charge density, and momentum).

Jach4• 14 years ago

I'm more of a fan of having multi-character symbols than to try and pigeonhole every concept into one symbol. Have full or partial names, just like we do in programming. We started down that path with writing things like sin x, cos t, div f, |rad|, etc. We should continue! If people really have problems distinguishing we can go the Perl route and add sigils.

FrojoS• 14 years ago

Fellow Mechi here. Yes, they can do that tau revolution without us. In the dynamic equations, tau is also used for the "generalized forces" (a vector/array which contains all the forces and torques that act on the generalized coordinates). At the same time, they are usually full of trigonometric functions (with 2*pi).

I can only recall one time in my academic career where someone used pi for something else than a time constant or circle constant. It made me feel so uneasy ...

orbitingpluto2• 14 years ago

I always enjoy these little diatribes.

However tau will never beat out pi because of one very important reason. The chicken scratch that passes for writing these days makes tau unreadable. Tau can be a: r, 7, t, tau, etc...

For the sanity of TAs everywhere, please stop this madness.

Jach4• 14 years ago

It's just a pi with one leg in the middle. It's pretty easy to read. Also engineers (who don't have very good writing) have been using tau for things like a time constant (among other uses) for a long time.

The bad handwriting problem is solved very easily, TAs should complain to the professor to make it a requirement that if you guys can't read it, it's wrong. I've had classes like this, I just use LaTeX.

DrHankPym2• 14 years ago

Yeah, I remember using Tau mainly for time constants back in school.

http://en.wikipedia.org/wiki/Time_constant

orbitingpluto2• 14 years ago

I did a symbolic dynamics course that required the letter xi ad nauseum in lowercase. I still can't write that damn letter.

madhadron2• 14 years ago

Really? I just took off over half the points for probles that I couldn't easily read until handwriting improved. Students with illegible chicken scratch suddenly developed perfectly readable handwriting overnight. End of problem.

For my part, I write two very clear faces, an italic for my prose, and a block for my math. Writing clearly isn't difficult. The vast majority of people are already capable of it. Most of them at some point internalized that they had bad handwriting, perhaps rationalized because they were writing fast, and so write poorly because it's an element of their identity.

As for tau, I could see using a symbol for 2pi removing a class of error when I'm calculating, so I might adopt it. However, I doubt I'd use pi. More likely ð.

woodall• 14 years ago

So π is a reduced version of τ?

msluyter• 14 years ago

For a quick overview of the pedagogical significance of tau, compare figures 6, 7, and 8. I learned figure 6 in high school trig, but by "learn" I mean I basically memorized it without a deep understanding of what was going on. Figures 7 & 8 unveil what I had been missing.

brudgers• 14 years ago

Is it that time of year again?

InclinedPlane• 14 years ago

Nope, this is about 3 and a half months early.

angersock5• 14 years ago

well played good sir/madam

bartl2• 14 years ago

March 14. Yep.

oflannabhra• 14 years ago

This argument gets pretty complex, but the thing I like about pi is that it is fundamental. Sure 2 * pi might occur more commonly in important expressions, but 2 * pi is only special because pi is special.

ianterrell3• 14 years ago

Is it circular reasoning to suggest pi is only special because 1/2 tau is special?

See what I did there?

mcdaid• 14 years ago

Hi posted this earlier, but it didn't get any traction. I guess the people here might be interested.

http://www.visnos.com/demos/pi#launch

haberman3• 14 years ago

I find the argument for tau very convincing. I remember having similar thoughts when I learned about radians and had a hard time understanding why they would define pi to be half of a turn, particularly since the diameter equation was d=2πr. The fact that A=πr² seemed like a counterargument, but the Tau manifesto makes a good case that having 1/2 in this equation helps show its similarity to other physics equations.

I think pi's best argument is that the area of the unit circle is pi. I actually find it very enlightening to think of the unit circle as having an area of pi but a turn of tau radians.

It makes sense to me to use both pi and tau; whichever makes a particular set of equations more clear. I certainly don't think that pi should be preferred just because it's older, or that pi's prevalence should prevent introduction of a new symbol.

adambyrtek2• 14 years ago

Occam's razor to the rescue. Using two distinct constants which differ only by a factor of two sounds more confusing than choosing either one and sticking to it.

T-zex• 14 years ago

Diameter is 2*radius, so 2pi as another constant isn't that bad. pi being expressed in terms of diameter and tau in terms of radius.

scythe3• 14 years ago

See also: h and ħ in physics.

haberman3• 14 years ago

Occam's razor pertains to competing hypotheses that attempt to explain some phenomenon, not the question of which of several equivalent formulations is best.

If you argue that there can only be one, I say tau is better. But to me it's less confusing to have both tau and pi, simply because both of these graphics seem very fundamental to me:

http://tauday.com/images/figures/tau-angles.png http://www.thepimanifesto.com/areas.png

chimeracoder• 14 years ago

> Using two distinct constants which differ only by a factor of two sounds more confusing than choosing either one and sticking to it.

Physics is incredibly unusual in that it has very few overloaded terms, and so almost everything that is expressed in daily use is expressed with as much precision and as little ambiguity as possible. Most scientific disciplines are not so lucky.

It does still happens in physics (h and h-bar, as noted in another response below), but it happens all the time in statistics, so much so that it's incredibly frustrating to read a new text for the first time.

𝜀? Probably refers to the error of a regression, assumed to have mean zero and be independent.

e? Uh-oh. Possibly refers to the error term in a regression, as above. But it could also refer to the residuals, which always have mean zero and are never independent. Very different.

ê? Okay, once I see this, I know that the author probably doesn't use 𝜀 as well, so this narrows it down somewhat. But not entirely.

And don't get me started about 'standard error'. I have heard that term used in reference to a sample mean, a sample mean divided by the square root of the population, or the standard error of a regression (which is more complicated than I care to describe in plain English).

Do you want to get started about economists, who often use π as a variable? Or computer scientists, who use it to refer to a process calculus?

How about Σ, which can be used to indicate a sum, but can also be used to indicate a covariance matrix? Which, by the way, can also be expressed in terms of either S or Q, depending on who you ask.

Ooh! How about Λ? That's the precision matrix, so it has to be precise, right? Well, yes, except that all we're doing is writing the inverse of Σ, so it's an unnecesary letter altogether[1].

Right now, we're dealing only with conflicts within a given field, but we could open Pandora's box and talk about the fact that λ is an anonymous function in computer science, but a constant that (partially) defines the stationary point for a given optimization problem.

Or letters that look like each other - can you really tell the difference between ν and v? Or ω and w? What about when I write them out by hand?

[1] Unless you're using the term to implicitly declare that the covariance matrix is invertible, but if you think that substituting one capital Greek letter for another is a clear way of telling me that a matrix is full-rank, we need to have a much longer conversation.

EDIT I must say, I'm very impressed that HN handled all that unicode beautifully. Kudos to pg (& co.?)!

valgaze• 14 years ago

Are there any punny desserts/treats/snacks that one can make with "Tau"?

maaku2• 14 years ago

pi is “wrong” in the same way that mobile electrons being the carrier of negative charge is “wrong”

RuggeroAltair5• 14 years ago

Nope. Although they are both conventions, in the electric charge there is really no natural choice. While mathematicians can argue whether or not a factor of 2 is a natural choice for a convention.

maaku2• 14 years ago

Isn't the “natural” choice to have the direction moving charged particles to be the same as a moving charge? In everyday life it is electrons, not protons that move. When we say "charge is moving from high voltage to low voltage" we are actually saying "electron particles are moving from a position of low voltage to high voltage"--the exact opposite. How is that natural?

Even if that weren't the case, it's still a matter of opinion. Minus signs show up in a number of pretty unnatural positions as a result of the negative electric charge. The convention in Physics is that minus signs convey semantic information (reversal in direction, slowing down, etc.). The negative electric charge upsets this convention, resulting in un-semantic minus signs.

abrahamdemoivre• 14 years ago

Good article, and your arguments are strong. As long as we're being contentious, though, I would like to point out that e is a more interesting number than τ or π. Sure, you can make circles with π (or τ) and do angular calculations galore, but without e you've really got nothing more than that. Rate of growth? Compound interest? Hidden things of the universe? Fuhgedaboutit.

mhartl4• 14 years ago

The Tau Manifesto author here. Don't yell it too loudly, but I strongly agree: e is by far the most important of these numbers. But e is the natural choice, so I don't have any bone to pick with it. (Bob Palais made an analogy in "Pi Is Wrong!" Suppose e were defined as 1/2.718281828... Then there would be confusing negative signs everywhere. Exponential decay would have a positive exponent, etc. Such is the case with τ and π, with a factor of 2 in place of a factor of -1.)

bartl2• 14 years ago

In math, you see 2π everywhere, for example in the Fourier Transform. So in that regard it'd make more sense if that was the constant.

Visually it looks wrong, pi looks like 2 tau, not the reverse.

mhartl4• 14 years ago

The revision of The Tau Manifesto that launched today includes a wealth of new material, including a rebuttal of the "Pi Manifesto" (http://tauday.com/tau-manifesto#sec:the_pi_manifesto_a_rebut...) and a new section on the volume of a hypersphere:

http://tauday.com/tau-manifesto#sec:volume_of_a_hypersphere

If you thought the original quadratic-form argument for circular area was good, prepare to be blown away by n-dimensional spherical volumes.

chalst2• 14 years ago

Nonsense!

The Leibniz sequence is the most simple way to describe the equivalence class under multiplication by non-zero rationals of numbers that contains both pi and tau:

1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - ...

which sums to equivalently pi/4 or tau/8.

The fundamental simplicity of the Leibniz sequence makes this value the only natural choice from that equivalence class. Let us call it lambda, for Leibniz.

Then the area of the unit circle is 4 lambda, the diameter of the unit circle is 8 lambda, the volume of the unit sphere is 16 lambda, the surface of the unit sphere is 16/3 lambda, etc.

Should I start selling the t-shirts?

RuggeroAltair5• 14 years ago

I wouldn't call it nonsense. If the comparisons with the virial theorem and energy isn't enough to consider it at least as a reasonable choice, at least you should consider it as such because John Baez agrees with it.

And you don't mess with John Baez ;-)

chalst2• 14 years ago

I'm attempting irony. The tone of the manifesto is a bit breathless, putting up interesting enough considerations in a overwrought manner, with half-hearted efforts made at looking at the issues raised from more than one point of view.

Draft one of my lambda manifesto shows that the only consideration of conceivable value, namely ease of defining a transcendental by means of a convergent series, leads to only one possible choice of fundamental trigonometric constant, lambda.

Of course to gain widespread acceptance, I will have to cover up having written the indulgently tolerant http://www.advogato.org/person/chalst/diary/275.html

adharmad• 14 years ago

Euler's identity: e ^ (i * pi) = -1 will be uglier: e ^ (i * tau/2) = -1

ianterrell3• 14 years ago

Maybe read the linked page? Maybe see section 2.3?

pfanner• 14 years ago

The next 100 years people will ask "what is tau?" and one will say "it's 2pi".

scythe3• 14 years ago

I am a graduate student, first-year, so I have a lot of problem sets to work on. I've been thinking about tau every so often since the last time this was posted here, and so every time I work a problem, I think to myself: would this be clearer if I used tau? And the answer is usually "no". Sometimes 2 is next to pi because of the natural reason appealed to in the tauifesto, and other times it's there for some other reason, other times it isn't there at all; there's usually a huge constant factor out front anyway, and often a single equation goes across the whole sheet of paper, so even making it one character shorter doesn't accomplish much.

Not that I haven't learned notation that really helps. Gaussian units and Einstein notation are a godsend. If you could standardize introductory physics courses using these, I think it could help significantly, especially when people struggle to work out the curl of a cross product or some-such strenuous vector calculation. All hail the Levi-Civita tensor! But we haven't even been able to agree on this.

Furthermore, as a test grader of nearly-a-year, the work done by undergraduates is nigh-inscrutable in the maximally acceptable way already, and so the tau "rebellion" promoted here would just make my job harder. It's not too much of an issue: only the technically-inclined, who already do pretty clear work, are likely to use it; still, telling people "just start doing your work like this, let them figure it out!" means that I have to know if that scribble is a tau or a T or what-have-you, two hundred times. Pi is a very recognizable character.

Plus, tau is the letter I reach for whenever I need to introduce an adjusted time of some sort, such as proper time; it's also the natural temperature in stat mech (though thermodynamic beta is itself more natural and usually better), it's torque, it's a common time constant, &c. People generally avoid pi as a character which does not represent 3.14159, though it is the prime counting function and an adjusted momentum, in which case it usually has a half-arrow, so you can tell what it's doing. We don't avoid tau at all; it's everywhere.

So the tau-switch, as notational improvements go -- and math has had many over the years -- seems like a relatively large-pain, small-gain deal. Many things must change, since pi is all over the place, but few are greatly improved. So I don't see much reason to use tau in my work, or for my students to use it in theirs. If it works for you, though, good!

haberman3• 14 years ago

To me it has almost nothing to do with "making it one character shorter," it's about making it easier to see fundamental relationships intuitively. It's not a notational improvement as much as a model improvement.

It's sort of like how angles can be measured using degrees or radians, but the derivative of sine is cosine only when using radians. It's not the notation of radians that is better, it's the model.

loup-vaillant2• 14 years ago

The biggest advantage of Tau over Pi probably lies in learning trigonometry for the first time. That's not something you can judge for yourself, as you can't learn something for the first time twice.

We should perform experiments, but I strongly suspect that using Tau (or 2Pi as if it where a single symbol) would significantly reduce confusion in the heads of the pupils. That would count as a "big advantage" in my book.

Now does Tau have an actual disadvantage besides clashing with time constants and such? If not, we should keep in mind that switching is a one-time cost while the cost of using a worse notation is unbounded (proportional to the number of uses, actually).

I have to reckon however that making an effort to switch away from Pi probably shouldn't be our first priority. I don't know Gaussian units nor Einstein notation, but if they do "really help" then we probably should take care of that first.

kiba• 14 years ago

The biggest advantage of Tau over Pi probably lies in learning trigonometry for the first time. That's not something you can judge for yourself, as you can't learn something for the first time twice.

If you can live long enough, you could literally forget your trigonometry. After all, humans are not exactly the pantheon of long term memory or even...reliable memory.

cygx2• 14 years ago

Einstein notation is nice when you actually have to compute something, but it doesn't help in understanding the geometry of the problem.

In fact, it might be more of a hindrance because it encourages thoughts along the general line of I don't need to care about the geometry of the problem as long as I know how to calculate the stuff I'm interested in, or as a more specific example It doesn't matter if there's an upper or lower index as I can always contract with the metric tensor.

A personal pet peeve is when tensors are introduced as entities with given transformation laws (transforms like a vector in each component, etc) without ever mentioning specific geometric meanings.

Thing is, classical tensor calculus makes everything look the same - even things that aren't. I prefer the 'modern' coordinate-free notation of differential geometry (which has been around since at least the 60s), and it's easy to introduce Einstein notation on top of it...

scythe3• 14 years ago

For me, Einstein notation had the major advantage in electrostatics of reeling in the complexity of really long integrals that show up in boundary-value problems in electrostatics as well as being able to rederive formulas like "curl of curl is div-grad minus laplacian" by hand in seconds (which I always forget -- I took Calc ).

I had only a relatively cursory introduction to general relativity last semester; I have at best a vague understanding of Christoffel symbols, to give you an idea. So if Einstein notation can at some level become a way to fling symbols around and forget you're doing physics, I guess I haven't gotten there yet. I would like to think, though, that were it introduced alongside vector calculus instead of several years later, people might connect the adscripts with their meaning more easily.

Usually, if I want to understand the geometry of a problem, though, I find the best tool is a diagram, if at all possible.

archgoon2• 14 years ago

Let me guess, Biology grad?

vibrunazo• 14 years ago

The benefit of the easier formulas tau might provide. Minus. The confusion of having different generation of students being taught differently.

Is it worth it?

The author tries to dismiss this by saying "it's easy, they'll get it quickly, we don't need to rewrite all textbooks if you can just say 'let tau = 2pi'". But it seems he's rushing in that conclusion. The imagine the confusion between trying to convince students who are used to a whole set of formulas to use new ones will be huge. The confusion caused by different generations trying to communicate seems huge. You might not need to burn old books but you would need to write new ones, which again does sound like a huge endeavor.

I'm not even questioning whether he's right about tau formulas being easier. That seems irrelevant to me. Just the problems you're creating with the confusion seems not worth the small benefits. It just doesn't seem the pros outweigh the cons.

tfm• 14 years ago

Aw dang, I missed Pi Day again! Stupid timezone. I guess tau will be getting my support this year ...

randomibis2• 14 years ago

Tau in Python: http://bugs.python.org/issue12345 (rejected)

Tau in Ruby: http://bugs.ruby-lang.org/issues/4897 (ignored)

verve2• 14 years ago

Yeah, yeah, and degrees are wrong for expressing angles, and steering wheels are wrong for driving cars.

Incidentally, there's already a symbol for \tau: 2\pi. Same number of syllables and characters as 14, so what's the problem?

verve2• 14 years ago

Why tau, BTW? Why not winnie, after the Wonder Years character portrayed by the eminent mathematician Danica McKellar?

randomibis2• 14 years ago

Your question is answered in the article...

RockofStrength• 14 years ago

Both pi and tau form infinite strings of prime numbers with their digits. See if you can get the first twelve!

mitko• 14 years ago

math background and then studying robotics for a bit, where all we have are models of the reality, I now laugh at such comparisons.

Both Tau and Pi are models for mathematics that work- to argue which is true is nonsense. They are both correct but neither is true, or god-chosen. We made them up, folks!

No, really, pi is wrong: The Tau Manifesto

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