
No, really, pi is wrong: The Tau Manifesto - llambda
http://tauday.com/tau-manifesto
======
scythe
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!

~~~
loup-vaillant
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
_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.

~~~
loup-vaillant
If we ever come to that, I want a brain update.

------
chalst
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?

~~~
RuggeroAltair
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 ;-)

~~~
chalst
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>

~~~
RuggeroAltair
+1 for irony anyways...

------
haberman
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.

~~~
adambyrtek
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.

~~~
chimeracoder
> 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.?)!

~~~
wging
Would you mind telling me these characters?

>the author probably doesn't use 𝜀 as well,

>𝜀? Probably refers to the error

I'd like to know which my browser (latest Chrome on up-to-date Win7) is
failing to render.

~~~
ghayes
It's a lowercase Epsilon character.

~~~
wging
Thank you.

------
orbitingpluto
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.

~~~
Jach
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.

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

~~~
Dove
One of my math profs in grad school called lower case xi "a scribble". And
when he had to write it on the board, it was.

~~~
ars
Pic: <http://en.wikipedia.org/wiki/File:Xi_uc_lc.svg>

------
andylei
<http://www.thepimanifesto.com/>

~~~
ddlatham
Be sure to check out the rebuttal.

[http://tauday.com/tau-
manifesto#sec:the_pi_manifesto_a_rebut...](http://tauday.com/tau-
manifesto#sec:the_pi_manifesto_a_rebuttal)

~~~
onemoreact
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.

~~~
baby
> I like showing +Pi and -Pi both point away from 0.

What would be cool is to see two different points showing what the direction
is.

c = Pi/2

+c would point upwards, -c would point downwards. 4c is a pizza.

------
mhartl
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...](http://tauday.com/tau-
manifesto#sec:the_pi_manifesto_a_rebuttal)) 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.

------
msluyter
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.

------
angersock
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
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).

~~~
Jach
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.

~~~
zokier
Now that I have been programming for longer than doing maths, I just must
wonder why math community seem so entrenched in using single letter
variable/function names. How confusing larger source codes would be if
programmers did the same.

~~~
angersock
You've never looked at academic or numerical code, have you? It's totally
fugly. Go skim the first edition of Numerical Recipes in C--it'll give you
nightmares.

------
ianterrell
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
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.

------
bascule
The best part of this is the area of a circle, which is the integral with
respect to r of r _tau, or tau_ r^2/2.

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

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

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

~~~
RuggeroAltair
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?

:-)

------
gbaygon
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.

~~~
sophacles
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
_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?

~~~
sophacles
I don't know. Perhaps they are still at the "convince someone to actually try
it for an entire n-year math curriculum" stage? It's not like using Tau is
something that you can do in class for a week and expect better results. Even
if you try the lesser "here's tau, always substitute 2pi with tau" every
couple years on a group of students and track them through their math
learning, it takes time and a lot of effort to get those numbers.

In the mean time, there is lots of anecdotal evidence of people understanding
a lot of concepts easier with it, suggesting these studies be done in a
scientific way.

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

~~~
RuggeroAltair
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.

~~~
maaku
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.

~~~
RuggeroAltair
I use a different definition of natural. The fact that the electrons,
negatively charged, were moving wasn't known when they assigned the positive-
negative labels. But still, although what you are saying is certainly true in
proton/electron systems, it would be completely reversed in
antiproton/positron systems, where the charge would be moving together with
the particles. Also, don't forget of ions, or particles in space, or in
particle physics experiments, where both charges move.

The point is that the natural choice that you assign would be natural just
because of some contingent conventions, but it's not more natural in terms of
some more fundamental/mathematical meaning. While the pi vs. tau is.

------
vibrunazo
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.

------
brudgers
Is it that time of year again?

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

~~~
angersock
well played good sir/madam

------
randomibis
Tau in Python: <http://bugs.python.org/issue12345> (rejected)

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

:(

------
verve
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?

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

------
mcdaid
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>

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

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

~~~
polemic
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.

~~~
philwelch
Tau is better than pi because a lot of useful expressions have 2pi as a term;
SI is better than imperial because we happen to have ten fingers. Tau is a
universal simplification like Planck units; SI is merely an anthropocentric
simplification.

~~~
Skalman
I think the greatest advantage of the SI system is that it's exponential. I
don't care _that_ much about what base is used, but switching is just
confusing (12" = 1', 3' = 1yd).

------
bartl
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.

------
mitko
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!

------
4ad
_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
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.

~~~
4ad
Yes, the article is correct and I agree with it, I just feel the wording is
unfortunate.

~~~
breckinloggins
The wording is marketing.

~~~
mhartl
True, and it's not even my marketing; it comes from Bob Palais' original
article "Pi Is Wrong!".

------
tzaman
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.

~~~
archgoon
> 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?

~~~
huxley
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.

------
Dove
‎#define tau 6.2831853

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

~~~
hdevalence2
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.

~~~
Dove
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.

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

------
abrahamdemoivre
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.

~~~
mhartl
_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.)

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

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

------
woodall
So π is a reduced version of τ?

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

~~~
ianterrell
Maybe read the linked page? Maybe see section 2.3?

------
oflannabhra
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.

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

See what I did there?

