No, really, pi is wrong: The Tau Manifesto
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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!
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.
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.
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.
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...
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.
Specialization sometimes makes sense, but it's non-obvious (at least it wasn't to me) that when checking if a force field is conserved by computing it's rotation, you're doing the same thing as when computing the derivative of a differential to see if it belongs to a conserved thermodynamical potential, or that the difference between a bra and a ket is the same as between a covector (lower index in Einstein notation) and a vector (upper index) - things look so different that it's hard to see when they are the same.
Another example is the relation between Newtonian and Lagrangian mechanics. In the lectures I took, it was presented as if Lagrangian mechanics is somehow special because you have an invariant formulation using generalized coordinates, wheres Newtonian mechanics was only ever done in Euclidean or Minkowski space.
It turns out that Newtonian mechanics is as invariant and general as Lagrangian mechanics (however, it's possible to further generalize Lagrangian mechanics, whereas as far as I can tell, you're pretty stuck with second-order system when doing Newtonian mechanics):
The Euler-Lagrange-equations are Newtonian equations and the differential of the Lagrange function dL is just a funny way to write down a force field - ie the main difference between Newtonian and Lagrangian formulation is that you require your force to be derived from a generalized potential (more formally: every hyper-regular Lagrangian system is a Newtonian system, any Newtonian system where the force maps to a closed form under the isomorphism T* TM ~ TT* M is locally Lagrangian).
In my opinion, lectures on theoretical physics are somewhat broken, and that's a more serious problem than the non-issue of whether to use τ or 2π…
PS: Please don't get me started on Christoffel symbols if you're not prepared for another rant ;)
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?
And you don't mess with John Baez ;-)
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
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.
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
You should also note that the circumference of the unit circle is 1 tou.
Not only is it a great answer to the original question ("How many angels can dance on the head of a pin") but it is a great engineering principle as well.
So I agree with the GP who says that use of two competing variables which differ only by a factor of two is redundant and runs up against this principle.
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.?)!
>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.
And it also represent a null character in linguistics, empty set in some set mathematics, null-pointer in some computer science texts, etc.
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.
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.
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.
http://tauday.com/tau-manifesto#sec:the_pi_manifesto_a_rebut...
C = 2 * Pi * r = Tau * r
A = Pi * r ^2 = Tau / 2 * r
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.
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.
EDIT: And shear stress. Derp. That's why I write software now. :)
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).
Also, that's why we have subscripts.
They do use subscripts though.
Mind you, if I get deep into integration and the only reflexes from my Jackson days kick back in, the notation reverts entirely. At some point it magically switches back after the act of integration is done, though.
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 ...
http://www.kickstarter.com/projects/ianterrell/say-no-to-pi
*Free if I can get it to you with a stamp in the US.
I'd buy a "say no to tau" patch.
Sure, pi*r^2 is nice and easy to remember, but also remember even the mnemonic is wrong... pies are round!
Also, e ^ (-i * pi) + 1 = 0
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?
:-)
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.
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?
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.
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.
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.
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.
Tau in Ruby: http://bugs.ruby-lang.org/issues/4897 (ignored)
:(
Incidentally, there's already a symbol for \tau: 2\pi. Same number of syllables and characters as 14, so what's the problem?
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.
Anywho, the point stands. They're simplifications for our benefit.
Visually it looks wrong, pi looks like 2 tau, not the reverse.
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!
> 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.
Kudos to author though, I think most of mathematicians out there just take math as it is. Lack of entrepreneurial spirit I'd say.
Really? People whose job is continually creating new math daily are just taking math as it is?
math builds on earlir work
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.
If you haven't gotten around to it, today's a good day.
#define M_TAU = (2.0*M_PI)
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.
See what I did there?