After conferring with some fellow mathematicians, nobody had heard of the "tau movement", and nobody cares. Tau is for people who want to feel like they're taking stances on things that matter in math without actually putting in any of the work to do math.
I think tau is really just motivated by people who like things to be "clean". You see them in all walks, wishing programming language syntax, math notation, spelling, grammar, etc, had been chosen to be cleaner or make more sense from the beginning.
I don't have a strong opinion, but from what I've seen I think tau makes the most sense. These things are fun to think about, because it requires that you think about how pi is used in mathematics and contemplate how we organize and use the concepts related to pi. It's a good brain exercise.
But how much cleaner is using tau, beyond grade-school geometry? Euler's equation is arguably the most beautiful relationship in mathematics: e^(ipi)= -1. I'm afraid tau would dirty it up.
Not only that. If you think about multiplying any number z by e^(ia), where a is a real number, as rotating z by a radians in the complex plane, then i think the tau version of the Euler's equation conveys a more fundamental meaning: rotating any complex number by a whole turn yields that same number, it's the same as multiplying it by the multiplicative identity (i.e. doing nothing).
e^(iτ) = 1
And, as rotating something by one whole turn is the same as rotating it by any number of whole turns, you can get a nice intuitive series of equivalences that are not as pretty if you where to use pi:
Can people really not separate out a factor of 2? There's weirdness in every language. If I started SUDDENLY speaking English with a more uhh... orthogonal grammar, people would maybe get what I was thinking, but also deem me a self-righteous asshole. This is EXACTLY how mathematicians see tau. Math is no more flexible than any other widely spoken language.
Yes, once you have learnt the basics, factoring out a 2 is trivial. But that doesn't make it right to make children that still haven't learnt that wonder "what's the angle of 3/4 turns?". One and half pi? Why? It's 3/4 tau. It should be trivial. There shouldn't need to be any conversion by a factor of 2, especially for people that are starting with that stuff (angles and trigonometry is where lots of kids get lost at maths).
I think it's not too much about doing maths, but instead about teaching maths. It's much easier to explain to a kid that the angle of a slice of pizza is tau/8 (that is, if you cut your pizzas in 8 slices), and a quarter of a pizza has a tau/4 angle, and so on, instead of their pi equivalents. The idea that 1 tau = 1 turn is very powerful and simple.
anecdote, N=1: I was explaining some basic trig to a child and while I was at it also explained the concept of tau. The kid said he liked tau better :) (I was careful to also explain that his schoolteachers probably didn't know about tau so he needed to know 2pi as well)
I can see the utility in programming, since defining Tau to be 2 * pi potentially saves a lot of multiplication operations (although a good compiler would optimize these away anyway).
But I agree that it's a distraction, and I don't think the radial relationship is that important. I got over my annoyance of typing 2 * pi by thinking about it as the ratio of the diameter to the circumference.
I neither know nor care. It doesn't make geometry any more difficult to conceive of it that way, so I've been happy to do so. I do a lot of compass & straightedge type geometry for fun so it wasn't hard to adapt.
I disagree, but setting aside the radius/diameter debate, consider the simple equation: area=pir^2. How is it not more difficult to think of the constant in terms of diameter, when the rest of your equation is in terms of the radius. Unless you use area = 1/4pi*d^2, which I have never seen.
I'm not telling you what to do, I'm telling you what I like to do. Yes, I do sometimes use (pi * d^2)/4. It's not more efficient, and I didn't claim it was (though if I were concerned with efficiency I could argue that pi * d is a pleasingly simple way to calculate circumference).
I'm sorry that my personal idiosyncrasy bothers you so much.
I use tau every time I do something that would typically involve pi. It is simply cleaner. I view the tau movement more as a symbol of cleaning up math, and pi/tau is simply low hanging fruit.
Clearly the idea of pi and tau are both wrong, since they make the 2d circle the reference instead of the 1d circle.
Notice the formula for the volume of an n-ball is proportional to pi^(n/2)r^n! Define tau properly, and could become (tau r)^n!!
Also notice how many other cases outside of 2d geometry uses sqrt(pi) or sqrt(2pi). The gamma function, normal distribution, stirling approximation etc. are all full of it, and when you rarely need pi or tau, adding a ^2 is much nicer than always having to draw a big ugly sqrt.
I don't mind tau especially, but it is seems the wrong battle to fight. Sqrt(pi) also has mythical powers: http://www.squarerootofpi.com/
I'm not against being open to things like ancient aliens, but I think the fact that squarerootofpi references that and astrology, etc puts it on completely different footing from tau, which the post we are discussing right now mentions being included in programming languages and scientific papers. [edit: obviously this says nothing about square root of half tau and its uses!]
I put tau into my math-related code last year, and I have enjoyed the results (which are much the same as before, of course).
You can construct a new series such that every term in the old one is there. However, if you try and rearrange the terms you end up with a backlog that coverages to the difference between the series. Which is why the actual series converges in the first place.
I don't think so. The stupid thing about pi is that it is the only place in math that you ever talk about the diameter of the circle, everything else is expressed in terms of the radius.
Admittedly, the radius does feel like the natural way to define a circle (all points r away from the center), however that is not the only way to define a circle, and we have no other intelligent species to use a reference for how objectively natural that way is.
We're saying the same thing. I'm using that as an example of why it seems like radius is the right one. Of course we have no examples, but we do know that an alien intelligence would be free from the historical accidents of humanity. "Mathematics is the only truly universal language", as Sagan would say.
Here's a similar argument: In mathematics, do we measure angles in radians or diameterans? Why is that?
>that is not the only way to define a circle
Ok, let's hear 'em! What rigorous definition of a circle is more simply expressed in terms of diameter?
The only one I can think of is, "the smallest volume shape that can contain a length D line segment at any angle", but that seems contrived to me. I shutter at using the solution to optimization problem as a definition, except when there is no other choice. What's worst, it seems like a trivial derivative of the radial definition, just with extraneous concepts tacked on.
The most natural, diameter based, definition I can think of is, for any given point on the circle, the distance to the farthest point is a constant, D. You probably need to add additional restrictions to make this rigorous (such as requiring the circle to be form a continuous, closed loop).
You could also imagine rotating a line segment about its center. This has a nice sense of symmetry, as we are rotating about the center, not an arbitrary end. This also implies a generalization to off center rotations.
Wow this is hilarious. Truly a "manifesto" to set a constant for 2pi. It definitely has a great point, and I totally agree that it's a better constant than pi, but it's not the worst historical accident of mathematics.
The thing is, tau is really my go-to variable when I need a 2nd constant to compare with t. It is already used as a time constant, a dummy integration variable that substitutes for t... The manifesto has a pre-made counter-argument to this... But I'm not sure it's convincing.
The thing is that we like to say some variation of e^(-tau) a lot, and we also like to say e^(2 pi i) a lot. And often we combine these (rotation and exponential decay), and we get e^(-tau+2pi i). And this would ruin tau notation! It would be e^(-tau'+tau i)...
I really like using tau as a time constant, too, and I'm sorry to lose it when there are pi's in the picture. It's tough picking good notation, though, and unfortunately no choice is conflict-free.
Before writing The Tau Manifesto, I tried several other candidates before settling on tau. After using it for a while, tau started to "feel right" in ways that other choices didn't. Then, as noted in the manifesto, I found out that I wasn't alone:
Since the original launch of The Tau Manifesto, I have learned that Peter Harremoës independently proposed using τ to “π Is Wrong!” author Bob Palais in 2010, John Fisher proposed τ in a Usenet post in 2004, and Joseph Lindenberg anticipated both the argument and the symbol more than twenty years before!
Yeah, I've literally never seen lower-case pi used as a variable (thank goodness!). tau is a good choice because it looks like pi, and is possible for normal people to write (I would have had a fit if it were lowercase xi!).
I like the explanation page for why tau a lot, and I think that page is much more important than actually using tau in real mathematical works. It's a great explanation of how 2*pi is the more natural quantity, and how it comes into play in the different functions. But it's just much easier to deal with factors of 2 than it is to explain alternate notation.
In fact, I think a lot of making mathematical works easy to understand is using commonly accepted notation...
Why didn't you make the circle constant a circle in your manifesto? I'm not sure but it seems the function composition symbol could be safely reused without confusion or perhaps a larger circle with a drawn radius. Wouldn't a circle have avoided some of the Tau criticisms?
The major problem is that a circle is too ambiguous a symbol; you'd need to add a notch or line or something to make it clear that you're referencing a constant. And when you do that, it becomes harder to write.
I always favored a variation on ◷, something like ◯̶ or ◯̵ or ○̵. Unfortunately, combined glyphs are representationally very unstable.
Yea, the thing about a circle constant (whether pi or tau) is that it's so ubiquitous you can't use the symbol for anything else. It's pretty rare to use lower case pi for anything in physics and math, and most of the exceptions including something like boldface or a vector hat to distinguish it.
There are no available symbols. The only option is to introduce a new one, like ת suggested by samatman, or possibly by using a weird typeface (like \mathfrak in LaTeX).
Incidentally, my friend has pointed out to me that sigma is a much better choice than tau if we're going with existing symbols. Who cares about tau sounding like a "t" for "turn", when sigma actually looks like someone trying to measure the circumference of a circle with the radius of the circle: σ.
σ has potential, but there's a pretty bad conflict with standard deviation. The circle constant shows up an awful lot in statistics.
Who cares about tau sounding like a "t" for "turn"
Pi comes from perimeter, phi comes from Phidias, etc. There's a long tradition of naming constants using a linguistic root (typically Greek). To my knowledge, there's no precedent for using a symbol based on its visual appearance to a geometric object. (Of course, you could always break with tradition.)
If you want it to look round, uppercase theta IMO is a much better option, especially some of its older forms http://en.wikipedia.org/wiki/Theta. There may be some collisions with its use to denote 'an angle', but that must be a solvable problem given that there are 29 (sic) different theta-like Unicode code points (and that ignores the archaic variants)
Why not just start in on Chinese characters? There's plenty of those and I'm sure Chinese mathematicians could suggest some likely candidates based on their long mathematical traditions.
The use of infinitesimals in calculus! Things like integrating against "dx" or taking a derivative of a function "df/dx" make calculus seem like something it's not. Worse, it really introduces a whole new language whose rules for manipulation are unclear.
Like if we have a function f(x,y), we often write the "total" derivative of f as
df = (df/dx) dx + (df/dy) dy
And the rules for manipulating a differential like this are quite strange (e.g., dx on top cannot cancel the dx on bottom).
There's a language of differential forms that makes this notation a bit more rigorous, but I still think the notation is very misleading and confusing, especially for those who begin to learn calculus.
To be fair to mathematics, in your example the proper notation is df = (del f/del x) dx + (del f/del y) dy, which does not suggest the possibility of cancelation.
Pi is just fine. It makes just as much sense as tau, has history on its side, is simpler for many formulas, etc. The Pi Manifesto has good counterarguments and heaps of examples: http://www.thepimanifesto.com/
