Tau replaces Pi - Numberphile

PHIL MORIARTY: We're talking about something called tau, which is equal to 2 pi. So there's been a huge amount of debate and discussion about this right across the internet. There's even a Tau Manifesto. So one of the real advocates of tau versus pi is ViHart, and I really recommend you go and visit her videos. Brady might put the link in somewhere-- maybe down here or up there or over there-- somewhere. For some people, it's not an issue. And initially, my response was what are people arguing about this? This is total nonsense, because if you want to use tau, then you just go tau-- I'll repeat it-- tau is equal to 2 pi, and then proceed. It's more than that though, because, with maths and with physics, what you want to strive for all the time is clarity, particularly when you're teaching this stuff. It's far too easy, if you've been teaching this for 15 years like I have, not to be able to put yourself in the mindset who's doing this right from the start. And so if you're got schoolchildren who are coming to trigonometry, who are coming to angles, who are coming to pi and circles and circumferences and radius for the first time, you do not want to confuse them. You want to minimize the confusion. And moreover, in maths and in physics, a really important property is symmetry. And you want to try and have the most elegant, concise description of something. Here's my very wonky circle. And let's put our center there. And that's our radius r. And from there to there, you all know, I hope-- let's do it in a different color-- is d. And we define pi, very simply, as the ratio of the circumference to the diameter. And this holds true for any circle, regardless of its size. Pi crops up in very many different ways. And the first thing we've got to think about is how we define angle. So let's draw another circle on this very crumply brown paper. If we start here at this point and we move around and come back to the same point, that's how many degrees, Brady? BRADY HARAN: 360. PHIL MORIARTY: 360 degrees. So there's a much more natural way of defining an angle. Let's take a circle again. Here's our radius. We're going to call that r again. And let's say that, instead of walking all the way around, we instead walk a distance r along here. So that's r as well. And we call angles, conventionally, in physics-- you can call it whatever you like, but conventionally, we denote it by theta. That angle, when this arc length is equal to the radius, is equal to 1 radian. OK. So that's a much more natural definition, because here, we're basing it on just fundamental properties of the circle. Instead of going, well, that's 360 degrees, we're saying that when we go around a certain arc length which is equal to the radius, that's a radian. Now, the question is, how many radians do we get if we go all the way around the circle? How many radians does that turn out to be? Well, it turns out to be 2 pi radians. And the reason is-- because you remember that the circumference length is 2 pi r. All right? So 2 pi radians. Here's where the debate kicks in. Some, particularly Bob Palais-- I hope I've pronounced that right. It all stems from this wonderful paper, "Pi is Wrong"-- Bob Palais. What he argued-- well, isn't this rather confusing? So you go around one turn, and instead of being pi radians, it's 2 pi radians. I have a daughter who's nine and another daughter who's six. And my daughter who's nine is starting to look at shapes and starting to look at angles, doing basic geometry. And I can see how this is going to be a big problem. I'm looking around for the paper. And what we're going to do now is we're going to look at a number of different angles. So we define angles counterclockwise. So we go that way rather than that way. Quite why we do that, I have no idea. It's convention. Here's where it gets interesting. 0 radians-- exactly the same. We go around a quarter of the circle. That angle is not pi over 4. It's pi over 2. That's already confusing. We go around the circle a bit more. Instead at of being pi over 2, because we've gone half way around, it's pi. And here, it's 1 and 1/2 pi-- 3 pi over 2-- and here, it's 2 pi. A natural unit is one turn. A natural unit is you go around once. And yet, when you go around once, that's 2 pi. Now, I think that's confusing, not just at a basic elementary level, but that use of 2 pi instead of pi or instead of something which represents one turn, which represents one revolution-- well, let's just redefine that. Let's say that tau is equal to 2 pi. One turn is equal to 2 pi. And let's just put the pi thing to one side. So that's one turn. So we start off with 0. How far have we gone? Well, we've gone a quarter way around. We've done a quarter of a revolution. That's tau over 4-- much more natural. This is tau over 2. Because we've gone halfway around, it's natural. What's this one, Brady? BRADY HARAN: That's going to be 3/4 of tau. PHIL MORIARTY: Exactly. And this one's going to be tau, and then 2 tau and 3 tau and 4 tau and 5 tau. And that's much more natural. And of course, any professional physicist or mathematician or scientist is going to go, well, it's straightforward. If I want to use that, I just write that and I continue my equation, my analysis. I continue my derivation using tau. But that's not the point. It's not about the professional physicists. It's about teaching this stuff. 2 pi appears throughout physics, all the way from the very large scale, all the way right down to quantum mechanics. And I'll get on to quantum mechanics in a sec. Let me show you a simple demo, first of all. Mass on a spring-- we let that bob up and down, it performs something which is the bane of so many first year undergraduates' lives in physics, which is called simple harmonic motion. It goes up and down. What describes that? Oh, we call it circular motion. And what describes that is a sine wave. And the period of that motion, again, is 2 pi. In terms of how atoms vibrate, atoms vibrate all the time in molecules. We describe that in terms of an angular frequency. We describe in terms of a frequency, but often, we use something called omega, which is an angular frequency, which is 2 pi times f. And this is where it gets really, really interesting, because even as physicists, we even extract out that factor of 2 pi, because we have something called Planck's constant. It's a fundamental constant that crops up throughout quantum mechanics. And we actually redefine it. And we have got something called h-bar. And h-bar is equal to h over 2 pi. And one of the reasons this 2 pi factor crops up time and time again throughout physics is that when we get down to the quantum level, matter behaves like a wave. It's got oscillatory motion associated with it. It moves back and forth with a period. We can describe that in terms of waves, which have a period of 2 pi. It crops up absolutely everywhere. Here's my favorite equation. That is called Fourier transform. And as you'll see, you've got 2 pi here. But, not only do you have 2 pi there-- that's an omega, which is 2 pi f-- you've got a 2 pi factor and there's another omega up here. What that does-- BRADY HARAN: That's got 2 pis everywhere. PHIL MORIARTY: That's got 2 pis absolutely everywhere. Now, what we could do-- and you can just replace this instead of having 1 over root 2 pi, have 1 over tau here, have 1 over root tau. That's not the point. I really don't think that's the point. If you go to the various websites-- where are they? The tau manifesto and the pi manifesto-- and they have this game where they bat each other back and forth. Look, this equation's more simple if you put tau and this equation's more simple if you put pi in. That's not the point. It's about the fundamental teaching of this. For professionals physicists, we're not going to change this. I really do not think we are going to change this. If you open up any textbook, there's a 2 pi. 2 pi there, actually. There's one already earlier on. So open a book up at random. You're very, very likely in a first-year physics textbook or in any physics textbook to come across a 2 pi. We are not going to change these textbooks. That's not the point. The point is when you're teaching this stuff, tau is a better representation, is a better concept. The idea that you do one turn and that's tau, as opposed to 2 pi radians. That's just a much better way of explaining it. BRADY HARAN: If no one's going to change, why are we even talking about it? Are just we just saying this is what we should have done? PHIL MORIARTY: Yes, it is what we should have done. Euler, I think, possibly had an opportunity to change this hundreds of years ago. We should have done it. We haven't done it. Perhaps we could, from this point onwards, think about writing textbooks and at least mention tau. I think what we need to do is you can-- perhaps in a textbook like this, throughout it's going to talk about 2 pi. But we could say right at the start that there is this concept of tau-- many students find it very helpful-- and introduce it. And perhaps, then, there will be a gradual evolution towards tau. I doubt it. I really doubt it, but it might be nice to think so. -You've got to keep in your mind when you're doing calculations-- and actually, if you're a seasoned mathematician-- PHIL MORIARTY: --the sampling and the effects and frequency shifting to get it into the right pitch. But I'll play you Vi's vocals again, because they are really quite neat, if I can find the channel. -And also, that 1/2's here, because the half tells you something. -This works out just as nicely.