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Tau

The mathematical constant pi is under threat from a group of detractors who will be marking “Tau Day” on Tuesday.

Tau Day revellers suggest a constant called tau should take its place: twice as large as pi, or about 6.28 – hence the 28 June celebration.

Tau proponents say that for many problems in maths, tau makes more sense and makes calculations easier.

It makes more sense but does it make it easier? Well, not really. I think anyone who can even pretend to be able to hack trigonometry really can cope with a few factors of 2 kicking around.

“I like to describe myself as the world’s leading anti-pi propagandist,” said Michael Hartl, an educator and former theoretical physicist.

I once knew someone who played a sport internationally for England – unicycle polo. You gotta have a hobby.

“When I say pi is wrong, it doesn’t have any flaws in its definition – it is what you think it is, a ratio of circumference to diameter. But circles are not about diameters, they’re about radii; circles are the set of all the points a given distance – a radius – from the centre,” Dr Hartl explained to BBC News.

Yes, admitted. τ does perhaps have that elegance and such thoughts had occurred to me when I were a lad but… nobody who handles π outside the context of Greggs knows that Dr Hartl. There are other logically equivalent definitions of a circle but if you ask any student 999/1000 that is the definition they’ll give despite that evil π.

By defining pi in terms of diameter, he said, “what you’re really doing is defining it as the ratio of the circumference to twice the radius, and that factor of two haunts you throughout mathematics.”

I think “haunts” is putting it a bit strongly. As I hinted before I struggled learning a lot of the concepts and machinery of mathematics as a student (everyone does apart from geniuses and liars) but a factor of 2 kicking about in there was the least of my worries. Indeed in a field such as complex analysis which has more &pis; than John Prescott’s freezer to complain about those pesky factors of 2* when you got poles at infinity (something The Express would approve of) and Laurent series and even the dreaded Bromwich contour to worry about is like complaining about the in-flight meal when the aircraft is at nearly π/2 radians from the horizontal and both engines are on fire. “Stewardess, I don’t think you understand, I specifically requested the vegetarian option at check-in!”.

Dr Hartl reckons people still use degrees as a measure of angle because pi’s involvement in radians makes them too unwieldy.

Er, no. People use degrees because 360 is a very easily divisible number. π or τ aren’t being (a) rather small and (b) transcendental and therefore irrational. In order for Dr Hartl to be right on this point you’d have to imagine this scene on a building site with the foreman upbraiding an underling over a wonky door fitting, “Does that look like τ/4 radians to you pal?” It is not going to happen. Can you even conceive of the mathematical joinery posse? Imagine the discussions! “Well yeah, but that’s only if we accept Euclid’s fifth…” And that’s just applied mathematicians. Let the logicians and set theorists loose and they’d be bijecting the set of screws with the set of holes…

Dr Hartl is passionate about the effort, but even he is surprised by the fervent nature of some tau adherents.

“What’s amazing is the ‘conversion experience’: people find themselves almost violently angry at pi. They feel like they’ve been lied to their whole lives, so it’s amazing how many people express their displeasure with pi in the strongest possible terms – often involving profanity.

I don’t condone any actual violence – that would be really bizarre, wouldn’t it?

Well for a certain value of “bizarre” possibly but having met a lot of mathematicians it’s hard to say really. I’d love to see the expression on the face of a beak who was more used to dealing with yobs who got into a barney because someone had called someone’s “pint a puff” deal with a pair of dishevelled looking men with the leather patches hanging forlornly from their torn tweed jackets because one had called the other’s τ a 2π.

The (real, apocryphal, whatever) disagreements of medieval theologians are often mocked by our modern men of science as being an exercise in playing les buggeurs risible but dear me! (I can’t believe this but I’m actually going to start getting serious here). The history of mathematics has indeed seen some massive changes in notation which have dramatically improved the usability of mathematics. The Great Grandpappy of course is Indian numerals rather than Roman and before that even place-ordering (thank you Babylon!). But you also have Leibniz notation over Newtonian for The Calculus*** and things like Boolean algebra and vector and tensor notation**.

Maybe τ is neater (sometimes – τ/2r2?) but Hell’s teeth I’m typing this on a QWERTY keyboard which is hardly ideal is it? It ain’t going to change. Anyway π is a cultural icon par excellence. It symbolises mathematics in a way both subtle and profound becuse it bestrides both the pure and the applied. It is to paraphrase my Grandmother one of the few things that separates use from things “eating shit in the trees.”

And it goes without saying that re-writing the whole corpus of mathematics for τ rather than π would be a monumental and confusing Children’s Crusade. An almost Borgesian endeavour.

Anyway I’d have to stop referring to people I disagreed with totally as being π rads off kilter. τ/2 rads doesn’t have the same kerching.

*What’s wrong with 2? Damn fine number if you ask me.
**I am aware that the last three aren’t just new notation but the idea of tackling Maxwell’s equations without vector or tensor notation is really scary.
***Newtonian is still used in some areas.

23 Comments

  1. Roue le Jour says:

    But tau chart isn’t a pune. Us computer types are very keen on our bad jokes.

    Obviously when he says people still use degrees he doesn’t mean plumbers, chippies, metal bashers, etc. etc. you know, all those dreadful tradesmen, he means proper people who have an indoor job with no heavy lifting at an educational establishment.

    Here’s one for ya, prof. Take this ruler and go measure the radius of that pipe.

  2. I think you’re being unfair. The Tau manifesto is incredibly convincing.

    For those of us who already understand pi and radians, you’re right, it’s simply quibbling. However, at the time of teaching, pi is an added confusion in an already confusing subject.

    I’m not really sure you’ve understood:

    “Does that look like τ/4 radians to you pal?”

    That’s the sort of thing he has to say in pi-world. In tau-world, he’d actually say, “does that look like a quarter of a circle to you?” Much nicer. Radians would simply be “percentage of a whole circle” (having been redefined to be in terms of tau not pi).

    Tau is so natural that a child could understand it; and understand it far better than they do 360 degrees. Since the way you begin with degrees is by saying “360 degrees is a whole circle” — in other words, degrees are taught in terms of tau already.

    Obviously no books will be rewritten; but tau is what pi should have been. We should lament the missed opportunity.

    One period is one tau is one circle. Lovely.

  3. john in cheshire says:

    He’s obviously a socialist and that should be sufficient to condemn his blatherings. Don’t argue with them, for goodness’ sake; it’s not worth the effort.

  4. CountingCats says:

    tau is what pi should have been.

    Absolutely true.

    Who cares?

  5. View from the Solent says:

    Angels dancing on the (circular) head of a pin. Something to do with pinheads, anyway.

    (btw, 360 degrees is Babylonian again; they used base 60)

  6. Roue le Jour says:

    I can’t see how using pi rather than tau stops you saying ‘quarter of a circle’, you could say that now, and indeed some of us practical types do, in the form of turns. Inductors, for example are so many turns. A very small one might be say, two and a half turns. Many mechanical adjustments are ‘half a turn clockwise” etc.

  7. View from the Solent says:

    An afterthought.
    Tau would make Euler’s identity lose its elegant beauty.

  8. I’m not foaming-at-the-mouth about this; I just like the idea. It’s hard not to appear like a lunatic on the Internet.

    However, since I seem to be the nominated defender for this thread…

    Tau would make Euler’s identity lose its elegant beauty.

    Actually, it gets better.

    e^i*t = cos t + i*sin t

    Evaluate at pi; e^i*pi = -1.
    Evaluate at tau; e^i*tau = 1

    See? 1 instead of -1. Nicer.

  9. NickM says:

    Roue,
    Degrees work fine for building a shed or an oil tanker or whatever. But nobody can use anything other than rads in the context of math physics (and indeed stuff like electrical engineering). Whether it is demarcated as pi or tau is irrelevant in much the same way uibbling over whether it is three hundred feet or 100 hundred yards is. The world is full of unit systems for all sorts of purposes and with all sorts of histories: magnitude numbers for stars (a really weird log scale going back to the anciencts), f/stops on cameras (proper cameras – not the ones also available in a range of colours*).

    Onus,
    No. There would still be the same number of radians in a circle. It wouldn’t percentalise it. It can’t because true angle measure like what works in equations is a transcendental number.

    John,
    if you would care to enlighten us as to in what way a choice of multiple of circular constant is indicative of political orientation we would be extremely grateful.

    *There was a vile advert for Tesco(?) just befgore Christmas where someone was looking for a Samsung camera in pink – no other spec specced. Call me an old git but I’d recently bought a DSLT and the last thing that occurred to me was the sodding colour. It’s black, obviously. I ws thinking all that techie shit. And another thing. Cameras without viewfinders! What theatrical fucking gayness is that about? You see twats with them everywhere holding them at arm’s length looking like a small child trying to drive a bus. Wankers. And the people who think the flash from a friggin’ Nikon Coolpix has a range measured in sodding kilometres…

  10. Well yeah, I can see the appeal of Tau over Pi, but “Pi x r squared” (a calculation which I have used often enough) rolls off the tongue (and is intuitively correct – it’s a bit less than the area of a square with the same distance across, i.e. divide the big square into four smaller squares), what’s the equivalent in Tau?

    “Tau x half x r squared” I suppose. Doesn’t have the same ring to it, does it? And you have to remember that it’s not “[half r] squared” it’s “r squared times a half”.

  11. A picture paints a thousand words (and in support of the comment by Onus Probandy).

    http://xkcd.com/179/

    Best regards

  12. Roue le Jour says:

    Just trying to be funny, Nick. Bugger is, I’ll have to go back and change all my 3D code and replace the #define PI2 to #define TAU…

  13. On the subject of pi*r-squared…

    Yes, that one does get a little more complicated:

    half * tau * r-squared

    But… physicists and mathematicians will actually find that one more appealing. You see, that form is common in many physical equations; and turns up a lot in maths. The reason is that it is the integral of an equation of the form Ax is 1/2*A*x^2. (I skip a lot of steps below, as the calculus is for math-weenies):

    Kinetics: velocity = acceleration * time
    integrating to obtain distance: distance = 1/2 * acceleration * time^2

    momentum = mass * velocity
    integrating to obtain kinetic energy: energy = 1/2 * mass * velocity^2

    If you want to derive the equation for area of a circle, you can do so using calculus.
    Circumference of a circle, radius r is tau * r (by definition). To get the area of an infinitesimally small band, dr, it’s tau * r * dr
    Integrate from 0 to R to get the area of a circle of radius R:

    INT[ tau * r dr ] = 1/2 * tau * R^2

    Shocker… the same form as all the other equations.

    I’m afraid to say that 1/2 * tau * r^2 is the more appropriate form; as lovely as pi * r^2 is; it doesn’t fit into place.

  14. APL says:

    NickM: “*There was a vile advert for Tesco(?) just befgore Christmas where someone was looking for a Samsung camera in pink ”

    Off topic; pie and tau a little above my head.

    But advertisements that really annoy one.

    My current bete noire is the Barclays series for financial products, they witter on about shoes, little origami birds, railways that take you in one direction or another, the latest being balloons you can shoot with an arrow.

    Anything but the f*****g interest rate!

  15. NickM says:

    APL,
    Agreed.

    Barclays have a long track record of producing adverts which seem singularly designed to get on my tits. Recall the Tony Hopkins “Big Bank” one? It’s the smug arrogance of them. And yes, HSBC, you are lso guilty as charged. In contrast the Nat West ones actually say something about their services

  16. NickM says:

    Onus,
    I was wondering when someone would bring up that. I’ll give it you except…

    In pi formulation it’s the same but the twos cancel and if I go back to my original point, if you can’t cancel 2s without even thinking then you have no business integrating anything!

  17. Kevin B says:

    Having just received a stament from my super dooper bonus savings account from which I earned a massive £26.50 in interest last year, (after HMG took its chunk), I concur with the general irritability at bank adverts Not to mention the gubmint’s new ad which profers financial advice to the masses. I don’t want your fucking advice! Just stop spending our money on fucking boondoggles for your fucking pals!

    On topic.

    Mmmmmm, Tau.

    Nah. Just don’t work.

  18. As I, and Dr Hartl, said — the distinction really doesn’t matter for mathematicians.

    Of course the two’s cancel, that’s maths. The question is which is the more fundamental constant, pi or tau? The Tau Manifesto convinced me, Tau is the one. Of course I don’t expect anyone to change; I don’t really expect myself to change. That doesn’t mean I can’t think he’s right. For me, tau is the circle constant and pi is just a multiple of tau.

    I’m also convinced that tau would be more natural for mathematical neophytes to get to grips with because radians = k*tau where k is the percent of a full circle. Anyone who has ever worked with radians and not had to stop for a second to think “pi/2, is that half a circle or an eighth of a circle… oops, wrong way, it’s actually a quarter of a circle” is far too clever for their own good and isn’t in a position to appreciate the difficulties of people less clever than them anyway.

  19. NickM says:

    Onus,

    The point is the only people who ever measure angles in radians are mathematicians (or related fields) so agitating for an alternative is a bust flush. And as I said earlier – much though I appreciate your view on the pedagogics of mathematics I really don’t see it as an issue.

    Let’s take an excursion shall we? ℏ gets used a lot in physics. that is h/2pi. That takes out a lot of the factors of two anyways in Quantum Mechanics. The 2 either doesn’t matter (users of degrees), is easily cancelled or doesn’t even rear it’s ugly head in QM. If you use angular frequencies then it is irrelevant.

    As to the teaching of math at that level then if I had one wish it would perhaps in terms of The Calculus and not confusing the fuck out of everyone by flitting between using infinitesimals and limits with gay abandon. Now that is confusing.

  20. NickM,

    Ah, but does nobody use radians because they are too hard to understand? Perhaps if we’d all been taught tau, we wouldn’t need degrees at all?

    Anyway, as I said, I’m not foaming-at-the-mouth about this. It really doesn’t matter that much to me; I just noticed I was the only pro-Tau chap in the discussion.

    I’m happy to leave you, and the world, to your pi.

  21. CountingCats says:

    Onus, you are not the only pro tau chap. I can see value to it, its definition is slightly more elegant, but so what? Anyone who seriously thinks the use of tau over pi is of anything more than trivial benefit is an idiot. An individual who can’t deal with both concepts shouldn’t be dealing with either.

  22. CountingCats says:

    Onus, you are not the only pro tau chap. I can see value to it, its definition is slightly more elegant, but so what? Anyone who seriously thinks the use of tau over pi is of anything more than trivial benefit is an idiot. An individual who can’t deal with both concepts with equal ease shouldn’t be dealing with either.

  23. Jan says:

    Real mathematicians prefer complex numbers.

    They would like to say that the most fundamental constant
    is 2 pi i = tau i, but this is complicated by the fact that “i” is not quite unique, because it cannot be distinguished from “-i”.

    On the other hand, tau has the appealing property that its
    square root is equal to the product of all natural numbers:
    1.2.3.4… = the square root of 2 pi.

    Isn’t this cool? It is even better than
    1+2+3+4+ … = -1/12.

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