Pi, Tau and Eta

Recently, I’ve heard a lot about the number τ, and I find the whole thing a bit odd.

Here’s how it goes:

The number π is the ratio of a circle’s circumference to its diameter. It’s been known about for thousands of years and is an extremely useful number which appears in all sorts of unusual and unexpected places. It’s not only irrational but also trancendental, which means you can’t write it down exactly using fractions or even square roots. Its decimal expansion begins 3.14159… and a not-too-bad approximation using fractions is 22/7.

People are so enamoured with π that they celebrate π day (14th of March), and π approximation day (22nd of July) — in fact, I will be celebrating π approximation day by writing the digits of π on the street in Adelaide.

But here’s the thing: some people claim that π is not the best number to use as your fundamental circle constant. This is because, if you represent angles as distances around circles (which is what mathematicians do), then π only represents half of the circle. Therefore, these people claim that you should use instead 2π — which they call τ. Vi Heart gives a very impassioned talk on this on YouTube:, and others have launched τ day (28th of June) as an alternative to π day.

Included in their reasoning to throw out π and embrace τ is a claim that it’s pedagogically more sound — that it’s confusing for the fundamental constant to only represent half a circle, and that many more formulas are easier to work with and easier to remember with τ rather than π. For example, they cite the trig functions and how they repeat themselves every τ as opposed to every π.

But this is my main bug-bear: of course it’s not easier! The switch in people’s minds from degrees to radians is such a huge jump that whether you use π or τ is really not going to make all that much of a difference! And while many formulas are nicer with τ, others are just uglier (in my mind!).

It just says to me that you can be passionate about something loudly enough and lots of people are likely to agree with you.

But I have one more thing to add: If you were going to work with a new circle constant, I think you should use not 2π, but π/2 — let’s call it η. You see, η represents a right angle, which to me is an extremely fundamental thing in our modern lives. And moreover, it represents the ratio of a semicircle to its diameter. That is, if you want to go from A to B, it’s how many times further you go if you go around a circular path as opposed to in a straight line. That makes a lot more sense to me than either the circumference/diameter, or the circumference/radius. Finally, the trig functions repeat their shape (if not their orientation) every η so the very constant you use would remind you of this simple fact. Yes, if you were going to define a new constant, I reckon η makes heaps more sense than τ.

But of course, I don’t care quite enough about this to make an empassioned speech about it on YouTube, so it’s unlikely anyone will listen. 😉

[NOTE: I do actually respect Vi Hart very much and wholeheartedly support her work in the physical and musical representation of maths, and also her use of YouTube to encourage play in maths rather than rote learning. I just don’t agree with her opinions about π.]

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20 Responses

  1. David Butler says:

    Ok, so maybe I was wrong about not caring enough to make an empassioned speech on YouTube:

  2. Karl Medlicott says:

    … but why η (ἦτα)?

    • David Butler says:

      Hi Karl, I used eta because it seemed like it wasn’t used for much yet, especially not being used for an angle often, and because the capital is H for half, but mostly because it makes a nice pun: “eta pi”. 😉

  3. Karl Medlicott says:

    … but why “η”?

  4. Karl Medlicott says:

    … & didn’t the original French metric system use the right angle [the hectograde = 100 grades] as the measure of circular arc? Brilliant!

  5. Karl Medlicott says:

    mesures d’arc de méridien

    hectograde (Hgr) = quart du méridien terrestre
    décagrade (Dgr)
    grade (gr) degré centésimal (°) ≈ décamyriamètre
    décigrade (dgr) ≈ myriamétre
    centigrade (cgr) minute centésimal (′) ≈ kilomètre
    milligrade (mgr) ≈ hectomètre
    décimilligrade (dmgr) seconde centésimal (″) ≈ décamètre
    centimilligrade (cmgr) ≈ métre

  6. Karl Medlicott says:

    …but if we wish to move beyond πι & ταυ
    we needs must stop talking or referring to either one of them,
    or defining or naming our own thing in terms of theirs
    (though puns are always fun).

    I’d much rather call our “new circle constant” — no, that’s not at all quite right —
    well, I’d rather call it “q”
    [actually a SMALL-CAP Q, (pronounced “qu”, as in ancient Latin)
    which looks rather like a Q, but smaller]
    which no one else is using for anything at all,
    & which stands for “quadrant”;
    & it’s Latin,
    not Greek!

  7. Karl Medlicott says:

    … I’ve just this moment read that, in 1958, the eccentric English mathematician Albert Eagle had π/2 as the circle constant — & he used the symbol τ!

  8. Karl Medlicott says:

    forget “Q”, or “q”;
    I’ll go
    with Albert Eagle’s circle constant:

    τ ≈ 1,570 796 326 794 896 619 231 321 691 639 751 442 098 584 699 687 552 910 487 472 296 153 908 203 143 104 499 314 017 412 671 058 533 991 074 043 256 641 153 32…

  9. Karl Medlicott says:

    Yes Albert was punning too.
    τ = 1/2 π.

  10. Karl Medlicott says:

    … & then there’s this

    “A few formulas should simplify by changing to the circle constant η = τ∕4.”!

    • David Butler says:

      Thanks for that Karl! Michael Hartl himself has referenced me in the newest version of the Tau Manifesto saying that η simplifies in particular formulas for volume/surface area of spheres in higher dimensions.

  11. Karl Medlicott says:

    Michael Hartl himself does more than reference you, he agrees with you, save for the “inconvenient factors”, writing

    “(I liken the difference between τ and η to the difference between the electron charge e and the charge on a down quark qd=e/3: the latter is the true quantum of charge, but using qd in place of e would introduce inconvenient factors of 3 throughout physics and chemistry.)”

    Thank-you for η!
    … & thanks for showing me that the original French creators of the metric system knew what they were about when they defined the grade as 1/100 of a right angle — this, with τ, had long disturbed me.

    … but what would be the canonical definition of η as an equation, mentioning neither π nor τ? Surely not C/r/4!

    η ≡ … ?

    • David Butler says:

      A possibility is to define eta as the ratio of the area of a circle to its inscribed square.

  12. Karl Medlicott says:

    I’m all with you (& Albert Eagle, & the French mathematicians who devised the original decimal metric system) about right angles — I want η to be the thing!

    … & then there’s this

    • David Butler says:

      *sigh* I don’t really want people to switch to eta. I’m saying that if they were going to make a switch anyway, then I’d prefer eta. What bothers me most is that he thinks he can convince people by telling them they’re stupid — I have never found that a healthy approach to things.

      • Hi, Dr. Butler! Thanks for taking the time to read!

        Honestly, despite my attempt at pithiness with the “Pi is Stupid” and “Be Smart, Use Tau” lines, my goal was not to tell people that they are stupid, but rather to say that we should do our best to prevent the obfuscation of mathematics. Just as I wouldn’t define a circle as ‘two semicircles which share a diameter and endpoints but which have opposite direction,’ I don’t think that the primary constant for describing circles should be defined as the ratio of its Circumference to double (or quadruple) its radius.

        Re-reading my article, I do now realize that it has an unnecessarily antagonistic tone to it (especially towards the end), and for that I apologize. I tend to get a bit more emotional about my geometry than a person probably ought to get.

        Thanks, again!

  13. Karl Medlicott says:

    Being but the village idiot,
    the best I now can do is

    η ≡ C/4r ≈ 1,570 796 …

  14. […] mathematician Albert Eagle just about made a career out of trying to replace Pi with Tau, which is merely the ratio between a […]