Today’s Big Controversy

All week, I’ve been looking forward to today, but I was mindful that it might be contentious and provoke rancor among my friends.  But, ultimately, truth is more important than partisanship, so, come what may, I want to wish you all a very happy Tau Day!

As you may remember from math class, pi is approximately 3.14159 and, every year, on March 14th, math geeks tend to celebrate.  Pi is what you get when you divide a circle’s circumference (the length around the outside) by the diameter (the length straight across.  And it’s deucedly inconvenient.

We don’t normally define a circle in terms of its diameter, we use its radius.  When you look at the ratio of a circumference to the radius you get tau = 6.283185…. = 2*pi.  And that actually turns out to clean up a lot of the formulas and conventions you learned in trig.  Look back to the video above to see how radians go from being that weird conversion you have to keep using your TI-89 to double check to a totally intuitive way to talk about angles.  Pi takes you halfway around a circle (in radians, pi is equivalent to 180 degrees).  Tau is equivalent to 360 degrees so if you want to talk about half a circle, you’re talking about tau/2.  Three quarters round a circle is 3*tau/4.  There’s none of this 3*pi/2 = 270 degrees nonsense (which I still had to look up to confirm, because it’s not intuitive).

For more examples and answers to pro-pi objections, you can check out the Tau Manifesto, from which I’ve excerpted the following.

It makes no more sense to have a separate symbol for τ/2 than it does to have a separate symbol for 1/2. Indeed, imagine we lived in a world where we used the letter h to represent “one half” and had no separate notation for 2h. We would then observe that h is ubiquitous in mathematics. In fact, 2h is the multiplicative identity, so how can one doubt the importance of h? But this is crazy: 2h is the fundamental number, not h. Let us therefore introduce a separate symbol for 2h; call it 1. We then see that h=1/2, and there is no longer any reason to use h at all.

This hypothetical scenario becomes reality in the case of circles: what is really going on here is that π is half of something. We have a standard symbol (π) for half a “circular unit”, but we have no standard symbol for the unit itself. Whether we use τ or some other symbol, the circular unit needs a name. If you ever hear yourself saying things like, “Sometimes π is the best choice, and sometimes it’s 2π”, stop and remember the words of Vi Hart in her wonderful video about tau: “No! You’re making excuses for π.” It’s time to stop making excuses.

In other words:

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  • Baron Korf

    The 1/2 symbol argument likewise attacks the importance of R since D=2R.

    • leahlibresco

      But R is 1 with regard to the circle. It is the unit of distance away from the center that defines all points included in the circle, sphere, hypersphere, etc.

      • Randy

        But that kind of kills your argument. I did think when reading the post that the diameter is much more intuitive than the radius. You ask how big a cake is you mean the diameter. When I do math questions involving a circle I have to stop and think. It says the radius is 5. That means the circle is actually 10 units across. But if you still need a unit circle of diameter 2 then you are still in radius world. So then pi*radius**2 makes sense.

        • Kodegadulo

          Although from a stone-age perspective the width or diameter of a circle might seem more “intuitive”, in fact this property of the circle is not at all definitive of what a circle is. To say a circle is a shape with constant diameter is not enough, because there are in fact an infinite number of shapes that have the property of constant width. Look up Reuleaux triangles for instance.They’re why a Mazda rotary engine can work the way it does. Some British coins are pentagonal versions of the same idea. They’ve got what are evidently corners, yet you can roll on them just like on circular wheels, without wobbling. So “diameter” is not definitive of a circle. But Euclid himself recognized that the real definition of a circle was a set of points all equally distant from a given point. That given point is the center, and that distance is the radius. And it turns out that the math dealing with a circle is much simpler in terms of the radius than in terms of the diameter. That includes trigonometry, especially when you break free of the initial SOHCAHTOA crutch based on right triangles and start looking at cosine and sine as “circle functions” mapping an angle to its x and y coordinates on a “unit circle” — a circle whose center is at the origin and whose _radius_ is a unit. And in fact we use the radius to give us the most mathematically simple unit of measuring angles around the circle: the radian. You take the length of an arc and divide it by the radius and that gives you the angle in radians. We don’t divide by the diameter and measure the angle in “diameterans”. Why not? Because when you dig deep into the math underlying sine and cosine, you discover that it all gets a lot simpler with radians. Any other angular unit, including “diameteran” or even degrees, introduces some additional scaling factor that just complicates things. So, now, how many radians are there in a circle? 6.28… And as for area, the 1/2 in A = tau (r^2/2) has nothing to do with the tau. It’s what you get when you start with the linear form C = tau * r for the circumference and do integral calculus to get a quadratic form. You know, raise the power of the variable by one and divide by the resulting power? When you get to three dimensions and you’ve got the surface area of a sphere as A = 2 * tau * r^2, and you want the volume, it’s the same idea: raise the r to the 3rd power and divide by the resulting power to get 1/3. So its V = 2 * tau * (r^3/3). Again the 3 has nothing to do with the tau. If you want to rearrange that to (2/3) * tau * r^3 fine but you still won’t be able to cancel that 3 with anything you pull out of tau — unless you invent some constant that means 1/3 turn in the same way that pi means 1/2 turn. Even if you do, what in the world would an angle of 120 degrees have to do the volume of a sphere? About as much as an angle of 180 degrees has to do with the area of a circle. Which is a nice definition of the additive identity element: zero. :)

          • Baron Korf

            It is still the most intuitive, If I were to give you a rod and a caliper, the intuitive dimension would be the diameter. As I did with Leah, I’ll grant you the mathematical elegance, but intuition? No.

          • Randy

            The definition of a circle as all points equidistant from the center is a bit of a math geek definition. If I ask my kids what a circle is they will define it based on shape. It is round but in a consistent way so it rolls easy. The fact that it is also the set of points that are the same distance from the center is interesting but intuitively the first thing we associate with circle. It is logically a fine definition because it defines exactly the right things to be circles and it is easy to state precisely but that does not make the radius intuitive.

      • Baron Korf

        I’ll give you the mathematical elegance that R=1 gives. That’s as far as it goes, however. In practice however, nothing else is measured from its center. Rather it is defined and measured from the limits of its bounded space.

        And really, the center is not an actual part of the circle. By definition, happy supplied by Kodegadulo, a circle is “a set of points all equally distant from a given point”; this excludes the center as part of the circle proper. The diameter is therefore measured from two of the actual points of the circle, meanwhile the radius is from 1 actual point and 1 derived point.

        • leahlibresco

          That’s also true for the foci of an ellipse, but they’re still pretty darn salient.

          • Baron Korf

            True. I grant its usefulness. But even an ellipse is bounded by a major and minor diameter (or axis depending on how you were taught). These bounded measurements (diameter and circumference) are based on the actual attributes of the object. As a result, the ratio that proceeds from the two is more deeply tied to the nature of a circle.

            I guess this is an irreconcilable difference between mathematicians and engineers.

        • Alex

          You haven’t responded to the reuleaux triangle criticism of defining a circle via it’s diameter.

  • Loud

    Ew, leah, don’t make me *enjoy* an article about math! That’s EVIL…..

    • Joanne K McPortland

      No, it’s theology! She’s teaching us to enjoy that which we can never fully comprehend! :)

      • deiseach

        You think there cannot be a connection between mathematics and theology? Puny Earthlings! Read this article on Thomists disagreeing about St. Thomas Aquinas’ approach to a philosophy of mathematics, arising from a commentary on the Holy Trinity :-)

      • Nick

        See also for… well, I don’t know, but it’s interesting at least.

  • Tara S

    THE PI IS A LIE!!!

  • Gilbert

    I’ll just link to the other side of this controversy.

    I’m a little surprised that Leah of all people is a tauist, because one of the mathematical areas where tau sucks more than pi is statistics.

    • leahlibresco

      But on the plus side, switching to tau means people will stop being confused about what pi means when we use it to denote the stationary distribution in Markov Chains.

      • Gilbert

        Yeah, but that confusion is necessary training. Otherwise, how are people going to get formulas that use italic “i” as a summation index and upright “i” as the imaginary unit?

        • Nick

          I am SO glad you reminded me, Leah. I made sure to make a Facebook post at 3:18:53. I caught your blog just in time….

          Happy tau day!

          • Nick

            Whoops, posted that in the wrong place. lol

    • Beadgirl

      I was all set to defend pi via Euler’s Identity, but this article did it for me (and in a much more eloquent way).

  • kenneth

    Trig is just such an evil, debased art that I would welcome any different system of notation that would make it more manageable. The whole exercise surrounding it in calculus and the classes leading up to it was torture for torture’s sake. Memorizing special angles, “identities” (because, you know, you can’t get through life without dealing without knowing several different ways to re-state inverse cotangents!) etc. Calculus is wonderful, elegant stuff, but the only way I ultimately got through was with a professor who favored algebraic functions and had a special fondness for polynomials.

  • Ted Seeber

    My response is completely off topic for Tau day. HOWEVER your wonderful stress on mathematics combined with the fact you have more atheist readers than I do, points to the real topic of my response.

    Thanks for posting THIS as the Controversy of the Day, instead of following all other blogers into that “believers in heaven commit more crimes” nonsense when the real study left out atheism and was really about “believers in heaven” (meaning people who think they’re going to heaven no matter what) and “believers in hell” (which means their entire data set was really Christians with either a Once Saved Always Saved confirmation bias, or a Calvinist bias, or a Traditional Nicene Liturgical Christian bias, where they conflated the last two groups).

    But that raises an interesting point- so if you send me an e-mail to remind me (since as author of this Patheos blog you can) I would like to put together a survey monkey that records a survey that is more accurate, and see if between your blog, my blog, the Magis Center of Reason and Faith page on facebook, and a few other strategic locals such as Catholic and Atheist blogs and even a few non-denominational Christian and Islamic and Jewish blogs, we can’t get a better dataset than two kids from the University of Oregon.

  • Cous

    And here I thought this was going to be about the SCOTUS ruling on the ACA.

    • leahlibresco

      Re that, I can only say: this

    • kenneth

      The SCOTUS ruling is small time. Empires and their petty problems come and go. The torments of trigonometry are forever….

  • math_geek

    Tau has some major disadvantages in calculation complexity.

    A = R^2*(T/2)
    30-60-90 = T/12-T/6-T/4

    30, 60, 90, and 45 are far more useful as angles than the larger angles Tau makes more useful. Area is a more useful measurement than circumfrence. And messing up Euler’s equation is frankly unforgivable. :-p

    • leahlibresco

      I think it’s quite nice as e^(i*T) = 1

      • math_geek

        Ugh, I can’t believe I forgot that.
        Shows how little I remember from complex analysis.

  • Christian H

    Leah! This comment has nothing to do with Tau. I’m just noting that I’m gone for two weeks and during that time you up and convert. Wowza. This will really distract me from the paper I’m supposed to be writing.
    Congrats, I guess? Hang in there against all the angry comments? Have fun with that? You should think about stepping into an Episcopalian church sometime; I’m not saying you shouldn’t go Catholic, but for the sake of ecumenism, I think you’ll benefit.
    Anyway, what I really came over here for was to tell you that, if you’re interested in computer games that tweak your sense of space (which, thinking about it, includes a lot of computer games), you might want to try this one: It’s on my school’s Graduate Student Society’s website. It’s a pretty easy-to-play platformer/puzzle-solver, with no violence. It doesn’t tweak space-sense as much as /Portal/ might, but it has a similar sort of rearrangement going on.

    • leahlibresco

      I wondered where you were! I’ll have to check out that link when I get home tonight.

  • Christian H

    Oh, also, yes, I totally support tau.

  • Jake

    I’m rejecting your arguments for Tau for the same reason we Yanks haven’t switched to the metric system yet: it would invalidate all the digits of Pi so many of us have memorized. And darn it, I spent valuable hours of high school memorizing Pi when I could have been out making real friends!

  • Jason Schalow

    The Traditionalist Society of St. Leonhard Euler rejects your ultramodernist heresy and insists on our right to force you to instruct your children in the geometry of our forefathers! Bending the knee to weak-kneed liberals who only want to make there petty formulas simpler is unacceptable! Pi R Squared!

    • (a+b^n)/n = x

      Tomorrow’s headline: Prominent matholic blogger converts to Tauism

  • Contrarian

    $\tau$ is in many cases notationally cleaner, but it obviously doesn’t have any information that $\pi$ doesn’t have. Regardless, why not just normalize units so that $\pi = \tau = i = 1$?

    PS- Leah, when are you going to get Mathjax so that you can really talk math?

  • Judy MacDonnell

    Hi Leah, Just discovered that you’ve accepted Christianity rather than atheism. Good on ya! However…don’t stop there, will you. A search for truth will lead you on. The Catholic church ealy adopted pagan doctrines from its converts (during the time of Emperor Constantine – read up on it). This church today is very different from the church of Jesus’ day. Keep going…back to basics. And God bless you.

  • Brent

    Engineer here. what manner of device do you use to accurately measure the radius of a circle? Would you use Vernier calipers and divide by two (kind of defeats the purpose of tau), or fiddle about with radius gauges? It’s really hard to even get to half a millimetre accurate that way.

    I’m just saying, I don’t know where you get this radius business from. Where I stand it’s diameter all the way down.