This is a “hot topic” among mathematicians recently, I guess. At the very least, a bunch of my fellow grad students have weighed in on the topic recently. So, I know nobody else may care what my opinion is, but I wanted to voice it SOMEWHERE. Facebook seemed like the wrong place for a long-winded response on a niche topic, so I thought my own personal blog was the most logical place to blab away. After all, if you don’t want to know what I think about things, why are you reading my blog?
Basically, around Pi Day and “Tau Day” this year (3/14 and 6/28, respectively), some major news outlets had little pieces about the importance of Pi, and about the emergence in math&science culture of an argument to eradicate Pi and instead define a constant called Tau, which would be equal to 2Pi.
Of the things I read:
News article 2: http://www.bbc.co.uk/news/science-environment-13906169
The Official-ish page for Tau: http://tauday.com/
Essentially, they claim that while Pi is historically speaking the conventional standard, it is not the most logical choice for the “circle constant.” The argument against Pi goes something like this:
Pi is defined as the ratio of the circumference (C) of a circle to its diameter (d). In other words, Pi=C/d, no matter what circle you’re talking about.
Since a circle is defined in terms of the radius (a technical definition of a circle is: “the set of all points a fixed distance [the radius] from a single point [the center]”), why is the circle constant defined in terms of the diameter? Wouldn’t it make more sense to relate it to the definition, the radius?
So, they suggest the circle constant should be in terms of C and r, not C and d. As you may remember from grade school, d=2r, so Pi=C/(2r) and C/r = 2Pi. So, set Tau=2Pi and start using that because it “makes more sense.”
I just read the “Tau Manifesto” (third link above) and while the author makes a few good points, his argument is not as irrefutable as he wants people to believe. So, I want to make a few points of my own…
First, his arrogance disgusts me. If he wants to be taken seriously, I think he should spend more time on persuasive arguments than on ending every other paragraph with “I bet you feel like a fool NOW, don’t you?” That does not make his argument stronger. In my opinion, these assertions weaken his argument because I cannot take someone seriously who is trying to use bullying techniques to revolutionize mathematics and science and change a centuries-old convention.
Second, I do like his initial appeal to the technical definition of a circle. Yes, circles are defined in terms of their radius. So, I can buy into the argument that other circle-related things should also relate back to the radius. But there are good reasons for using diameter in this situation – mainly that diameter is a measurable quantity. If you want the radius of some physical circular object, you do not measure it -you measure the whole way across the circle (the diameter) and divide by 2. So, when the ratio for the circle constant was originally discovered and calculated, they related two things that they could measure to each other. A practical definition for this constant was chosen, not a “wrong” definition.
Third, the only supporting evidence he had that I found convincing was that it makes sense to make the period of sine and cosine, and the measure in radians around a circle, its own unit, namely Tau. Currently we say the period of sine and cosine is 2Pi and the measure around a circle of radius 1 is 2Pi. He is all bent out of shape about a factor of 2 appearing in so many important places, saying that students hate to have to remember to multiply something by 2. (By the way, in the places in math where Pi – which is Tau/2 – shows up instead of 2Pi, I would argue that students hate fractions (namely 1/2) more than they hate multiplying something by 2…)
However, the assertion that teaching trigonometry using 2Pi instead of Tau is a “pedagogical disaster” is ridiculous. I can’t speak for anyone else involved in teaching pre-calculus or trigonometry (either as instructor of record or recitation instructor), but when I taught it last year, I did not find that thinking of 2Pi as the distance around the circle is what students struggled with. Saying One-Tau is the distance around the circle instead of Two-Pi is intuitive (ignoring the obvious problems of converting everyone alive that learned basic math with the traditional symbols to entirely new ideas); I can admit that. But saying that Two-Pi is so confusing that students can never get past their confusion with that concept to learn other things is just plain wrong. The biggest problems I saw in teaching trigonometry were with analytical thinking, BASIC algebraic manipulation of quantities, and learning new definitions. Students already know how to find the circumference of a circle, so thinking about a circle of radius 1 having a circumference of 2Pi is not difficult for them.
Fourth, his argument about the Eulerian identity (e^(i*Pi)=-1 OR e^(i*Tau)=1) is moot. It’s a beautiful identity either way, and measuring in terms of Pi or Tau is a matter of what you’re measuring angles with. I do not see any advantage to using Tau in place of Pi when thinking about this context exclusively.
Lastly, he considers the cherry on top of his argument to be replacing Pi with Tau in the formula for the area of a circle: A=Pi*r^2 OR A=(1/2)*Tau*r^2. I actually see this as his weakest point. Basically, he claims that because a bunch of physics formulas that model natural processes and forces have a factor of 1/2 in them, the formula for the area of a circle should have a 1/2 in it too. I call bullshit on that, especially after his whining about adding extra factors into calculations involving the unit circle and trig functions confuse students beyond repair and implies that they need years of psychiatric treatment to deal with such hardship. (Okay, now I’m the one exaggerating things a bit. But that just further proves my point that being over-dramatic makes you look less convincing.)
All in all, I think it doesn’t really matter. The point is, Pi naturally occurs in the relationship between Circumference and Diameter, and since diameter is easily related to the radius, with a simple substitution you can relate Circumference to Radius and call it something else (Tau, here). I don’t think they are fundamentally different (as the author of the “manifesto” claims) and I don’t think one is right and the other is wrong. As with much notation in mathematics, it is a matter of convention and Pi is the convention and has been the standard since the relationship was discovered. These people want a revolution and want to change convention. It has happened plenty of times before in society. But math isn’t a social science and I don’t really see why change is necessary. I think it’s cute for mathematicians to have something to argue about, but I think it’s a silly cause to take up. I also think his attitude is all wrong. If you’re going to convince a bunch of people who only listen to logical arguments, it makes little sense to spritz your manifesto with flowery language that tries to bully the reader into thinking the same thing you do; anyone who disagrees is perpetuating “unadulterated madness” (section 4.5).
He anticipates responses like mine, and addresses people that agree with my previous paragraph saying we are “making excuses” for Pi. To that, I say that people who agree with his argument are making excuses for being too lazy to multiply a constant (Pi) by another constant (2) when it is warranted, and to use just one constant (Pi) whenever that is warranted. Why is using Tau and sometimes dividing by 2 FUNDAMENTALLY SUPERIOR to using Pi and multiplying by 2 sometimes? I’m not sold on that.
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