Thursday, June 28, 2012

Happy τ day!

The mathematical constant τ, for those of you who don't know, is defined to be the ratio of any circle's circumference to its radius.  As a result, it is exactly equal to 2π, or about 6.28318...

The use of this new constant has been advocated by several people (including one example that predates the others by some twenty years), the most prominent possibly being The Tau Manifesto (read that now if you aren't already familiar with some of the arguments in favour of τ).  Now, in response to this, a rival manifesto called The Pi Manifesto (hereafter TPM) has also appeared, defending the use of π as the circle constant.  The Tau Manifesto (hereafter TTM) has done a decent job of responding to some of its points, but now I shall take the time to address some points it has made that I haven't seen addressed as of this writing, as well as commenting on a couple of other things. 

First, a quick word on the mathematics often used to justify τ:  TPM doesn't claim this explicitly, but it might as well have argued that τists use elementary mathematics more often than advanced mathematics to make their point.  Of course, after TPM came out, TTM put some fairly advanced maths in, and some of the other sites had them in there to begin with.  However, I argue that this criticism misses the point entirely.  The reason we use maths that are learned only by newcomers to mathematics is because those are the people we are trying to help out!  As has been pointed out, using π instead of τ is a pedagogical disaster. 
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Now, let's start with section 2.2, or "Other definitions of π."  Now, earlier today, TTM came out with a new section, claiming to show "what is really wrong with π."  I honestly wasn't convinced much when I thought of it in that light.  The argument is in Section 5 of TTM, but the main thrust of it is that there are three families of relevant constants here:  those giving the ratio of the n-surface area of a n-sphere to its radius (to the appropriate power), those giving the ratio of the n-volume to the radius, and those giving the ratio of the n-surface area to the diameter.  For n = 2 (a circle), we recover τ from the first family, and π from the third.  The argument goes:  in the n = 2 case, the second constant-family is also equal to π, but this is a pure mathematical coincidence, and since π therefore belongs to neither the first nor the second family, it isn't as fundamental as a constant that does belong to one of those families, such as τ.

When I read this argument expecting a knock-down argument against π, I wasn't impressed (again, you can read it for yourself).  However, after rereading TPM, I realized that it worked perfectly against one of its arguments.  TPM claims that π could just as easily be defined as the ratio of the area of a circle to its radius squared.  However, this constant is a member of the second family, not the third (to see why, take n = 2 and note what that means).  Therefore, the two definitions define two different constants—they may have the same numerical value, but they're still different.


Next, we come to what TPM calls a "silly argument" for τ, and probably one of the most-widely-used ones:  the unit circle in trigonometry.  TPM argues that, while τ may indeed define the circumference of the unit circle, π defines its area.  Since area problems are used more often than circumference problems, I guess π is the better constant, right?  Of course, TPM appears to be a bit tongue-in-cheek in proposing this, but let's respond to it anyway:  the arc lengths of the unit circle define not just circumferences, but also angles.  Angles, unlike sector areas, are universal across all circles.  If I cut my circle off at an angle of, say, τ/4 radians, that's true regardless of the radius of my circle.  Area, on the other hand, is completely dependent on radius.  But let's ignore that fact; note that the formula for sector area is: A = 1/2 θr2.  Plenty of arguments have already been given as to why the factor of 1/2 in the formula for circle area can't be avoided, and here's one more:  plug in τ (the angle measure of a full circle) for θ, and see what happens.

Speaking of which, TPM dismisses the idea that there is a missing 1/2 by simply saying that the traditional formula "is already a quadratic form preferred by mathematicians."  I'm not sure what exactly they mean by that, but if they mean to suggest that we use it the traditional way because it's the traditional way (which is what it means to have been preferred through the ages by mathematicians), they miss the entire point of TTM.


I shall end this post by enumerating the main reason TTM convinced me; it had to do with these angle measurements.  Less than a few weeks prior, I had been thinking about two of the major angle measurements used:  radians and cycles.  For example, in physics, you have frequency, measured in cycles per second (or Hertz), and angular frequency, measured in radians per second.  I had long thought that having these two measurements, both seemingly equally fundamental, was rather ugly (Degrees and grads are not fundamental, at least not in the same sense that radians and cycles are).  However, with τ, we have a nice unification of these two measurements; for example, 6 cycles is equal to 6τ radians, and τ/2 radians is equal to 1/2 cycles.


Anyway, just adding my two cents to this back-and-forth.

~Ian

P.S.  The Pi Manifesto claims that Euler's formula with τ is weaker than the original formula with π, and I agree.  However, the same formula can be rewritten with τ:  0 = 1 + e-i*τ/2.  As has been pointed out in Tau Before It Was Cool, this has the same strength as the original one with π, but is arguably more beautiful (at least, I find it more beautiful).  It contains the four arithmetic operators in standard order, the numbers 0, 1, and 2 (arguably the most significant integers) in numerical order, and e, i, and τ in alphabetical order.  Beat that, π partisans!  The reason I didn't include this in the main body was because it had already been addressed by someone else, and I was trying to be original.P. Whehe FF  First