Ah, friends, Tau Day draws ever closer.

Last time I talked a little about why some conventions are better .. *righter* .. than others, in a sense more meaningful than mere personal preference.

Circle constants are not just found in geometry. They are ubiquitous in mathematics. Everywhere you find yourself tripping over 3.1415somethings and 6.2831and-so-ons, in applied and theoretical stuff. You probably call those numbers and , and I’d rather call them and . The Tau Day claim is that the latter choice is *better* than the former.

There are lots of ways to make the case for the primacy of (and I’ll mention more as it gets closer), but the most concise is this: *There are radians in a full circle*.

This begs a question, of course. How many angle-units are in a full circle depends on what units we use. There are 360 degrees, or 4 right-angleses (much early Euclid-style geometric writing measured angles relative to the right angle), or 400 gradians, or 4 million myriogrades, if that’s your thing. So we’ve traded “Why ?” for “Why radians?”.

But that’s a relatively easy question, and certainly a well-known one. There are lots of reasons why radians are the most natural/desirable, and lots of places to read about them. Even the wikipedia article does a decent job. So let me just give you a few concise points.

- The radian measure of an angle can be defined intrinsically as follows. For any angle, draw a circle centered at the vertex, and compute the ratio of the arc length inside the angle to the radius of the circle. (Hence the term
*radian*; the angle measure in radians tells you how many multiples of the radius are in the arc length contained in the angle.) It seems like this might depend on how big you draw the circle, but a moment’s thought shows that it doesn’t, and you end up with a nonarbitrary angle measure. - The calculus-related behavior of the trigonometric functions is best when you work in radians. For example, if you work in radians, the derivative of is and the derivative of is . If you don’t work in radians, you get ugly correction factors.
- (This is actually very closely related to the previous.) If we work in radians, sine and cosine have the simplest possible power series, and it is easy to see (at least formally) the celebrated Euler’s formula , where that
*e*is the base of*the*exponential function, the one we met yesterday. - If we use radians, then the best first-order approximation to is, if is small enough, simply itself. (That is, if .) For any other choice of units, there’s a correction factor.

*A parting math-Zen koan*, especially for those of you who haven’t thought about this much. I have asserted (and do deeply believe) that there is a canonical, nonarbitrary *right* unit for measuring angles. What is the analogous best unit for measuring length?

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There is a different unit which is just as good as the radian: the Dark Radian.

1 Dark Radian is defined to be -1 radians. A circle contains -2pi, or rather -tau, Dark Radians.

This is related to the fact that it is impossible to distinguish the imaginary numbers i and -i from each other in any way besides arbitrarily choosing one.

(This comment is partially tongue-in-cheek. The exact tongue-in-cheek fraction is left to the reader to determine.)

Well played, Xamuel, well played. I’d have said “almost as good” rather than “just as good”, but that distinction can be absorbed into the constant of cheek-tongue-ality.

And, dear reader, estimating that tongue-in-cheek fraction is a very good exercise/koan, at least as good as the one I gave at the end of the post.

Doing so properly involves asking yourself what it is you want to measure when you measure angles (which is probably context-dependent). Is the distinction between clockwise and counterclockwise relevant? Are 450 degrees the same as 90 degrees, or not?

And after you’ve thought about that, it might be a good time to revisit how closely analogous (or not) measuring length and measuring angles are.

Step 1: Bend what you’re measuring into a unit circle, overlapping as necessary.

Step 2: Measure using radians.

Also, I have to point out a factual error in yesterday’s post: You claim that “Grape jelly isn’t as good as strawberry” isn’t fact, but it is.

Careful. “unit circle” presumes that you already have a unit.

It’s OK… When reasoning logically about circles, circular logic is very appropriate.

What parallel postulate do you assume?

Touché.

To make everything I said work out, we need to work in Euclidean geometry. (Remember that, in our class, we didn’t touch the topic of how to assign “lengths” to curves that aren’t line segments.)

I should add, though, that tau is an important constant even in situations that are not geometric on their face. For example, consider the differential equation , in some sense the simplest autonomous differential equation which does not lead to real exponential functions. All the nontrivial solutions to this differential equation are periodic with exact period tau. (The solutions are linear combinations of sine and cosine, as measured in radians, but my previous assertion does not depend on knowing that.)

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