Sunday, March 25, 2012

On Pi and Tau

My sister recently asked me to watch the video “Pi is wrong” which I initially had difficulty locating because it's actually called “Pi is (still) wrong” which sounds like a sequel to “Pi is wrong” rather than “Pi is wrong” itself.

Anyway, the basic overview of the video is that we shouldn't be using Pi (π) in math, we should be using 2pi which it suggests that we call Tau (τ). The reasoning is simple and it's something that you've probably thought of yourself, in math we almost always use the radius of a circle, yet Pi is calculated using the diameter. “What the fuck?” you might ask. Why do we pretty much always use the radius and then in this one single case use twice the radius? What possible purpose could there be for using 2r in one of our most important constants when we use r everywhere else? Pi = C/(2r) but why not just get Tau = C/r instead?

It doesn't make sense, it is confusing, and I noticed these things long ago. Then, like everyone else, I let it go and moved on with my life. In recent years people have decided to stop letting it go and start advocating for using C/r instead of C/(2r) which means using Tau instead of Pi.

I'm in a strange place with this, on the one hand I know exactly where they're coming from as I thought the same thing myself lo those many years ago, on the other I'm resistant to the idea. Not because I think it's inelegant or wrong, but because I like what Pi represents.

In radians to go Pi is to make a half turn, to go Tau is to make a full turn. Tau is the identity, adding a Tau is the radian equivalent of multiplying by one. Of course that makes Tau a more fundamental constant. But Pi is a more interesting one. Adding Pi is equivalent to multiplying by negative one, it's a 180 degree turn, it's motion and change. Tau is the same as standing still. Tau radians is actually equal to zero radians. It's nothing. It's stagnation. It's boring.


Consider one (1). Since it's the multiplicative identity, one doesn't generate anything with mutliplication. Think about it for a moment, what does one tell you? What does one create? What's one squared? It's one. What's the square root of one? It's one. (Yes, negative one is also the square root of one, but you can only say that if you already know negative one exists, but if you're dealing with just one then you wouldn't.) What's the square root of the square root of one? What's the cube root of one? What's the 10829074589274th root of one? One. Always one. Never anything but one.

Compare that to negative one. What's negative one squared? Why it's one. Already by considering negative one in isolation we've expanded our mathematical universe to include one. We've doubled our world. What's the square root of negative one? It's i. We've just discovered complex numbers. But more than that we've discovered bidirectionality. Because it's not just i, it's also -i.

If one moves counterclockwise one gets i, if one moves clockwise one gets -i, but they're both the same distance. They're mirror images of each other. Neither is more obvious, neither has primacy. We have difference without heriarchy. We've just stumbled onto the fact that different things can be just as good as each other.

And they generate each other. So if you never considered that you might move clockwise, if counterclockwise was the only thing that made sense to you, you'd have expanded your universe to -1, 1, and i. Could you really help but try to combine them? Multiplying anything by 1 does nothing and is boring, but that still leaves you with -1 and i to play with.

If we had started with 1 this would be boring as all hell. 1 is the most obvious square root of 1, so the equivalent would be having 1 and 1 to play with. No amount of multiplication can make something interesting come out of that combination.

On the other hand consider -1*i which is equivalent to i*i*i. That gives us -i. Which would be a new number we previously hadn't seen.

If we had gone clockwise first we'd have the mirror image of this relationship, and we would have used -i to discover i. Either way we started with one number (-1), now we've generated four of the roots of unity. And there's no reason to stop there. What about the square root of i? What about the cube root of -1? Asking questions like this can generate and infinite number of numbers, which are points in the complex plane, and it's already showing us the outline of an entire field of meaning.

One doesn't do anything like that. It is definitely more fundamental, but it's also more boring.

Identities just sit around doing nothing.


So what does all of this have to do with Pi and Tau?  Everything.  Everything I just said applies to math with radians.  Except we use addition instead of multiplication.  Pi is negative one. Tau is one. Check it out. Do the math. Draw a picture. When we're talking radians they're different ways of saying the same thing.

Adding Pi radians is equivalent to multiplying by negative one. Adding Tau radians is equivalent to multiplying by 1. Adding Pi/2 radians is equivalent to multiplying by i, the first clockwise square root of negative 1.

Radians are a way of changing multiplication into addition and exponentiation into multiplication. Adding π/X radians is the same as multiplying by -11/X. Pi is the negative one of the circle world. This is, in many ways, a strong argument in favor of switching to Tau. We use one, not negative one, as our fundamental number after all. One is more elegant than negative one. Tau is more elegant than Pi. But, Pi and negative one are more fun.

Look at a picture of a circle marked out in Tau radians. Tau = 0. It's nothing. What could be more boring than zero? I don't think anyone will argue against the utility of zero, it's an incredibly important number and an incredibly important concept. Imagine trying to model motion without the ability to describe stillness. It would be a gaping hole in your description and the entire thing would be a counterpoint without a point to be counter to. But poetry lies in the counterpoint.

Then look at a circle mapped out in Pi radians. It doesn't run from zero to Tau=zero. It runs from negative Pi to positive Pi, clockwise Pi to counterclockwise Pi. What's the top of the circle? Pi/2. What's the bottom? -Pi/2. The relationship between top and bottom, the vertical symmetry, is marked out in a way that Tau doesn't even approach. Oh, sure, you could show the same thing by running the Tau circle from -Tau/2 to positive Tau/2, but the truth is that Tau/2 already has a name. That name is Pi.

Pi forces us to confront the fact that there are two ways to travel around the circle. With the exception of ±Pi, one way will always be shorter and the distance we see in the most basic Pi-radians marking of the circle is the shorter distance, but the two directions are both there. Positive and negative. Clockwise and counterclockwise. Parity instead of a single slog. Tau-radians conceals this by only showing us the positive path. The negative is marginalized, forgotten, discarded.*

That's hardly unique. Hell, look at the fucking name. We call it negative. Is it really that negative? If you get a bill that says you owe -$1,000 is that a bad thing? It seems pretty damned positive to me. If the amount of time you had to spend waiting for something that you wanted were negative, that would mean you got it ahead of time. Wouldn't that be pretty positive?

The truth is that there's nothing negative about negative numbers and nothing positive about positive ones. Context determines that. It's just that the negative ones got the name with all the negative associations because that name was “negative”. The negative numbers get no respect.

The Tauists will actually tell you that a minus sign is ugly (those bastards) and posit that the reason Euler's identity is as it is is to hide the minus sign.  Euler's identity being an equation that relates e, π, i, 1, 0, exponentiation, multiplication, and addition. It is:

eπi + 1 = 0

Do you see the negative one in there? You should. eπi = -1. We could speculate, as Tauists do, that that was seen as ugly and, instead of being an elegant way to related 5 constants and 3 functions, the formula was nothing more than a way to hide the negative one. If that were the case (I don't believe it is) then negative one was and is getting shafted. What's their solution? To shaft it more.

Their proposed rewriting is:

eτi = 1

With a note to the effect that you could make it eτi = 1 + 0 if you really wanted the zero in there. See what's missing? The negative one. The first Tauist formula says “one equals one” the second one says, “one equals one plus zero.” If we started with negative one being hidden, now we've reached the point where it is erased entirely.


When we consider radians Pi and negative one are intimately related. They're two different ways of saying the same thing. Adding Pi*X radians is the same thing as multiplying by negative one to the power of X. And negative one is an interesting and important number.

Pi is the only place negative one gets respect. Taking that away just seems wrong.

Replacing Pi with Tau doesn't just take away the place in which negative one is respected, it actually makes it into a place of less respect than negative one normally gets. Negative one is equivalent to Pi and Negative Pi. They're clearly marked as things that matter.  They get their own symbol.  But in the Tau system it becomes one half Tau and negative one half Tau. They're not marked at all.

At least when we point out that one is important negative one gets some degree of recognition by virtue of being [minus sign] [important thing] It is set up as the one and only opposite of the important thing. It's the primary counterpoint and marked as such.

With the Tau system Pi just becomes [fraction of important thing]. Sure, ½ is an important fraction, so much so that my word processor just automatically inserted a special one half symbol, but it hardly jumps out as the most important counterpoint. No, the clear radian counterpoint to Tau is Tau itself. Negative Tau is positive Tau is zero Tau.

Which brings me back to motion. ½ Pi radians is half of the way from zero to Pi in the positive direction. -½ Pi is half of the distance from zero to Pi (which is equal to negative Pi) in the negative direction. These are completely intuitive things. They're intuitive because Pi does not equal zero. There is a distance from zero to Pi, and thus we can easily divide it in half and see how far it is. Pi is on the other side of the circle from zero, so half the distance is half way between the two sides to the circle. It's half way there.

The same motions said in Tau radians are ¼ Tau and -¼ Tau. But what do they mean? Tau radians, remember, is the same thing as zero radians. The distance between Tau radians and zero radians is zero. One quarter of zero is zero. ¼ Tau is not zero. Which means that ¼ Tau is not one quarter of the distance from zero to Tau in the positive direction. It's something else. Something strange. Something inefficient. If you want to get to someone who is on the other side of something, it makes sense to walk around that something, but if you want to get to someone who is on the same side of it as you, someone who is standing so close that you're touching them, why get to them by walking around the thing? Yet that is what we have here.

It's not a quarter of the distance between zero and Tau, it's a quarter of an arbitrary walk that begins and ends at Tau. Tau is stillness. Tau is unmoving. Tau is unchanging. Add a Tau and nothing is altered. Add Tau as many times as you like, infinite times if you like, things will always be as they were before you started.

Pi is dynamic. It's a complete 180. Literally. That's what it means. Pi radians is equal to one hundred and eighty degrees. Add it to something and you change everything.

Pi may be generally less efficient, it may be more confusing, it may be an oddball number with questionable credentials. But I like it. Pi does things instead of sitting on its ass. Tau is stagnation, Pi is as far as you can move without wrapping around and heading back the way you came. Pi is the maximum possible progress.

Pi is also a vindication of the much maligned negative one.

Pi probably is a worse choice because it does make less sense, but who needs to make sense all the time? Tau makes sense, but it's boring. Pi is inefficient, but it's fun.

That's my thought on it, at any rate.


*Even when it's shorter. Half the time the negative path will be the short path, but does the zero to Tau=zero notation show this? No. Not in the least. It just completely ignores that it's possible to go the other way. It forces you into a single mode of thinking instead of giving you the tools to see it from both sides and thus suggesting those things which it did not have space to show.


  1. Radius is also immensely more practical when considering segments, sectors and entire third dimension of space.

    1. I don't actually know of anything that radius isn't more practical for. In my experience diameter is almost never used.

      I can make something up*, but I've never really bumped into such a thing.

      * If you happen to need the upper limit for the distance of a straight line inscribed in a hypersphere** then that's the diameter, but radius is so much more useful in so many contexts you might as well just call that upper limit 2r.

      ** Of which circles and spheres are examples.

  2. A late addition: