mrfb asked: What are your thoughts on the pi v. tau debate?
I’m actually extremely pro-tau, but only under certain conditions. I’ll explain.
Warning: The following is my personal take on these subjects. I’m no authority. This is pretty much late night armchair philosophy and ramblings of a madman. It’s just how I make sense of some of these ideas, and it’s the first time I’m trying to put these into words. Hopefully, they’ll make some sense and I won’t look like a complete nut.
On the merits of the debate
Mathematics thrives on conventions. Being able to symbolically convey a very precise idea is one of its greatest triumphs and strengths. For that, we have developed a set of (ideally unambiguous) conventional notations. Notation is pretty damn important. Learning this mathematical language takes a lot of effort, and it is a skill we should respect. A lot of important knowledge is being carried by these crazy symbols, knowledge built upon centuries of intense thought and research by some of the smartest people who have ever lived.
The use of the Greek lower-case letter pi (π) to denote a particular irrational number is one of such conventions developed in mathematics. As a convention, it is extremely valuable as it is. There’s little reason to change it. The fact we agreed with π as the ratio of a circle’s circumference to its diameter is of little consequence to any underlying mathematics, it changes nothing, so this isn’t really a point to be argued. The important thing is merely consistency.
In this respect, the tau vs. pi debate is a waste of time in my view. Saying equations are prettier because of a factor of two somewhere is missing is a bit ridiculous and non-mathematical, and entirely misses the point of having a constant defined in the first place. Are we arguing over mathematics or typography/aesthetics?
However, conceptual differences are important. This is where the debate can be fruitful, I think, so dismissing it completely can be (and I believe it is) a bad thing. Oddly nobody else seems to be making this particular point, at least not how I’m going to expose it here.
You see, π shows up everywhere in math by itself, no factors of 2 attached at all. It’s a pretty remarkable number on its own. It shows up even when things don’t seem to be related to circles. For instance, the integral of the Gaussian is √π, which is a surprising result (and it’s one of my current math animation projects). The sum of the reciprocal of squares, also known as the Basel problem, is π2/6. No circles in sight here.
But whenever tau (τ = 2π) shows up, people like to talk about circles. They’re missing the point, I think. 2π isn’t the circle constant. It’s the ANGLE constant. The circle just happens to be related to the concept of angles, and not the other way around.
The most mathematically natural way of measuring angles is in units of radians. Everything works out so simply when we use radians that it’s tempting to call it the one true way of doing so.
"Dimensions" vs. "units"
Now, before I go a little bit more into this argument, I need to clear something up. A lot of people say “radians don’t have units”, but that’s an incorrect use of terms. What these people are trying to say is that radians don’t have a dimension, that is, they are a dimensionless unit. See how we can use both terms together and still make sense? That’s because they have distinct and precise meanings.
A unit is a standard we use to measure other similar things. For instance, you can measure length in several units: meters, feet, the nearest spoon’s length, light-years, (toenail growth rate)·century, (your own name here)’s nipple-to-nipple distance, etc.
What all these units have in common is that they have the same dimension: length, or simply [L]. The other base dimensions in nature are time, [T], and mass, [M]. There are other dimensions that are used, but these are the more basic ones.
One way to understand this is to think of [L], [T] and [M] as the “real” physical quantities, or kinda like how Nature “understands” these quantities. When you read “2 meters”, you should be seeing 2 × (“1 of something we use to define a meter” × [L]).
The “meter” has a certain amount of [L] hidden in it, you see, because we defined it in terms of something else that has a length dimension. The 2 in “2 meters” is just there telling us just how much of that something we are talking about. The 2 is a pure, dimensionless number.
Using these three dimensions, we can build all sorts of quantities. Here’s a few, and some example units:
- Force = [M][L][T]-2 → newton, pound-force
- Energy = [M][L]2[T]-2 → joules, calories, kilotons of TNT
- Frequency = [T]-1 → hertz, radians/seconds
Dimensions can be treated just like variables: you can multiply, divide, take powers and square roots of them, but they don’t “mix” together. You can even add different dimensions, though just like variables, you get nowhere with that: a+b is just a+b. While it makes little physical sense something with dimensions [L]+[M] (think, 1 metre + 1 kg), there’s no reason why you shouldn’t be allowed to have it, if you’re really into that kind of thing. Weirdo.
By the way, this topic is called dimensional analysis, and it’s a very interesting subject.
Radians are an unit defined as the angle enclosed by an arc around a circle that is as long as that circle’s radius. Here’s an animation I created that explains it:
It doesn’t matter what radius you pick (that’s why in the animation the radius is just a generic “r”), the angle is always the same because the arc’s length is also proportional to the radius, so the length of the radius always cancels itself out when you actually end up calculating radians.
But also, notice that the arc-length is a unit with dimension [L], and so is the radius. If you divide one by the other, the [L] dimensions cancel out, just like variables would. We end up with a quantity that’s just a number, a dimensionless quantity. A full turn has about 6.28 radians in it, that is, 1 turn ≈ 6.28 × (“1 of something we call radians” × [no dimension whatsoever]).
So, radians have no dimensions. We can treat them just like any other pure number. This is usually how everyone does it: they say it is a pure number, no meanings attached to it, and call it a day.
This is where my take on the subject takes a weird turn…
"Dimensions" vs. "Concepts"
But conceptually, these numbers are still measuring something. Two instances of the same dimensionless number can still represent two entirely different things.
For example, the number 2 is, by itself, conceptual-less. But “2 radians” and “2 (radius of the planet inhabited by humans in the early 21st century / radius of Earth)”, while dimensionally and numerically identical, are totally different conceptually.
So, here’s where my take on all these things gets weird: I think that beyond dimensions, we also attach “concepts” to numerical quantities and units, and these are also subject to a “conceptual analysis” similar to the dimensional analysis I mentioned up there.
While dimensions have a physical meaning, “concepts” are, well, abstract. (For consistency, I’ll denote concepts in single quotes from now on. E.g.: ‘angle’)
An ‘angle’ is such a concept, attached to the unit of a radian: 1 “radian” = 1 × (“something we call radian” × ‘angle’), where “something we call radian” is the same as “ratio between length of an arc of a circle and that circle’s radius”, that is, the definition of the radian. So, hidden inside a radian, is the concept of ‘angle’ being multiplied by the number, just like a dimension such as [L] would be.
In fact, in terms of concepts, we could say: ‘angle’ = ‘circular arc’ / ‘line segment’, so that we have: 1 radian = (1 × [L] × ‘circular arc’) / (1 × [L] × ‘line segment’).
In other words, what I’m trying to say here is that even if the dimensions cancel out in the definition of a radian, the concept of ‘angle’ shouldn’t really go away with the number we’ve got. The concept ‘angle’ is intrinsically in the “radian” unit, and it is not a dimensional quantity.
I love to play with this idea of “conceptual analysis”, and it has given me some weird and accurate insights before.
Dude, just get to the point
All right, all right, here’s my point. I think we should have two definitions:
- π = 3.14159265…
- τ = 6.28318530… radians
Notice the difference?
π is just a pure number, like 1, 2.5 or √2. It has no concepts attached to it.
Meanwhile, τ is a number attached to the units of radians, which means τ carries the concept of ‘angle’ with it everywhere it is used, always. Seeing τ immediately implies we’re talking about angles.
This is the important conceptual difference I talked about in the beginning. This is where τ really makes a lot of sense and where it would be useful.
"The Conceptu-tau Manifesto" (groooan)
So, here’s my crazy proposal: let’s adopt tau as THE ANGLE CONSTANT.
Let’s face it, π isn’t going anywhere. It’s already well-established way beyond the scope of circles anyway. It makes no sense to fight it, and it has earned its place.
But whenever we talk about angles and rotations, there’s no question that τ is the proper constant to use, just as surely as the use of radians instead of degrees for angles. A full turn is the important idea, not a half turn.
Here’s the same animation as the one above, except this time using τ for the full turn instead of 2π.
Notice that this time we can just keep using our unit of measure (the red arc of 1 radius in length) all the way around, counting each new whole radius (or radian) that fits, only adding a fractional bit at the end to complete the whole turn (the 0.28.. part). This makes much more sense, since that’s how we awalys used any unit of measurement: we count how many times our unit fits in the whole of the thing we’re measuring, not just in half of the thing.
With π, we are assigning a certain special name for a half-turn, even though it is the full turn the thing we are trying to measure. While this isn’t inherently a bad thing (a rotation of a half turn has a lot of importance in mathematics, hence why π exists), it is an odd special case that’s simply an arbitrary quirk of definitions.
The undeniable fact about all of this is that a full turn is more important than a half turn, so it deserves its own symbol.
However, notice that the foundation of that argument is not the numerical value of the full turn or half turn. That’s totally irrelevant, which is exactly why we’d like to use a symbol in place of these numbers! We don’t care about them! But for some reason, this is what most people seem to focus their attention on.
No. The foundation of that argument is in the word “turn”. It narrows down the single mathematical concept we are addressing in the discussion, and it’s in that context that τ really makes all the sense in the world, since it’s the one that represents a turn.
If you’re not convinced yet, just look at our language. We don’t even have a word (in common use, at least) to describe “half a turn”. We already talk about half a turn in terms of a full turn in our natural language. We all already use the definition π radians = τ/2 radians, but only when we talk about angles and rotation. It’s just how we naturally treat the concept, and it makes perfect sense that way.
If that doesn’t make it deserving of a mathematical notation, I don’t know what does.
An example of the conceptual use of τ as the angle constant
Now, imagine we live in an alternate reality where τ = 6.28… radians, as I proposed. What could math feel like?
The following is obviously incredibly biased (this is an opinion text, so that’s kind of the point here), but it’s pretty close to the thought process I had when I was trying to make sense of the same ideas.
Euler’s Identity: eτi = 1
You see that mathematical expression for the first time in your life.
You see τ in there. Your brain attaches to it the idea of a “full turn”, as you have been trained to. Your brain is now thinking of things rotating and angles.
You see a representation of a “full turn” multiplied by the imaginary unit, i. You try to make sense of that, and you fail. As you should. But now you’re thinking about the complex plane and what could a “complex full turn” possibly mean.
But your brain doesn’t give up. I hasn’t finished reading the expression yet. So it reads the exponential function. You already know the e0 = 1, that’s one of the key properties of this function. But now, the exponential of a “complex full turn” (whatever that is) is doing something new. What it is? You look at the right of the equals sign.
You see the number 1, the multiplicative identity. This is the same value as e0 that you have already thought of. So, the exponential of a “full complex turn” is doing the same thing as doing nothing.
Your brain makes the connection: the exponential of “a full complex turn” (whatever that is) is bringing you back at the same place as you started. You know something is rotating, and you know this is happening on a complex plane.
Aha! Your brain finally gets it. It’s the only idea that makes sense now: the exponential function is performing a rotation in the complex plane itself.
And if you know trigonometry and think just a little harder, you should deduce that eθi = cos(θ) + i·sin(θ), Euler’s formula.
So, call me crazy or whatever you may, but this actually sounds like a nice convention to have around.
To be honest, I feel pretty uncomfortable talking about these things. This notion of concepts attached to numbers may be a bit nutty, and I’m not familiar with this sort of approach to things anywhere. (Though a quick Google Search has brought up Bertrand Russell’s Theory of Descriptions), which sounds kinda alike)
But this is similar to the way my brain works, for better or for worse. This is how I learned to tackle math concepts, and this is the kind of approach I try to convey in all of my animations. I try to carry these ‘concepts’ around using things like matching colors and visual styles.
Since so many people are fond of my animations, perhaps this idea has some merits, and I’m not a complete lunatic.
Either way, I don’t think there’s a magic trick to it or anything. It’s just about making sure you are keeping track of what everything represents at all times. This is the key approach to learning mathematics. The more stuff you can connect and correlate, the better and deeper your understanding will be.
And best of all, it’s supposed to make sense, even when it doesn’t. Usually, when it doesn’t make sense, it’s your intuition that’s wrong. It’s an odd lesson to learn, but these are the rules we play by in math.
"But I don’t want to go among mad people," Alice remarked.
"Oh, you can’t help that," said the Cat: "We’re all mad here. I’m mad. You’re mad."
"How do you know I’m mad?" said Alice.
"You must be," said the Cat, "otherwise you wouldn’t have come here."
(from Lewis Carroll’s 1865 novel, Alice’s Adventures in Wonderland)