The Right Angle

loopspace

2014-6-15

# 1 Introduction

Every year, in mid-March, a small number of people who write their dates in a weird way get all excited about the date. A little later, in June, another group also get excited. The semi-annual debate about the true value of $\pi$ continues with neither side showing any sign of weakening.

I must admit that although I've read through the various arguments for and against $\pi$, I have a feeling that I'm missing something somewhere. Neither side's arguments are, to me, all that convincing. But perhaps the problem is that I have already made up my mind as to what is the correct angle and since this is, ultimately, a subjective question the likelihood of me changing my mind is small.

As far as I can tell, the arguments for $2\pi$ (some want to call this $\tau$) are that lots of formulae involve $2\pi$ and so using a symbol for $2\pi$ makes them clearer, and that a whole circle is somehow more fundamental than half a circle. The arguments for $\pi$ are essentially refutations of these.

As I said, neither is all that convincing to me. But also as I said, perhaps the problem is that I already know what the right angle is and it isn't either of these.

# 2 The Right Angle

At its heart, the question is a simple one. Do either of the numbers $3.14\dots$ or $6.28\dots$ deserve a special symbol? And is one more deserving than the other?

That we need a symbol is not disputable. A number gains a standard symbol if it is used often and there is no simple way to write it (the latter condition explains why $\sqrt{2}$ has no special symbol). Both $3.14\dots$ and $6.28\dots$ are frequently found so both qualify under the first rule. Both are also difficult to write simply so also qualify under the second rule. However, there is a catch in that once we have a symbol for one then the other does become easy to write so we only actually need one. And if we decide to only have a symbol for one then the other will always be relegated to "second-rate" even if the decision was a close one: there can only be one winner.

So it would be good to have a definite reason to choose one over the other. Nice formulae just doesn't cut it. To plump for one over the other, there has to be some deeper reason. Something conceptual that shows that one number is deeper than the other. This is what the "whole circle/half circle" part of the debate is (presumably) meant to address. There's also the appeal to pedagogy in that it makes more sense to divide the circle as fractions of a whole rather than fractions of the half. This would seem to tip the scales in favour of $6.28\dots$, but somehow it doesn't feel as though it tips them all that far. What is needed is a deeper concept. Something that clearly and unequivocally favours one over the other.

# 3 Conceptually Speaking

When we talk of $3.14\dots$ and $6.28\dots$ then we are in the realms of geometry. The "whole circle/half circle" part of the debate seems to fit in well with this so would look like a reasonable place to start looking for the so-called Silver Bullet. The trouble with this is that drawing circles is not really a major part of modern geometry. It's often an early part of what we teach, but order of teaching is not going to provide the killer blow.

Drawing circles is nice, and can occupy us for a short time, but we soon move on to segments and arcs and for those we need more general angles. Here, the case for $6.28\dots$ makes headway as often the arcs and segments are thought of as some proportion of the whole.

But again, we don't spend all that much time and effort on drawing arcs so basing our decision here is still a little unsatisfactory.

So what do we do in geometry? What lies at the geometric heart?

I've taught a couple of courses where I'm of the opinion that I've lain the foundation for modern geometry. My starting point is always angles; not because I view them as fundamental but because they are pedagogically accessible. Once I've talked a bit about angles and reminded the students of their school-level geometry, I ask two questions.

1. What is the angle between the positive and negative $x$–axes?

2. What is the angle between the positive $x$–axis and the zero vector?

Using these as motivation, I introduce what I call the scaled cosine angle. We wrap the angle up in a couple of functions which deal with the ambiguities of angles. It makes angles safe: we don't have to worry about periodicity, and there are no disallowed vectors. We can always recover the actual angle if we need it, but until we actually do then we keep the safety covering in place.

The scaled cosine angle turns out to be the inner product. We quickly learn that not only does it make angles safe to use, it's also remarkably well-behaved and easy to compute. It rapidly moves centre-stage, and the angle moves gradually further and further into the background until we have forgotten its existence altogether.

Well, not entirely. The inner product wraps angles in a safety covering and one side effect of this is that for almost any pair of vectors, knowing their inner product does not immediately tell you their angle. Neither $3.14\dots$ nor $6.28\dots$ can be easily seen from an inner product. But there is one angle that can be immediately known: the right angle.

Two vectors are at right angles if and only if their inner product is $0$. It's an amazingly simple test, and also an astonishingly important one. Knowing when two vectors are at right-angles (we say orthogonal) is extremely important. It says that they are not just independent, but completely separable. The two directions have absolutely no effect on each other in any shape or form. It all begins with Pythagoras:

 ${c}^{2}={a}^{2}+{b}^{2}$

and goes on into the heart of Fourier analysis.

This concept of orthogonality is fundamental to geometry. It is of far greater import to know when two directions are at right angles than to know if they are the same or opposite directions.

Thus the angle that deserves a symbol of its own is neither $3.14\dots$ nor $6.28\dots$ but $1.57\dots$. As for the symbol itself, well $\tau$ does look a lot like half of $\pi$ so it's a natural choice.

Join me, therefore, on the ${57}^{th}$ of January (if you're American) or the first of September in four years' time (if not) for a truly geometric celebration of the one angle that is more fundamental than any other: the Right Angle.