# The Unapologetic Mathematician

## Dual Billiards

I’m not sure when I’ll get to post tomorrow, so I’m giving a little extra tonight: something I realized at about 3 in the morning last night.

The last time I talked about billiards I was linking to Rich Schwartz’ paper on “outer billiards”. I noted that it seemed to me there should be some sort of “duality” between outer and inner billiards, turning problems in one into problems in the other. I think I’ve figured it out. I haven’t checked through all the details, but it looks good enough to satisfy my curiosity. If I were going to write a paper and use this fact, of course, I’d rake it over the coals.

So here it is: inner and outer billiards are related by projective duality. Those of you who know what this is probably are already thinking either “ah, I see” or “of course it is. you didn’t see that before now?” For the rest of you, I’ll skim a bit about the projective plane and formal geometry. I’m sure I’ll eventually come back and write more about them, but for now I can give enough of the gist.

First of all, projective geometry tweaks the familiar axioms from Euclid’s Elements. Euclid says that given a line $l$ and a point $p$ off the line there is exactly one line through $p$ parallel to $l$. In projective geometry, though, any two lines intersect, and moreover they intersect exactly once. That seems nutty at first, but we can make it work by adding a “line at infinity”, with one point for each direction parallel lines could run. If lines seem parallel, they’ll run into each other at that point.

The other ingredient is a formal approach to geometry. Remember when I defined the natural numbers, I said that we don’t care what it is that satisfies these properties, just that anything satisfying these properties will do whatever we say the natural numbers will. Well the same goes for geometry. We have an intuitive idea of “point”, “line”, and “plane”, but that doesn’t really matter. David Hilbert famously said that all of Euclidean geometry should still be true if we replaced “point”, “line”, and “plane” with “table”, “chair”, “beer mug”, wherever they occur. Here: “Any two tables intersect in a unique chair”.

So all the axioms of projective geometry do is set up a system of referents and relations like the Peano axioms do. Any things that fill those referential slots and relations between those things implementing the axioms will do. The points and lines of the regular Euclidean plane, plus those points and the line “at infinity”, satisfy the right axioms, and so everything projective geometry says will hold true for them.

Here’s the trick: The lines and points of the projective plane also satisfy those axioms. Did you miss that? The axioms for “points” and “lines” of projective geometry are satisfied by the lines and points of the projective plane. We can switch lines and points and everything still works out! For example, we’ve talked about the axiom that any two “lines” share a unique “point”. There’s also an axiom that any two “points” share a unique “line” through them. Switching lines and points swaps these two axioms. Any result for projective geometry is really two results: one for the points and lines and one for the lines and points.

Okay, here’s how this all ties back to billiards: don’t think of a ball moving along the table and bouncing off the edge. Think of the line the ball is traveling on and the line of the edge it’s moving towards. They share a unique point, where the lines intersect. Then there’s another line intersecting the edge line in the same point at the same angle, but “on the other side”. That’s the line the ball follows after the bounce, and so on. In outer billiards, we have a point and the edge point it’s heading towards. There’s a unique line between them, and another point on the same line the same distance away, but on the other side of the edge point. We interchange points and lines, lengths and angles, and transform inner billiards into outer billiards and vice versa.

Of course, the calculations strike me as being pretty horrendous in all but the simplest situations. I don’t know that it would be useful to use this duality in practice, but maybe it can come in handy. Actually, for all I know the experts are already well aware of it. Still, it’s nice to have figured it out.

March 9, 2007 - Posted by | Billiards

1. Hi John,

I don’t buy this connection between inner and outer billiards that you mention. I agree that projective duality exchanges some of the corresponding concepts used in the definition of the two dynamical systems, but I don’t think that this fact raises the supposed correspondence to anything like the level of actual mathematics.

Note that Projective dualities do not respect the metric notions
of angle and length. Inner billiards is defined using the concept
of the angle of incidence, and outer billiards is defined using the
concept of distance to the shape. These two notions are not swapped
by projective dualities, except in such trivial cases as the circle.
Your analogy seems to work well on a vague level – and perhaps
could inspire a general transfer of theorems between the two
subjects – but when examined closely it breaks down. (The experts

Here is one way to see that the connection you mention is not
really useful: It doesn’t produce anything like an equivalence
between any two examples of the dynamical systems. I mean
“equivalence” in the precise, mathematical sense.

There are different ways that two different dynamical systems could be equivalent. The strongest notion is one of conjugacy. One has maps
f1: X1->X1 and f2 X2->X2, and then a conjugacy would be a map
g: X1->X2 that made the obvious square commute. A weaker notion
would be orbit equivalence, in which the map g carries f1-orbits to f2-orbits.
A weaker notion still would be measurable orbit equivalence, where
everything in sight is measure preserving, and g carries almost all orbits
to orbits. And so on.

In order for your “correspondence” to really be of any value, you would
have to show how it induced some kind of useful equivalence
between dynamical systems of the one kind and the other. I can imagine
that this might work out just fine for circles, but I don’t see how it could
work in any other case. It would take just one nontrivial case to
convince me. Can you supply it?

I think it is misleading to imply that it is only “horrendous calculations”
that prevent your correspondence from being useful in practice. This
implies that the correspondence actually works, and that it is only a
matter of having the fortitude to implement it.

best,
Rich

Comment by Rich Schwartz | September 9, 2007 | Reply

2. Thanks for your thoughts on the matter. It was mostly a guess on my part, and I didn’t really go much further with it beyond tossing it by Jayadev. You’re far more the expert here than I am, so I’ll bow to your knowledge.

Still, I’m glad in a way to hear that this doesn’t work, and I’m not inadvertently sitting on top of a gold mine.

Comment by John Armstrong | September 11, 2007 | Reply