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 and a point off the line there is exactly one line through parallel to . 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.