Biquandles of Virtual Knots

I haven’t quite gotten to the notion of a quandle yet, but it’s just around the corner. Still, if you’re interested in knot theory, there’s a paper just up on the arXiv by David Hrencecin and his former advisor Louis Kauffman (who I’ll be seeing again in Ohio next weekend). It’s on an invariant of virtual knots, which is another thing that I should talk about.

More 5th graders

Finally I see another math question. The first one of tonight’s show and the woman playing drops out rather than try to answer “how many sides does a rhombus have?” She would have guessed ten, so it’s a good thing she quit.

Coincidentally, the next guy white-knuckled his way to guessing ten for “how many sides does a decagon have?” He also does a good job working his way through “how many cups are in five and a half gallons?” This one I find slightly unfair because he’s not allowed to work things out on paper, which amounts to multiplying 16 by 5.5 in his head to get 88: not easy.

Also this is the first time anyone made it to the $175,000 question with any cheats still available, and the girl there to help him got 89. I haven’t taught down at that level, but this strikes me as a very odd error to have made. I don’t see a way to have misremembered the numbers to multiply, or to have made a mistake in multiplication, and to get 89 instead of 88. What this screams out to me is that the kids have been told to supply wrong answers beyond a certain point. Thus the “save” and “copy” cheats are highly unreliable late in the game.

Psychoceramics

Every so often, as faculty in my department, I get unsolicited submissions from crackpots. Just tonight I got a real doozy, from a guy who’s sent me stuff before. I’m going to just post screen capture of the email because I don’t want to even try to replicate this formatting.

John Baez has a Crackpot Index for physics, but I don’t know of a good checklist for mathematics. I’ll just leave this to my intrepid readers to find the flaws.

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 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.