字幕列表 影片播放 列印英文字幕 I want to talk about a problem in mathematics called billiards study billiards and talk about maybe a couple of aspects of that these are called polygonal billiards you have some figure in the plane maybe it has straight sides or maybe not. You can either think of this as billiards or you could think of this as a room where the sides of the room have a mirror started a point and we move in a straight line starting at the point until we come to the side and then we bounce off the side with the angle of reflection equal [to] the angle of incidence so These two angles are the same and then we continue till we hit to another side we bounce off where the angle of incidence equals the angle of reflection and so forth when you get into a corner there's no way of saying what reflection means and so if you are a billiard player you maybe think of this as pocket billiards. We could also think of this, as I say, as a room with mirrors and you are moving along, maybe a light source. A beam of light comes to a mirror and then it reflects in the mirror and continues on in a straight line and so forth. And these things go forever? [Inaudible] Unless they hit a corner? These things go forever unless they hit a corner, yes. Ok so, I'm going to talk [inaudible] at least to start with a problem called an illumination problem. So maybe I have a new figure of a billiard table. Maybe think of that as a room. Now that's a kind of a funny-shaped room, but nonetheless. And I have put a candle at a certain point in the room that's my kind of, a light source. And the question is: does that candle illuminate every point in the room? So that means that if I have some other point in the room, is there a beam of light that will start at the candle and start bouncing off the walls of the room? Maybe many, many bounces until it finally hits that point. So in other words does the candle illuminate the entire room or are there places in the room which are dark, that the candle doesn't illuminate. The question is: moving in any direction, can you get to this point? The candle is radiating light out in all directions and do you illuminate every point? For example, if my room is just the usual - kind of a square - and my candle is anywhere in the room, I can certainly get to any other place in the room by just going in a straight line. So I can go, "here's my candle, here's my other point in the room." I don't have to do any bouncing off walls because I just go in a straight line. This kind of room is called convex, which means that you can join any two points by a line without bouncing off the sides. So convex rooms - it's obvious or clear that illumination happens. This room is not convex because if I'm at this point there is no straight line that gets me from the candle to this point because the straight line would, you know, leave the room and I'm not allowed to do that. The original problem about illumination was asked in the 1950's. One question was: does every point illuminate every other point? The first example where this was solved in a way and that was by Roger Penrose, who was an English mathematician and physicist. What he did was he draws the top half of an ellipse and he draws the bottom half of an ellipse. Same ellipse but doesn't connect them. And instead he draws a mushroom and then a similar mushroom on the other side. The red point here is the focal point of the ellipse, the top ellipse. And here's the other point, [which] is the focal point of the top ellipse. And then these points are the focal points of the bottom ellipse. And what Penrose found was that, for example, if his candle was placed right there, somewhere in the top half, these two regions were not illuminated. So in other words, there was no way of sending a light source from the candle and having it bounce into - to those two shaded regions. They're permanently dark if that is where the candle is. And what he showed was no matter where you put the candle, there will be regions in this ellipse which are dark that are not illuminated by the candle. [Brady] Oh there's no spot I can put the candle? [Prof. Masur] No spot where you can put the candle and illuminate the entire table. In 1995, a man by the name of Tokarsky found an example of a 26-sided polygon and a place where you could put the candle where it didn't illuminate some other place on the table. Why does this have to be so complicated? Well, if you have a polygon as I drew here, it's sort of hard to decide when you send the light source out where that beam of light is going to go. So this problem of what you illuminate is is possibly pretty complicated, and so that's why [in] this first example, there was a point right here. Put a candle [and] he found a single point which was not illuminated by that candle. Now the difference between this and the Penrose example was in the Penrose example there was a candle and there was a whole region two of them which were not illuminated in this example is somewhat different it and this was kind of the only point that wasn't illuminated you could get to any anyplace in this room except if you were standing right if you stood right there at that point you would be in the dark but everybody everywhere else you would see the light from the candle so it's a little different. [Brady] But Doc, it's an infintesimal point It, well it's a single point you can think of it as an infinitesimal but it's a single point as opposed to a whole part of the room and your person is standing right at that place that person is in the dark somebody standing right next to that person sees is in the light so it's a little it's a little different because it's a mathematical construction it's not literally a person standing at a point because after all our feet have width and so it's it's a it's more of a you know theoretical construct there has been research in just in the last two or three years that says that for any polygon and now i'm going to put a little kind of restriction what I want to look at is when the angles at the vertex are a fraction a rational number P over Q is a rational number times a 180 degrees so every angle is a rational number times a 180 degrees for example a square, a square room that angle is 90 degrees that's of course one half of 180 I want to talk about polygons where all of the angles are any rational number and degrees they're called rational polygons or rational billiards if i put a candle at any place the dark regions are these as you say infintesimal points and they're only finitely many so maybe two of them or something for example in