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  • 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