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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]

• 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