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 the

Tokarsky example there was just one

point there might be two or three or ten

but there isn't something that's a whole

patch of infinitely many points so it's

impossible to construct a rational

polygon that will have a patch of dark

[Doc] that's right

that's right you will not in this

example rational polygon you will never

have a whole part of the room which is

dark

that will never happen even a smaller

not even a very tiny one

what is an irrational polygon for

example let me draw in irrational

triangle and i'm going to make the angle

here 90 times the square root of 2

let's say 30 degrees and then here since

the total has to be 180

this would be 180 - 90

square root of 2 - 30

So obviously, Professor, that's convex [inaudible]

Yeah yeah that, so triangles of course are convex that

thank you for that observation so but

this is an example of this is uh this is

an example of a non-rational triangle

If I created a complicated polygon

that had a rational angles [Doc] Then I don't

think anything is known about the

illumination problem as far as i know the

things that are known as rational

polygons so that's the illumination

problem it it really started in 1955

and as a consequence of other

study of other things having to do with

rational billiards there's been new work

and progress on this illumination

problems i thought i would mention one

other problem that arises that you can

ask about polygons you could ask is

there what's called a periodic orbit

when I say orbit and that might be start

off at a point cloud

it's really difficult feels very

skillful well beautiful it rolled round

rather brands reminding me but i'm going

to bed