problem that arises that you can ask

about polygons. You could ask is there

what's called a periodic orbit? When I

say orbit, that might be "start off at

a point, you bounce off the walls and

after some long number of times; can you

always find a path that repeats itself?"

So let me give one very, very classical

example of that and that goes back to

the square – if you start, let's say on the

bottom, and move off at a right angle,

90 degrees, you move to the top. Now when you

get to the top, because these are

parallel, this angle is also 90 degrees,

and so the billiard path says you

rebound where angle of reflection equals

angle of incidence, and that means that

when you come to the side you will

bounce off exactly the way that you came

in; you will come to the side and

bounce right back. You come back along

this line and then you will bounce and…

repeat yourself – so that's what's called

periodic. There are other periodic lines.

So, for example, you could start at a

45-degree angle, everything would be

45 degrees, and you would come back right

where you started and then repeat

yourself. In fact, if you started any

point, at any angle that is a rational

number – a quotient of two integers –if you started with an angle of 27 degrees

starting here, you will come back after a certain number of bounces.

That's true for a square; and, in fact, one can prove

for any rational polygon, were all the angles are fractions of 180 degrees,

or P over Q times 180 degrees,

again I like to write it like that,

you can always find a periodic orbit.

[Brady]: From any point? [Professor Masur]: No not from any point.

There's some periodic orbit, from somewhere.

Some periodic orbit from somewhere.

There might be; so there will be

some direct… Some point, in some

direction, where there will be a periodic orbit.

[Brady:] For example on this side there are none, but there is one sitting over here somewhere.

[Professor Masur:] Well usually what happens if… It, it…

In most cases, that… That's possible;

in fact you can always find periodic

orbits that will hit every side. In most

cases a periodic orbit will hit a side.

There might be other points on that side,

where the orbit in that same direction, is not periodic.

So in other words, you start at a point on a side and move in a direction

it might close up, meaning periodic,

but if you move the point a little bit and headed off in the same direction,

what could happen is that it isn't periodic;

what might happen is it hits a corner

and then you don't know what to do.

That's different from the square. In the square no matter where I

started, if I went at 45 degrees, it would be periodic,

but in a general,

rational polygon that may not be the case.

I want to talk about triangles now

that are not necessarily rational, but just triangles.

If they are acute,

which means all of the angles are 90 degrees or less,

what you could do is: you could take the vertex of the triangle

and drop a perpendicular down to the opposite side to get a point,

and then drop this perpendicular down to the opposite side and get a point so these are right angles.

This triangle joining those points gives a periodic

orbit; meaning if I start here I go to

the line towards here this angle will

equal that angle and I'll bounce like that.

And then i'll get to this point and

this angle will equal this angle and

I'll bounce like that.

And I come back, this angle equal this angle, and I'll repeat myself.

This example was has been known for hundreds of years.

I don't think it was ever thought of as billiards, but anybody in high school or

even junior high could do this in… In trigonometry.

When you do obtuse triangles,

you can't drop a perpendicular because if I started at this vertex and drop the perpendicular,

it wouldn't be on that side.

That's because this angle is bigger than 90 degrees and, in fact,

for something as very simple as in a

triangle, obtuse triangle, one does not

know whether there are periodic orbits.

So this is a famous unsolved problem in

the subject of, what this is called is, dynamical systems.

Well so again if the obtuse triangle is rational,

then there are periodic orbits.

So I'm talking about obtuse triangles. I think I gave an example –

90 times square root of two – that particular one I would

not be able to tell you if it had a periodic orbit.

There are some…

Some reason where people, by very, very hard

work, have shown where if this angle is less than a hundred degrees,

then there are periodic orbits. But that was very,

very hard work and if it was a hundred and ten degrees;

again I'm assuming not rational;

then we don't know the answer.

If it's rational we always know, if all the

angles, if this is a hundred and eleven,

and this is 36.

What's left? 33 degrees there will be.

But if one of the angles is an irrational

number then we don't know.

[Brady:] So what, does greatness await if someone can crack that?

[Professor Masur:] Well, it's… It's um…

There is a very prominent, one of the most prominent

mathematicians in the United States or

in the world in the subject of

dynamical systems whose name is

Professor Katok at Penn State has

listed this as one of the five

outstanding problems in dynamical systems.

—But really there's nothing you

could do. You can try desperately to

solve it, but if it hasn't been solved

for a hundred years, you probably aren't going to.

And you know it's only given to one

person, so to speak, to solve a particular

one of these problems. So we're used to

it and here's an atmosphere of resignation.