[Professor Masur]: But 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, 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.