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 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