Light Beam Reflection

Date: 01/31/2002 at 07:02:08
From: Teri Lurie
Subject: Algebra

Please can you help me with this question?

Four mirrors form a rectangle 3 m by 2 m. A light beam is shone from A 
at 45 degrees. Which corner does the beam strike first?

Teri

Date: 01/31/2002 at 12:17:13
From: Doctor Peterson
Subject: Re: Algebra

Hi, Teri.

You can either do this in a very simple way, by just drawing what 
happens, or in a way that demands more insight but makes the problem a 
lot more interesting.

First, you can just draw it out:

   B             C
    +---+---o---+
    |   | / |   |
    +---/---+---+
    | / |   |   |
    o---+---+---+
   A             D

As I've shown, the beam will first hit the wall 2/3 of the way along 
side BC, at coordinates (2,2) if A is the origin. Then it will reflect 
at the same 45 degree angle back down toward side CD. Draw where it 
will hit that:

   B             C
    +---+---o---+
    |   | / | \ |
    +---/---+---o
    | / |   |   |
    o---+---+---+
   A             D

(It will be very useful to realize that a 45 degree angle will always 
take the beam one unit in the y direction for each unit it goes in the 
x direction.)

Keep on like this, and you will soon find the answer.

Now for some deeper insight:

Instead of thinking of the beam reflecting, you can think of the 
reflections of the "pool table" itself (that's how I think of it) in 
the mirrors:

    A---+---+---D---+---+---A
    |   |   |   |   |   |   |
    +---+---+---o---+---+---+
    |   |   | / |   |   |   |
    B---+---o---C---+---+---B
    |   | / |   |   |   |   |
    +---/---+---+---+---+---+
    | / |   |   |   |   |   |
    A---+---+---D---+---+---A

If you were standing on the "table," you would see many copies of the 
table (and yourself) over and over. And the path of a light beam would 
seem to be a straight line cutting through many of these reflected 
copies. I've shown its path from A to side BC and on to a reflected 
copy of side CD, matching what I showed before. Keep going, and it 
will eventually hit a reflection of a corner, and that will be your 
answer.

Now if you think about this a bit, you will see that the corner it 
hits will be at coordinates (x,y) where x and y are equal (so that it 
is on the path of the light beam), but x is a multiple of 3 and y is a 
multiple of 2 (so that it is a reflection of a corner). Can you see 
how to use this fact to solve the problem without having to follow the 
beam all the way? That would be very useful for solving the problem 
more generally, for different sized tables and different directions.

You can find some related problems in our archive at

   Bouncing Cue Ball
   http://mathforum.org/dr.math/problems/nichole10.29.96.html   

   Pool Table Algebra
   http://mathforum.org/dr.math/problems/roland10.21.98.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/