# Images between two mirrors

1. Mar 20, 2005

### fliptomato

Hi everyone--I've been mulling over a homework question from earlier in the term:

Suppose you are standinig in the middle of two mirrors of length $$L$$ separated by a distance $$D$$. ((i.e. if you are standing at the origin, the centers of the two mirrors are at $$x=\pm \L/2$$)) The mirrors are at a relative angle $$\phi$$. ((i.e. if the mirror behind you was in the yz plane, then the mirror in front of you will be rotated by $$\phi$$ in the xy plane))

The question is: how many images of yourself do you see in the mirror?

This seems like a purely geometry problem, given the constraints of $$L$$ and $$D$$, how many admissible angles $$\theta$$ (in the xy plane) will produce a ray that bounces around a certain number of times and ends up back at the starting point (with the added trivial subtlety that it must hit the front of your face, not the back of your head).

Anyway, I think I've got it down to a formula for $$d(\theta, n)$$ that is recursive in $$n$$, where $$\theta$$ is the angle at which you are looking at $$n$$ is the number of times (forward and back) that the ray is reflected. The problem is that this isn't particularly helpful to answer the question (how many images)!

The assigned problem gave the numerical values $$L, D = 1$$ meter, and $$\phi = 1$$ degree.

I have a copy of the solution for this case, but I can't even make heads or tails of it! (Indeed I'm skeptical if this is even the correct solution for this problem.)

I've posted this at: