Lasers in a mirrored room

[groza528]

You are part of an expedition team. On one of your digs, you discover an infinite square grid. No, I don't know where you would bury an infinite-- look, it's just flavor, ok?

Built on this grid you discover an odd building housing a most unusual room. The room is no particular shape that you can discern. However you do notice that all of the corners of the room are at right angles. Some of them bend in, others out, but all are 90 degrees rectangular (see post 7). Further inspection reveals that all of the corners of the room lie on gridpoints, though the walls do not themselves follow the gridlines. Most intriguingly, all of the walls are covered in ordinary and remarkably-preserved mirrors.

This building is found at the very center of the grid. Or so you suspect; you'd have to pace it out to be certain and that would literally take FOREVER. Because of its central location you are absolutely certain that this building holds some major significance, that it was purpose-built for a very important and esoteric reason.

Unfortunately this reason has been lost to antiquity so your team mostly uses the room to play with the big laser you all bought with last year's grant money.

Before you know it, you are posing problems to one another vis-a-vis the laser and the arbitrarily-shaped rectangular room. Finally your newest teammate, Jenkins, asks one that really gets you thinking... "OK guys," he says, "Put the laser on any gridpoint inside the room. Now point it due northeast, so that it cuts across one of the grid squares diagonally. Also make sure that it is perfectly level. As you can see, the beam reflects predictably off the walls until it hits a vertex of the room or the back end of the laser (but only the back end because we were smart enough to get a laser with perfectly transparent components that allow the passage of light through it in any other direction)."

"Prove or disprove: The beam will only cross itself directly over gridpoints."

[novice]

A few observations:
With all angles being 90 degrees, the walls will all have one of two directions, and the laser beams will also have one of two directions since they're bouncing off right-angled walls. See below for illustration on how a beam might bounce.



Since a + b = 90, a = a' and b = b'. Similarly for the right-hand diagram, the incoming and outgoing beams are parallel.

[Lepton*]

I've been thinking about this for a while, but continue to have difficulty quantifying it. I think we can (WLOG) reduce it to seeing whether a single square can exist that has the laser crossing both its diagonals. Then, if the laser begins in the bottom left corner, then it finishes either in the top left or the bottom right. In other words, a non-gridpoint crossing means that the beam needs to pass within a distance of 1 from itself at gridpoints. However, this isn't possible, because at a 45 degree angle the beam will move a (taxicab, aka in orthonormal directions) distance of 2 with each grid square it crosses, and no summation of positive or negative even numbers can give an odd number.

That's certainly not the elegant solution Groza has in mind.

[groza528]

Oh, I do NOT have an elegant solution in mind. I actually don't know the answer myself. I suspect that it is true, but I'm not sure how to prove it.

[groza528]

Maybe let's start with the simpler question of proving whether it's true for walls that *are* on gridlines. That one I think I can prove with relative elegance.

[Thok]

I'm confused by the "The walls don't follow the gridlines" comment. Does this just mean the walls are a proper subset of the gridlines? (The way it is phrased suggests walls can leave grid lines, but the other conditions make that impossible.)

I'm also wonder if the room can have an interior chunk of walls that isn't connected to the interior (so you could have a small diamond inside a larger square, for example.)

I think I can prove some version of the statement by using a checkerboard argument and the reflection principle to strongly limit the path the laser takes. But I'd want more info before I did the problem seriously.

[groza528]

So I made an example to show the walls being off-grid, and it turned out to be a counterexample to the question as posed So let's amend the previous problem statement to require a rectangular room.
Here's the example I made anyway. In this example, if we call the laser apparatus itself the origin, the beam continues until it gets to (14/3, 14/3) then it reflects back, passing through (0,4) where it eventually intersects itself, and continues in this fashion, crossing itself twice more at (3,3) and (4/3, 4/3) before it finally stops at the internal corner of the room at (-1,1).
I notice, however, that the beam doesn't break the "only cross at gridpoints" pattern until it reaches the small alcove at the right hand side, hence the proposed edit.



Also, I didn't intend to allow interior walls, though that would be an interesting offshoot... The non-rectangularity would probably cause the same result, though.