Wednesday, May 1, 2013

Pinball with disks and rays

Joseph O'Rourke proposed a beautiful problem on Math Overflow:

Let every point of $\mathbb Z^2$ be surrounded by a mirrored disk of radius $r\lt 1/2$, except leave the origin (0,0) unoccupied by a disk.
    Q. Is it the case that every disk can be hit by a lightray emanating from the origin and reflecting off the mirrored disks?

Here is an example

I liked the solution I gave, so I would like to share it here. It assumes the radius to be about $1/3$ or less. The idea is that, instead of light ray and reflection, to think in terms of rings and ropes connecting them. We assume the rings and the rope satisfying suitable idealizations. The following picture shows that any ring in the lattice $\mathbb Z^2$ is reachable.

Moreover, we can use this method to connect with the origin all rings in the plane, with a single rope.

Of course, if the radius gets close to $1$, the rope becomes overlapped with the rings, and this solution will no longer work. But we can still use it to find a correct solution. We start with a radius of $1/3$, then gradually increase it. At some point, the rope will become tangent to one or more rings. In this case, just wrap it more, using the moves in the following picture