Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$,
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?
Lightrays are composed of (infinitely thin) segments, and reflect off the disks
with angle of incidence equal to angle of reflection.
For example, here is one way (of many ways)
to hit the $(0,2)$ disk when $r = \frac{1}{4}$ with two reflections;
it clearly cannot be reached directly, with zero reflections:
I believe the answer to my question Q is Yes, but I would be
grateful for confirmation from the dynamical systems experts.
(Forgive me if I have not learned sufficently from my previous, related question,
"Pinball on the infinite plane.")
It occurs to me it might be interesting to color the disks according to the minimum number
of reflections needed to hit each…
Best Answer
Douglas Zare's shortest path idea seems to me very well-suited for this.
Intuitively, we can view the circles as being rings, and the reflected ray like a rope going through the rings. We pull to obtain the shortest rope (considering the rings fixed, and other suitable idealizations).
The picture below shows how a path connecting $(0,0)$ with $C(m,n)$ may be. Normally, one should be able to calculate the precise contact points and the reflection angles, from the initial angle and $r$, but I am too lazy to do this.
Added
Douglas Zare's comment, that when the radius is close to $1/2$, the things get more difficult, is right, as it can be seen from the illustration provided by Joseph O'Rourke in another answer. So here's how I think we can use the solution I presented above, to handle any possible $r<1/2$.
Start with the above solution, which works for, say, $r_0=1/3$. If $r<1/3$, one can decrease the radii of the circles, until the desired radius is reached, without aby problem. The solution will still hold. The difficulties appear if the radius is larger. We gradually increase the radii of the circles, until the string becomes tangent to at least a circle. Then, we wrap once again the string. We continue to gradually increase the radius, and when the string becomes tangent, wrap it more, until we get the desired radius. The two main cases we can encounter with the solution I presented above, along with the proposed "moves", are represented below.