TITLE: Can every $\mathbb{Z}^2$ disk be pinball-reached?
QUESTION [32 upvotes]: 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...

REPLY [3 votes]: Permit me to draw attention to a new expository survey related to this question:

Alex Wright. "From rational billiards to dynamics on moduli spaces." Apr. 2015.
  (arXiv abstract.)

He discusses the "Wind Tree" model, also known as the Ehrenfest model,
which is essentially the model I was considering, but with square (or rectangular)
"trees" rather than disk trees: