TITLE: Forbidden mirror sequences
QUESTION [21 upvotes]: Let $\cal{M}$ be a finite collection of two-sided mirrors,
each an open unit-length segment in $\mathbb{R^2}$,
and such that the segments when closed are disjoint.
A ray of light that reflects off the mirrors determines
a mirror sequence or mirror string
consisting of the mirror indices in order of reflection.
Obviously no mirror string can contain $a a$ as a substring.
I am wondering if mirror sequences may be characterized
by a list of forbidden substrings.

          

For example, consider the special case of parallel mirrors,
none collinear.
Orient them all vertical, and label them sorted left to right.
Then for labels $a < b < c$, these are forbidden substrings:
$$ b * a c * b $$
$$ b * c a * b $$
with $*$ being any string (including the empty string).
So in the above example, after 415 appears, 4 cannot occur again.
Perhaps these are the only forbidden patterns for parallel mirrors?
Edit. Apologies for not initially phrasing a clear question.

Q. Are there strings of mirror indices that cannot be realized by some ray reflecting
  among some collection $\cal{M}$ of mirrors?  Is there a list of such strings
  that characterize all the realizable sequences?

In contrast to the parallel-mirrors example above, I am primarily interested in mirrors without
constraints on their placements or orientations.
Perhaps analogous sequences have been studied before, maybe in another context?
Pointers appreciated!

REPLY [5 votes]: Here is a possible way of producing such forbidden configurations. Suppose you have $(ab)^k$ for some large $k$. Then I'd like to claim that $a$ and $b$ must be nearly parallel (see Thurston's answer). So produce a sequence with three mirrors containing $(ab)^k$, $(bc)^k$, $(ac)^k$ and a forbidden configuration for parallel mirrors such as the one you gave above (which you then have to prove also is forbidden for nearly parallel mirrors).