3 Corrected a typo in the 1st sentence.

Let $\cal{M}$ be a finite collection of two-sded 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.

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!

2 Tried to clarify my question.

Let $\cal{M}$ be a finite collection of two-sded 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.

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 having difficulty identifying forbidden strings for arbitrarily oriented primarily interested in mirrors without constraints on their placements or orientations.

Perhaps analogous sequences have been studied before, maybe in another context? Pointers appreciated!

1

# Forbidden mirror sequences

Let $\cal{M}$ be a finite collection of two-sded 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?

I am having difficulty identifying forbidden strings for arbitrarily oriented mirrors. Perhaps analogous sequences have been studied before, maybe in another context? Pointers appreciated!