![]() (The cases that it is vertical or horizontal are trivial). ![]() Proof: One may assume that the beam is coming from the left and angling upward. Furthermore, the number of possible sequences of length $n$ grows only as a polynomial in $n. I won't try to write out a complete regular expression, butĬlaim: For any number of parallel mirrors, the set of possible sequences (as the positions of mirrors and the initial beam are vaired) is a regular language. matches any symbol.įor three parallel mirrors, you can have a sequence such as (ab)+(cb)+(ab)+, where + denotes 1 or more occurrence of the previous term, but this sequence implies that c is sandwiched between a and b, so you can't have any more c's after that. ![]() This is equivalent to not(.*aa.*) and not(.*bb.*) where. Thus, for two mirrors a and b, the language is b?(ab)*a?, interpreted with the usual regular expression convention where ? means 0 or 1 occurrence of the preceding term and * means 0 or more occurrences. For some presumably easier cases, the set of arrangements of mirrors can be specialized to range over some particular subset, such as parallel mirrors, or mirrors all at angles of the form $k \pi / n$ for some $n$. Perhaps a better formulation of the question is to ask whether the set of mirror sequences is a regular language, and if so, to describe it. It reminds me of having to use a word such as "industrious" in a sentence in 2nd grade. (There might be subtle issues of what happens when a light beam hits exactly at the edge of a mirror: in this case, if you allow both outcomes that are the limits of its perturbation, it gives a closed set).īut I don't think this is what you meant. On some level its obvious that the set of mirror sequences are characterized by the list of finite substrings that can never occur, if you allow lists to be infinite, and if you count the closure of the set of mirror trajectories as mirror trajectories. Perhaps analogous sequences have been studied before, maybe in another context?Įdited 12/27 to try to clarify, and to correct regular expression formatting (\ is needed so * doesn't turn into italization)* In contrast to the parallel-mirrors example above, I am primarily interested in mirrors withoutĬonstraints on their placements or orientations. That characterize all the realizable sequences? Let $\cal$ of mirrors? Is there a list of such strings
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