Inspired by recent work on refraction billiards in dynamics, we introduce a notion of refraction for combinatorial billiards. This allows us to define a generalization of toric promotion that we call \emph{toric promotion with reflections and refractions}, which is a dynamical system defined using a graph $G$ whose edges are partitioned into a set of \emph{reflection edges} and a set of \emph{refraction edges}. This system is a discretization of a billiards system in which a beam of light can pass through, reflect off of, or refract through each toric hyperplane in a toric arrangement. Vastly generalizing the main theorem known about toric promotion, we give a simple formula for the orbit structure of toric promotion with reflections and refractions when $G$ is a forest. We also completely describe the orbit sizes when $G$ is a cycle with an even number of refraction edges; this result is new even for ordinary toric promotion (i.e., when there are no refraction edges). When $G$ is a cycle of even size with no reflection edges, we obtain an interesting instance of the cyclic sieving phenomenon. This is joint work with Ashleigh Adams and Jessica Striker.
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