*Update*. I answered too quickly. The construction I describe traps a ray whose source is inside. > Mitchell, Zachary, Gregory Simon, and Xueying Zhao. "Trapping light rays aperiodically with mirrors." *Involve, a Journal of Mathematics* 5.1 (2012): 9-14. ([Journal link](http://msp.org/involve/2012/5-1/p02.xhtml).) > **Abstract**. We construct a configuration of disjoint segment mirrors in the plane that traps a single light ray aperiodically, providing a negative solution to a conjecture of O’Rourke and Petrovici. We expand this to show that any finite number of rays from a source can be trapped aperiodically. <hr /> [![SegMirrors][1]][1] </sup> <hr /> To obtain a polygon, one would have to connect their disjoint segments into a path, but I think this would not be difficult. *Update*. Apologies. Now that I found their construction, which mimics an irrational sloped billiard path reflecting inside a square, it is not immediately evident how to inject the ray from outside the construction... Incidentally, it is not possible to trap light rays from a continuum of directions, even with curved mirrors: > Dawson, RJ MacG, B. E. McDonald, J. Mycielski, and L. Pachter. "Light Traps." (1996). ([PDF download](http://129.69.211.95/pdf/mit/lcs/tm/MIT-LCS-TM-560.pdf).) [1]: https://i.sstatic.net/AHdia.jpg