In the early 1950s, Ernst Straus asked: is every region illuminable from every point in the region?
Here illuminable means that there is a path from every point to every other by repeated reflections.
In 1958, Penrose used the properties of an ellipse to describe a room with curved walls that would always have dark (unilluminated) regions, regardless of the position of the candle.
40 years later, Tokarsky came up with a counter example.
This idea is closely related to billiards. If you have a table that is the same shape as the polygons above, and you have a cue ball at one of the point and an object ball at another point. There is no way to hit the object ball even if there is no friction.
Future idea: I would certainly like to see a simulation of such a room.