A Room of Mirrors
You are looking at a room where all of walls are covered in mirrors. The only light source is a candle sat somewhere in the room. Is there a room we can design so that part of the room is in darkness even though the light will keep bouncing around forever? In terms of assumptions let's assume that the mirrors reflect all light that hits them, with the angle of incidence equally the angle of reflection and that the candle is small enough that the body of the candle blocking part of the light isn't going to be enough to affect the answer.
If we model the room as a 2D shape by thinking of it from a birds eye view then there are a few such rooms that people have come up with which satisfy the problem. The first was by Roger Penrose (of Penrose Tiling fame) in 1953 and was based on an elipse:
But curved mirrors are hard to construct and so we wondered whether we could make a room where there were a finite number of flat sides. After 42 years we finally managed to find a solution which fit this critria of being a polygon:
We are still unsure about the minimum number of sides needed so that even an infinite number of bounces will keep part of the room in darkness, but if you think about four and then five sides you should be able to convince yourself that more are needed. With four anything convex loses straight away and anything concave bounces around the corner at some angle or another.
Would it be possible to create this in real life, where you could walk around the corner and go from complete darkness to lit up, even though the walls are mirrors? The answer is no. Real life mirrors aren't completely flat and so will refract the light slightly. Similarly there is plenty of stuff floating in the air which would reflect some light and you yourself would be bouncing photons all over the place. And so the list continues, of dozens of extra, messy exceptions. This is why I am a mathematician not a scientist.
One final thought on this problem: by thinking of the shapes as billiard tables and the light source as the initial position of the ball, then these are tables where an infinite number of bounces is still insufficient to reach some parts of the table.