The only constraint on the polygon is that the angles are rational multiples. There all kinds of crazy non-convex shapes that satisfy this constraint, which is why the result is noteworthy.
I think this is also interesting when it comes to physics. A dark "point" physically means that the electromagnetic fields permeate the entire room except over a finite set of Dirac delta functions. If you were to try and put something physical in the dark point, you'd quickly turn this into a quantum mechanics problem. I don't think the dark points matter in real life because confining yourself to a single point would require infinite momentum.
This theorem assumes perfect mirrors and Newtonian light. In real life, I predict that the entire room would be illuminated. There's a path of least action for the light to take, but there is a probability of taking other paths. Scattering is not precise. I would think that would add a fudge factor to the paths.
