Illuminating piecewise-flat manifolds with geodesics

I would like to propose a simple example of a flat surface of genus $3$ with dark points. This also gives an example in dimension $3$. It is based on the following simple observation.

Observation. Consider the torus $T^2=\mathbb R^2/\mathbb Z^2$. Suppose that there is a geodesic segment that joins points $(0,0)$ with $(\frac{1}{2},\frac{1}{2})$. Then it passes through one of four points $(\pm \frac{1}{4},\pm\frac{1}{4})$.

Now consider the double ramified cover $S$ of $T^2$ branching at points $(\pm \frac{1}{4},\pm\frac{1}{4})$. Then on $S$ there is no geodesic segment that goes from any of two preimages of the point $(0,0)$ to any of preimages of the point $(\frac{1}{2},\frac{1}{2})$. Indeed if there were such a segment it would project on $T^2$ to a segment that joins $(0,0)$ with $(\frac{1}{2},\frac{1}{2})$. Hence on $S$ it should pass through a branch point, which is forbidden.

Added. One can desribe give an alternative description of this example. Namely, we can take $8$ copies of squares of size $\frac{1}{2}\times \frac{1}{2}$ and glue a surface of genus $3$ from them in such a way, that at each vertex $8$ squars meet.

If you want a 3-dimensional example just multiply this example by $S^1$. But it should be of course possible to construct examples that are not products, using similar idea.

This illumination problem has been studied for special kinds of polygonal surfaces, called (pre-) lattice translation surfaces. See

http://front.math.ucdavis.edu/0602.5394.

For these surfaces, the paper proves that the set of non-illuminated points is countable.

The illumination problem for translation surfaces was solved in this paper by Lelievre, Monteil and Weiss here (http://arxiv.org/pdf/1407.2975.pdf). In particular, they answer the "Conjecture 1" in your comment to Alex's answer above.