Illuminating piecewise-flat manifolds with geodesics

Question

If one issues a geodesic in every direction from a point $p$ on a piecewise-flat 2-manifold, will it necessarily illuminate the entire surface? I know the answer is 'No,' but I would like to explore the question further.

I am using the term piecewise-flat manifold in the sense that David Glickenstein uses it, e.g., in "Introduction to piecewise flat manifolds:" a gluing of Euclidean triangles edge-to-edge. This is also called a polyhedral manifold. For the purposes of this question, whether it is embedded in $\mathbb{R}^3$ is not relevant. More generally, these manifolds are formed by edge-to-edge gluings of planar polygons (each of which could be triangulated). The manifold is flat everywhere but at a finite number of vertices (or cone points) at which the surrounding angle differs from $2 \pi$.

Because geodesics do not pass through vertices (or, more accurately, I stipulate they cannot), it is conceivable that there is some $p$ from which geodesics shot in every direction fail to reach every point of the manifold. This was established in a rather different context in the paper by George Tokarsky, "Polygonal Rooms Not Illuminable from Every Point" [Amer. Math. Monthly, 102:867-879 (1995)]. Mathworld has a nice description, including this figure:

If you glue two copies of either of these polygons back-to-back, it forms a polyhedral 2-manifold with the property that geodesics (light rays) from one red point cannot reach the other red point.

One can ask many questions here, but these three interest me:

1. Tokarsky's example is a doubly covered polygon. If one generalizes instead to arbitrary polyhedral manifolds, are there other, perhaps more straightforward examples where from some $p$ not all the manifold is covered its geodesics?
2. I conjectured long ago that the measure of the "dark points" is zero. Is there an example (of a polyhedral manifold) where more than isolated points are unilluminated? Could a segment be unilluminated? A region of positive area?
3. Are there examples of these same phenomena in piecewise-flat 3-manifolds (gluings of Euclidean tetrahedra)?

Edit. In response to Henrik's example below, I should have said that ideally two further conditions should be satisfied: (a) $p$ is not at a vertex (so it is surrounded by $2\pi$ of surface); and (b) the manifold should be closed, without boundary. This is not to say that $p$ at a vertex and a manifold with boundary are not of interest!

Addendum: Thanks for the interest and help! I have much to learn on the topic of translation surfaces!

Answer 1

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 $8$ squares 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.

Answer 2

This illumination problem has been studied for special kinds of polygonal surfaces, called (pre-) lattice translation surfaces. See Modular Fibers And Illumination Problems by Pascal Hubert, Martin Schmoll and Serge Troubetzkoy. For these surfaces, the paper proves that the set of non-illuminated points is countable.

Answer 3

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