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:

- 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?
- 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?
- 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!