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][1]:" 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][2], including this figure:
<br />
![alt text][3]
<br />
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:
<ol>
<li>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?</li>
<li>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?
</li>
<li>Are there examples of these same phenomena in piecewise-flat 3-manifolds
(gluings of Euclidean tetrahedra)?</li>
</ol>

<b>Edit</b>. 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!

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

  [1]: http://math.arizona.edu/~asp/2010/piecewiseflatmflds.pdf
  [2]: http://mathworld.wolfram.com/IlluminationProblem.html
  [3]: https://i.sstatic.net/yxa2r.gif