Let $P$ be the surface of a closed convex polyhedron in $\mathbb{R}^3$.

. Does every non-closed geodesic $\gamma$ fill $P$ densely?Q

Of course $\gamma$ cannot pass through a vertex of $P$, but it could come arbitrarily close.

Here is an informal way to phrase the question.
One can have non-closed geodesics on smooth convex surfaces in $\mathbb{R}^3$
that do *not* fill the surface. E.g., on ellipsoids,
as in the MO question
"Surfaces filled densely by a geodesic":

^{ (Image from GeographicLib.) }

So: Is there a polyhedral analog?

(One could ask the same question for nonconvex polyhedra.)

Veech surface & V. dichotomy. Thanks! $\endgroup$