# Non-closed geodesics on a convex polyhedron in $\mathbb{R}^3$

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

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

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.)

• I expect that the words "Veech surface" and "Veech dichotomy" may be relevant here, but I am not an expert in these things and can't say anything more without spending some time thinking about it, or consulting someone who is an expert. (Unfortunately at the moment I have neither an abundance of time nor a conveniently placed expert.) Dec 13 '14 at 2:23
• @VaughnClimenhaga: Very useful key phrases, Veech surface & V. dichotomy. Thanks! Dec 13 '14 at 2:57