Is there a reason why there are no triply-ruled surfaces found in spatial geometry? Does it have to do with the fact that there are at most two dimensions/parameterizations for a surface? If that's so, then do 3D hyper-surfaces in a 4D+ space allow for triply ruled surfaces?
To be more clear, I am trying to find a proof of sorts that can demonstrate this fact, and whether or not the proof can be generalized to higher dimensional surfaces and spaces.
Well, one can show that the only doubly rules surfaces are the plane, the hyperbolic paraboloid, and the single-sheeted hyperboloid, and none of these is non-planar and triply-ruled.
One way to think about the fact that no surface in space can be triply ruled as follows:
Firstly, one sees that a surface $S$ of degree $3$ or higher is not ruled at all; a cubic surface can contain at most $27$ lines in total, and higher degree surfaces typically contain no lines at all. (One way to prove this is via an argument with incidence varieties; see the sketch in this tricki entry.) So a non-planar ruled surface has to be a quadric (i.e. cut out by a degree $2$ equation).
On the other hand, if $\ell$ is a line passing lying on $S$, passing through a point $s \in S$, then $\ell$ lies in the tangent plane to $S$ at $s$. Since $S$ is a quadric, when you intersect it with a plane, the intersection is a (possibly degenerate) conic section, and so can contain at most two lines. Thus there are at most two lines on $S$ passing through any given point $s$, and hence a quadric is at most doubly ruled.
