A beautiful, relatively recent result is that,

Every simple arrangement $\cal{A}$ of $n$ lines in the plane is induced by a simple $n$-gon $P$.

In a *simple arrangement*, every pair of lines intersect in a point,
and no three lines intersect in a common point.
A polygon $P$ *induces* $\cal{A}$ if $\cal{A}$ is obtained by extending its
$n$ edges to lines. Thus $P$ "visits" each line of $\cal{A}$ exactly once;
it is a Hamiltonian-like cycle:

This is proved in the paper,
"On Inducing Polygons and Related Problems."
Eyal Ackerman, Rom Pinchasi, Ludmila Scharf, Marc Scherfenberg.
*Algorithms-ESA 2009*.
Lecture Notes in Computer Science, Volume 5757, 2009, pp, 47-58.
(PDF link
)

Two natural question occur to me, neither of which is addressed in the paper:

Q1. Which arrangements $\cal{A}$, $n>3$, have auniqueinducing polygon?

Q2. Does the theorem extend to $\mathbb{R}^3$, or higher dimensions? I.e., does every simple arrangement of $n$ planes have an inducing simple polyhedron of $n$ faces?

It could be the answers are relatively easy: *none* and *no* respectively...?
If anyone sees quick arguments, I'd appreciate hearing them.
Thanks!

**Addendum**.
Here is an attempt to illustrate Gjergji Zaimi's idea, as I interpret it.
The hexagon induces the arrangement of lines in the horizontal plane,
and the polyhedron "attached" to the hexagon would be the intersection of the two tetrahedra.