  Consider the following inductively defined set. Given a collection $U$ of
unit magnitude complex numbers, and a set initially containing just 0 and 1,
through each point in the set, draw lines whose angles with the real axis are
in $U$. Add every intersection of such lines to the set. Upon taking the
closure, we obtain $R(U)$. We investigated for which $U$, $R(U)$ is a ring.
  Our main result holds for when $1 \in U$ and $|U| \ge 4$. If $P$ is the set
of real numbers in $R(U)$ generated in the second step of the construction,
then $R(U)$ equals the module over $\mathbb{Z}[P]$ generated by the set of
points made in the first step of the construction. This lets us show that
whenever the pairwise products of points made in the first step remain inside
$R(U)$, it is closed under multiplication, and is thus a ring.

Rings and Algebras