  Let $S_{g}$ denote the closed orientable surface of genus $g$. In joint work
with Huang, the first author constructed exponentially-many (in $g$) mapping
class group orbits of pairs of simple closed curves whose complement is a
single topological disk. Using different techniques, we improve on this result
by constructing factorially-many (again in $g$) such orbits. These new orbits
are chosen so that the absolute value of the algebraic intersection number is
equal to the geometric intersection number, implying that each pair naturally
gives rise to an origami. We collect some rudimentary experimental data on the
corresponding $SL(2, \mathbb{Z})$-orbits and suggest further study and
conjectures.
