  Roughly, a conformal tiling of a Riemann surface is a tiling where each tile
is a suitable conformal image of a Euclidean regular polygon. In 1997, Bowers
and Stephenson constructed an edge-to-edge conformal tiling of the complex
plane using conformally regular pentagons. In contrast, we show that for all $n
\geq 7$, there is no edge-to-edge conformal tiling of the complex plane using
conformally regular $n$-gons. More generally, we discuss a relationship between
the combinatorial curvature at each vertex of the conformal tiling and the
universal cover (sphere, plane, or disc) of the underlying Riemann surface.
This result follows from the work of Stone (1976) and Oh (2005) through a rich
interplay between Riemannian geometry and combinatorial geometry. We provide an
exposition of these proofs and some new applications to conformal tilings.
