  We show that every open Riemann surface $X$ can be obtained by glueing
together a countable collection of equilateral triangles, in such a way that
every vertex belongs to finitely many triangles. Equivalently, $X$ is a
\emph{Belyi surface}: There exists a holomorphic branched covering $f\colon
X\to\hat{\mathbb{C}}$ that is branched only over $-1$, $1$ and $\infty$, and
with no removable singularities at the boundary of $X$. It follows that every
Riemann surface is a branched cover of the sphere, branched only over finitely
many points.
