Non-compact Riemann surfaces are equilaterally triangulable

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 Belyi surface: There exists a holomorphic branched covering f:X→Cˆ that is branched only over −1, 1 and ∞. It follows that every Riemann surface is a branched cover of the sphere, branched only over finitely many points.