In this talk, we consider the following question. Suppose that we glue a (finite or infinite) collection of closed equilateral triangles together in such a way that we obtain an orientable surface. The resulting surface is a Riemann surface; that is, it has a natural conformal structure (a way of measuring angles in tangent space). We ask which Riemann surfaces are *equilaterally triangulable*; i.e., can arise in this fashion.

The answer in the compact case is given by a famous classical theorem of Belyi, which states that a compact surface is equilaterally triangulable if and only if it is defined over a number field. These *Belyi surfaces* - and their associated “dessins d’enfants” - have found applications across many fields of mathematics, including mathematical physics.

In joint work with Chris Bishop, we give a complete answer of the same question for the case of infinitely many triangles (i.e., for non-compact Riemann surfaces). The talk should be accessible to a general mathematical audience, including postgraduate students.