  Brane tilings, sometimes called dimer models, are a class of bipartite graphs
on a torus which encode the gauge theory data of four-dimensional SCFTs dual to
D3-branes probing toric Calabi--Yau threefolds. An efficient way of encoding
this information exploits the theory of dessin d'enfants, expressing the
structure in terms of a permutation triple, which is in turn related to a Belyi
pair, namely a holomorphic map from a torus to a P^1 with three marked points.
The procedure of a-maximization, in the context of isoradial embeddings of the
dimer, also associates a complex structure to the torus, determined by the
R-charges in the SCFT, which can be compared with the Belyi complex structure.
Algorithms for the explicit construction of the Belyi pairs are described in
detail. In the case of orbifolds, these algorithms are related to the
construction of covers of elliptic curves, which exploits the properties of
Weierstrass elliptic functions. We present a counterexample to a previous
conjecture identifying the complex structure of the Belyi curve to the complex
structure associated with R-charges.
