  We study and classify regular and semi-regular tessellations of Riemann
surfaces of various genera and investigate their corresponding supersymmetric
gauge theories. These tessellations are generalizations of brane tilings, or
bipartite graphs on the torus as well as the Platonic and Archimedean solids on
the sphere. On higher genus they give rise to intricate patterns. Special
attention will be paid to the master space and the moduli space of vacua of the
gauge theory and to how their geometry is determined by the tessellations.
