Through creating undergraduate course materials, Grothendieck was led to define a new notion of regular polyhedra that holds over arbitrary base rings. He asked in Esquisse d'un Programme for a classification of them over finite fields. In this talk, we will give an answer using the theory of triangular modular curves. Through this we also compute some new cases of monodromy groups of triangular modular curves. This is joint work with John Voight.