A note on regular polyhedra over finite fields

Grothendieck proposed a theory of regular polyhedra over finite fields in Section 4 of $\mathit{\text{Esquisse}}\mathit{\text{d’un}}\mathit{\text{Programme}}$. He isolates certain key parameters from the automorphism groups of regular polyhedra, which can be extended to any genus and specialized to various rings. In this note we give an interpretation of his sketched theory which explains some of his observations. We are able to compute some explicit examples and address a question Grothendieck raised about them in connection to dessins d’enfants. Finally, we highlight some of Grothendieck’s observations which remain unexplained by our current approach.