  We point out an explicit connection between graphs drawn on compact Riemann
surfaces defined over the field $\bar{\mathbb{Q}}$ of algebraic numbers ---
so-called Grothendieck's {\it dessins d'enfants} --- and a wealth of
distinguished point-line configurations. These include simplices,
cross-polytopes, several notable projective configurations, a number of
multipartite graphs and some 'exotic' geometries. Among them, remarkably, we
find not only those underlying Mermin's magic square and magic pentagram, but
also those related to the geometry of two- and three-qubit Pauli groups. Of
particular interest is the occurrence of all the three types of slim
generalized quadrangles, namely GQ(2,1), GQ(2,2) and GQ(2,4), and a couple of
closely related graphs, namely the Schl\"{a}fli and Clebsch ones. These
findings seem to indicate that {\it dessins d'enfants} may provide us with a
new powerful tool for gaining deeper insight into the nature of
finite-dimensional Hilbert spaces and their associated groups, with a special
emphasis on contextuality.
