The study of which regular (i.e. symmetric) graphs can be the skeleton of a regular map (an embedding into a surface) has a long-ish history. We have theorems saying for which values of $n$ the complete graph $K_n$ is symmetrically embeddable. Ditto for $K_n,n$ and the $n$-dimensional cube.
The generalizations to HYPERgraphs and HYPERmaps are much less familiar. We will examine symmetry in these more general cases, and we will find, after a suitable introduction, that they have an unexpected charm.