A polyhedral embedding of a graph on a surface is an embedding without edge crossings such that all the faces are bounded by simple cycles, and any two faces share a common vertex, share a common edge, or do not intersect at all. Moreover, all the faces are disks.
Given a graph $G$ with a polyhedral embedding on a surface $S$, one can define a dual graph $G'$ embedded in the same surface $S$, by creating a vertex in $G'$ for each face of $G$, and adding an edge between two vertices of $G'$ for each edge that the corresponding two faces in $G$ share.
I have a feeling that this will be simple, but I don't know how to prove it, so here it goes anyways. Question: The dual embedding that one obtains for $G'$ is also polyhedral?