Where can I find examples of polyhedral embeddings of simple graph with large face-width, such that all the faces have the same length?
By polyhedral embedding I mean an embedding of the graph on a closed 2-dimensional surface, where the boundaries of all the faces are simple cycles, and the intersection of any two faces is either empty, a vertex, or an edge.
The examples should be as non-homogeneous as possible. For example, the Archimedean solids are very homogeneous objects, because all the vertices are equivalent (for any two vertices, there is an isometry that takes one vertex to the other). I admit that I am not using "non-homogeneous" in a clearly defined sense. I am just asking for examples that do not have obvious symmetries.
Note : The face-width of the embedding is the smallest integer $k$ such that there exist $k$ facial walks whose union contains a noncontractible cycle. A facial walk is a walk that bounds a face of the embedding. The length of a face is simply the length of its face walk. Since we are dealing with polyhedral embeddings, all face walks are simple cycles.
