Petrie dual

In topological graph theory, the Petrie dual of an embedded graph (on a 2-manifold with all faces disks) is another embedded graph that has the Petrie polygons of the first embedding as its faces.
The Petrie dual is also called the Petrial, and the Petrie dual of an embedded graph formula_1 may be denoted formula_2.
It can be obtained from a signed rotation system or ribbon graph representation of the embedding by twisting every edge of the embedding.
Properties.
Like the usual dual graph, repeating the Petrie dual operation twice returns to the original surface embedding.
Unlike the usual dual graph (which is an embedding of a generally different graph in the same surface) the Petrie dual is an embedding of the same graph in a generally different surface.
Surface duality and Petrie duality are two of the six Wilson operations, and together generate the group of these operations.
Regular polyhedra.
Applying the Petrie dual to a regular polyhedron produces a regular map. The number of skew "h"-gonal faces is "g"/2"h", where "g" is the group order, and "h" is the coxeter number of the group.
For example, the Petrie dual of a cube (a bipartite graph with eight vertices and twelve edges, embedded onto a sphere with six square faces)
has four hexagonal faces, the equators of the cube. Topologically, it forms an embedding of the same graph onto a torus. 
The regular maps obtained in this way are as follows.
There are also 4 petrials of the Kepler–Poinsot polyhedra: