Petrie dual


In topological graph theory, the Petrie dual of an embedded graph 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 may be denoted.
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 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/2h, where g is the group order, and h is the coxeter number of the group.
For example, the Petrie dual of a cube
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: