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.
[2] It can be obtained from a signed rotation system or ribbon graph representation of the embedding by twisting every edge of the embedding.
Like the usual dual graph, repeating the Petrie dual operation twice returns to the original surface embedding.
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[4] hexagonal faces, the equators of the cube.
Topologically, it forms an embedding of the same graph onto a torus.