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 polygon of the dodecahedron is a skew decagon. Seen from the solid's 5-fold symmetry axis it looks like a regular decagon. Every pair of consecutive sides belongs to one pentagon (but no triple does).
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.
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