Petrie dual

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).

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.^[1]

The Petrie dual is also called the Petrial, and the Petrie dual of an embedded graph ${\displaystyle G}$ may be denoted ${\displaystyle G^{\pi }}$.^[2] It can be obtained from a signed rotation system or ribbon graph representation of the embedding by twisting every edge of the embedding.