These polygons and projected graphs are useful in visualizing symmetric structure of the higher-dimensional regular polytopes.
They form the faces of another embedding of the same graph, usually on a different surface, called the Petrie dual.
In periods of intense concentration he could answer questions about complicated four-dimensional objects by visualizing them.
Coxeter explained in 1937 how he and Petrie began to expand the classical subject of regular polyhedra: In 1938 Petrie collaborated with Coxeter, Patrick du Val, and H. T. Flather to produce The Fifty-Nine Icosahedra for publication.
[4] Realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes.
In the images of dual compounds on the right it can be seen that their Petrie polygons have rectangular intersections in the points where the edges touch the common midsphere.
Each pair of consecutive sides of the 3-cube's Petrie hexagon belongs to one of its six square faces.