# Petrie Duals

Petrie duality is a symmetric relationship between pairs of regular maps. Every non-chiral regular map has a Petrie dual. In some cases its Petrie dual is itself; it is then described as "self-Petrie dual".

If a regular map is not chiral, and is described by
{p,q}r   V   F   E
(meaning, each face has p edges, each vertex has q edges, each Petrie polygon has r edges, it has V vertices, F faces and E edges), then its Petrie dual exists, is a regular map, and is described by
N:{r,q}p   V   Fp/r   E.

This relationship is reflexive: if A is the Petrie dual of B, then B is the Petrie dual of A.

ARM denotes Petrie duality by π

For example, the Petrie dual of the tetrahedron is the hemicube. A regular map can be self-Petrie dual: an example is S2:{8,4}.

If a regular map is chiral, then its Petrie dual exists, but is not a regular map. For example, {6,3}(1,3) is chiral. Its Petrie dual is S3:{14,3}, which is not a regular map.

Duality swaps p and q in {p,q}r, while Petrie duality swaps p and r. Applying various combinations of these two operations allows us to generate all six possible permutations of the set (p,q,r). The six regular maps thus generated may be all distinct, or only three of them distinct, or only two of them distinct, or they may all be the same. In any case, the Petrie dual of the dual of the Petrie dual of A is necessarily the same as the dual of the Petrie dual of the dual of A.

### The name "Petrie dual"

Petrie duality, like duality, is a symmetric relationship, hence the "dual" part of the name. Petrie duals however are based on Petrie polygons, which were named after the mathematican John Flinders Petrie by Coxeter.