# 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.

Other relationships between regular maps

General Index