# Relationships between pairs of Regular Maps

This page lists some relationships that occur between pairs of regular maps,
and links to pages describing them.

The following table summarises some properties of these relationships.

**Symmetric ** means that if the relationship holds from A to
B, it also holds from B to A.

**Same manifold ** means that A and B are in the same manifold.

**Guaranteed ** means that if A has the properties that allow
B to exist, then B must exist and be a regular map.

**Unique ** means that for any A there is at most one
B having this relationship to A.

### Remarks

* The Petrie dual of a chiral regular map exists but is not a regular map.

† The rectification of a regular map is itself a regular map
if and only if the original was self-dual.

‡ The alternation of a regular map exists only if its faces
of the original have an even number of edges. It is only regular if the faces of the
original have four edges, or if the faces of the original have twice as many edges as
the valency of its vertices.

If the rectification of A is B, then
the holes of B have the same number of edges as the Petrie
polygons of A.

If A and B are dual, their Petrie
polygons are the same size.

If B is the dual of the Petrie dual of the dual of
A, it is also the Petrie dual of the dual of the Petrie dual
of A.

General index to regular maps