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.

typeSymmetricSame manifoldGuaranteedUnique
Petrie dualityYNY *Y
rectificationNYY Y
alternationNYY Y
full shurikenNNNY
half shurikenNNNY
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.


* 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