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.
| type | Symmetric | Same manifold | Guaranteed | Unique |
|---|---|---|---|---|
| duality | Y | Y | Y | Y |
| Petrie duality | Y | N | Y * | Y |
| rectification | N | Y | Y † | Y |
| alternation | N | Y | Y ‡ | Y |
| splitting | N | N | N | N |
| covering | N | N | N | N |
| diagonalisation | N | Y | N | Y |
| pyritification | N | Y | N | Y |
| full shuriken | N | N | N | Y |
| half shuriken | N | N | N | Y |
| truncation | N | Y | N | Y |
| triambulation | N | Y | N | Y |
* 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