Regular Maps and other Maps

A regular map is defined as an embedding of a graph into a compact 2-manifold in such a way that

the manifold is partitioned into regions (known as "faces");
each face has the topology of a disc;
the resulting embedding is highly symmetric (specifically, it is flag-transitive).

What happens if we relax the last requirement, and allow asymmetrical embeddings? We get things like "regular maps", but generally not regular. These are called "maps", and are the subject of these papers J01, J02.

This page is about these maps. Some are shown below.

Duality, and Petrie duality

Two regular maps are said to be "dual" if the faces of each correspond to the vertices of the other. Every regular map has a dual, which is necessarily also be regular. The dual of a regular map may be itself.

The concept of duality applies also to maps, with the same definition. Every map has a dual, which must also be irregular. The dual of a map may be itself.

Similarly, two regular maps are said to be "Petrie dual" if the faces of each correspond to the Petrie polygons of the other. Every regular map has a Petrie dual, which must also be regular. Some regular maps are self-dual. The concept of Petrie duality likewise extends to maps.

Some authors use a slightly broader definition of "regular map", requiring dart-transitivity rather than flag-transitivity. Thus they allow chiral regular maps. The Petrie dual of a chiral dart-transitive map is not dart-transitive (nor chiral). For example, the Petrie dual of the chiral, but otherwise regular map {4,4}_(2,1) is the non-chiral but irregular C5:{10,4}.

Hexads

The two operations, duality and Petrie duality, both permute the set (faces, vertices, Petrie polygons). So between them, they generate all six permutations of this set. Therefore regular maps can be grouped into "hexads", which usually have six members, but may be degenerate, with three, two, or one.

This is also true of maps.

A dual pair of maps must have the same number of edges, and must be in the same manifold. A Petrie dual pair of maps must have the same number of edges, and be an embedding of the same graph. Therefore each hexad is made up of six (or three or two or one) maps all with the same number of edges. It is therefore useful to group maps by their number of edges.

Another reason to group them by number of edges, rather than number of vertices or number of faces, is that there are only finitely many distinct maps for any particular finite number of edges, whereas for any positive number of vertices, or of faces, there are infinitely many maps.

All Maps with 0, 1, 2, and 3 Edges

There is one map with 0 edges, 3 with 1 edge, 11 with 2 edges, and 63 with 3 edges.

The first table below shows all maps with 0, 1, and 2 edges. The second table below shows all maps with 3 edges. In these two tables, a green border indicates that the map is regular, and clicking on it links to a page on that regular map.

The blue lines of each table partition it horizontally according to the embedded graph, and vertically according to the embedding manifold.

This page does not show hexads. To see the hexads, see the pages maps with 0, 1, and 2 edges and maps with 3 edges. On those pages, the green borders still imply regularity, but do not indicate links.