Definition

For the purpose of these pages, a "regular map" is defined as an of embedding a graph (a set of vertices and edges) in a compact 2-manifold such that

• the 2-manifold is partitioned into faces,
• each face has the topology of a disc,
• its rotational symmetry group is dart-transitive: for any two darts, there is a rotation of the whole thing that takes one face to the other.

This definition of "faces are regular" and "vertices are regular" does not imply that we can rotate a face or vertex so as to bring each edge to where the next was ‒ see S³{14,3} for a counterexample.

Note also that this definition excludes star-polyhedra.

For the reasoning behind this choice of definition, and a statement of an alternative definition, see What do we mean by "Regular" for Regular Maps?

For the sphere, this definition gives the five regular maps usually known as the five "platonic solids" or "regular polyhedra", and some other things. For manifolds of higher genus, it gives some things which which have a pleasing amount of symmetry, but will be less familiar to many readers.

Further restrictions

Further, optional, criteria are listed below. Regular maps violating these criteria are listed on these pages, with red marks indicating the violations.

1.   Each face has at least three edges
2.   Each vertex has at least three edges
3.   A face may not share a vertex with itself, equivalently a vertex may not share a face with itself.
4.   A face may not share an edge with itself, equivalently an edge may not share a face with itself.
5.   An edge may not share a vertex with itself, equivalently a vertex may not share an edge with itself.
6.   It is "flag-transitive", with full symmetry including reflection, not chiral

Replete, and Singular, Regular Maps

A regular map is said to be replete if it has a symmetry which does not fix all faces, and a symmetry which does not fix all vertices. Such regular maps are indicated by a green letter R on the index pages:

For example, in the sphere, there are five replete regular maps, the tetrahedron, cube, octahedron, icosahedron and dodecahedron. The hosohedra and bipolygons are not replete.

A regular map is said to be singular if its vertex-multiplicity and its face-multiplicity are both 1. Such regular maps are indicated by a green letter S on the index pages: