Definition

For the purpose of these pages, a "regular map" is defined as an embedding of 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 dart to the other.

Note 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 many things which which have a pleasing amount of symmetry, and will be less familiar to most 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
7.   It is not a "polyhedral map"

Index to Regular Maps
Glossary for Regular Maps