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, optional, criteria are listed below. Regular maps violating these criteria are listed
on these pages, with red
indicating the violations.
If a regular map is shown with one or more red blobs, some people
may consider it not properly a regular map.
- Each face has at least three edges
- Each vertex has at least three edges
- A face may not share a vertex with itself, equivalently a vertex may not share a face with itself.
- A face may not share an edge with itself, equivalently an edge may not share a face with itself.
- An edge may not share a vertex with itself, equivalently a vertex may not share an edge with itself.
- It is "flag-transitive", with full symmetry including reflection, not chiral
- It is not a "polyhedral map"
Index to Regular Maps
Glossary for Regular Maps