# 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 S^{3}{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.

- 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

If a regular map is shown with one or more red blobs, you may
choose to ignore it, deprecate it, or describe it as "degenerate"
or "pathological". Or you may reserve your contempt for those with
what you consider the more severe red blobs.
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: