Regular map (graph theory)

In mathematics, a regular map is a symmetric tessellation of a closed surface.

More precisely, a regular map is a decomposition of a two-dimensional manifold (such as a sphere, torus, or real projective plane) into topological disks such that every flag (an incident vertex-edge-face triple) can be transformed into any other flag by a symmetry of the decomposition.

Regular maps are, in a sense, topological generalizations of Platonic solids.

Regular maps are classified according to either: the genus and orientability of the supporting surface, the underlying graph, or the automorphism group.

Regular maps are typically defined and studied in three ways: topologically, group-theoretically, and graph-theoretically.

It is a crucial fact that there is a finite (non-zero) number of regular maps for every orientable genus except the torus.

of flags, generated by three fixed-point free involutions r0, r1, r2 satisfying (r0r2)2= I.

that fixes both a vertex v and a face f, but reverses the order of the edges.

The following is a complete list of regular maps in surfaces of positive Euler characteristic, χ: the sphere and the projective plane.

[2] The images below show three of the 20 regular maps in the triple torus, labelled with their Schläfli types.

Regular maps exist as torohedral polyhedra as finite portions of Euclidean tilings, wrapped onto the surface of a duocylinder as a flat torus.

Here's an example {4,4}8,0 mapped from a plane as a chessboard to a cylinder section to a torus.

The hexagonal hosohedron , a regular map on the sphere with two vertices, six edges, six faces, and 24 flags.
The regular map {6,3} 4 , 0 on the torus with 16 faces, 32 vertices and 48 edges.
The hemicube, a regular map.
Branko Grünbaum identified a double-covered cube {8/2,3}, with 6 octagonal faces, double wrapped, needing 24 edges, and 16 vertices. It can be seen as regular map {8,3} 2,0 on a hyperbolic plane with 6 colored octagons. [ 7 ]