The Genus-3 regular Map {6,6}

This regular map has four hexagonal faces, all meeting twice each at three of its four vertices. It has 12 edges, and an Euler characteristic of -4. It is shown to the right.

Its symmetry group is A4×C2.

It is self-dual. It is the double cover of S²:{6,6}. Its Petrie dual is C⁴{4,6}.

Each face shares three of its vertices with itself. Some readers may consider that this invalidates it as a map.

The edges of this map can be three-coloured, as shown to the left.

Its Petrie polygons have four edges, and are Hamiltonian. One is shown in red above.	Its holes have three edges. One is shown in red above.	Its 2nd-order Petrie polygons have four edges. One is shown in red above.	Its 3rd-order holes have two edges. Two are shown above, one in red and one in green.

The rest of this page describes a related, but irregular, map. It can be derived from the regular one above by putting a twist of π into each of the three "tunnels" of the genus-3 surface.

A less regular map S3:{6,6}

The figure shown and described below is not a regular map. Some of its Petrie polygons have two edges, some have four. It is not edge-transitive: any edge can be rotated while preserving the map, but the edges fall into two classes such that members of different classes cannot be interchanged by a rotation (they can be by a reflection).

This genus-3 map has four hexagonal faces, each meeting each of the four vertices, one of them three times. It has 12 edges, and an Euler characteristic of -4. It is shown to the right.

Its symmetry group is A4.

Each face shares three of its vertices with itself. Some readers may consider that this invalidates it as a map.

The edges of this map can be three-coloured, as shown to the left.