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 S^{2}:{6,6}.
Its Petrie dual is C^{4}{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.

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.

Other regular maps on the genus-3 oriented surface.

Index to other pages on regular maps.

Some pages on groups

Copyright N.S.Wedd 2009