The Genus-2 Regular Map {8,8}

This genus-2 regular map, shown to the right, has one octagonal face, meeting itself eight times at the single vertex. It has four edges, and a Euler characteristic of -2.

It is self-dual. Its double cover is S³:{8,8}. Its Petrie dual is the 4-hemihosohedron. It can be cantellated to produce S²:{8,4}.

Its rotational symmetry group is D8.

The face shares all its vertices and all its edges with itself. The edges join a vertex to itself. Some readers may consider that this invalidates it as a regular map.

Its holes have four edges. Its Petrie polygons have two edges.

Antipodal Faces and Vertices

The face is antipodal to the vertex, and vice versa. The four edges form a single antipodal set. Rotating any one edge about its centre causes every other edge to remain where it is and rotate about its own centre: this is the involution of its rotational symmetry group.