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}₂. Its Petrie dual is the 4-hemihosohedron. It can be cantellated to produce S²:{8,4}.

Its rotational symmetry group is C8.

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 Petrie polygons have two edges. Its holes have four edges. Its 2nd-order Petrie polygons have two edges. Its 3rd-order holes have eight edges. Its 4th-order holes 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.