The Genus-3 Regular Map {8,8}

This genus-3 regular map, shown to the right, has one 14-gonal face, meeting itself six times at each of its vertices. It has seven edges, and a Euler characteristic of -2.

Its dual is S³:{7,14}. Its Petrie dual is the 7-hosohedron.

Its rotational symmetry group is D14.

the face shares all its vertices and all its edges with 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

Each vertex is antipodal to the other; the seven edges form a single antipodal set. Rotating one edge about its centre causes every other edge to remain where it is and rotate about its centre: this is the central involution of its rotational symmetry group.