The Genus-3 Regular Map {8,8}₄

This genus-3 regular map, shown to the right, has two octagonal faces, each meeting four times at each of its two vertices. It has eight edges, and a Euler characteristic of -4.

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

Its rotational symmetry group is the modular group of order 16.

Its Petrie polygons have four edges, its holes have four edges, its order-2 Petrie polygons have two edges, its order-3 hole is Eulerian, with eight edges, its order-3 Petrie polygons have four edges, and its order-4 holes have two edges.

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

Antipodal Faces and Vertices

Each face is antipodal to the other; each vertex is antipodal to the other; the edges form antipodal foursomes.