The Genus-3 Regular Map {12,4}

This genus-3 regular map, shown to the right, has two dodecagonal faces, six vertices, and 12 edges. Its Euler characteristic is -4.

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

Its dual is S³:{4,12}. Its Petrie dual is S²:{6,4}. It is the result of cantellating S³:{12,12}.

Its rotational symmetry group is D24.

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

Antipodes

The faces form antipodal pairs, the vertices form antipodal pairs, the Petrie polygons form antipodal pairs, there are antipodal sets of four edges and two holes.