The Genus-2 Regular Map {6,6}

This genus-2 regular map, shown to the right, has two hexagonal faces, each meeting three times at each of its 2 vertices. It has six edges, and a Euler characteristic of -2.

It is self-dual. Its double cover is S³:{6,6}. It can be cantellated to produce S²:{6,4}.

Its Petrie dual is the 6-hosohedron.

Its rotational symmetry group is D12.

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

Its Petrie polygons have two edges. Its holes have six edges. Its 2nd-order Petrie polygons have two edges. Its 3rd-order holes have two edges.

Antipodal Faces and Vertices

Each face is antipodal to the other; each vertex is antipodal to the other; the six 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.