The Genus-3 Regular Map {4,12}

This genus-3 regular map has six square faces, each of which meets twice at each vertex. It has two vertices, 12 edges, and an Euler characteristic of -4.

Its Petrie polygons have six edges, and its holes have four edges. Its Petrie dual is S⁴{6,12}.

Its dual is S³{12,4}.

Its rotational symmetry group is D24.

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

Another diagram

To the right is another diagram of it. This diagram is poor. It is not at all symmetrical. It is neither attractive or instructive. This is because it was drawn by direct conversion of the diagram of the dual, replacing faces by vertices and vertices by faces, and then drawing in the edges, which as you can see loop repeatedly around the corners of the diagram.

If you create a diagram directly from the diagram of the dual, it often looks like this. I generally try to find a better way to present diagrams. I have included this one to show the consequences of creating a diagram directly from the diagram of its dual.

The figure shown to the left is of a S³{4,12} which is not a regular map. Each face borders three other faces, one of them twice. The edges between two faces with two common edges cannot be interchanged with those between faces with only one common edge, so it is not edge-transitive.