A Genus-3 Map {3,12}

There is a regular map S³:{3,12}: it is the dual of S³:{12,3}. However ...

The figure shown and described below is not a regular map. If we look for cycles of four faces, each bordering its two neighbours in the cycle, we find exactly three such cycles. The four remaining faces are not members of any such cycle.

This genus-3 map has four vertices and 16 triangular faces, of which 12 meet at each of its four vertices. It has 24 edges, and a Euler characteristic of -4. It is shown to the right.

Its Petrie polygons have 12 edges. Its holes have 12 edges.