The Genus-3 Regular Map {8,4}

This regular map has four octagonal faces, all meeting at each of its eight vertices. It has 16 edges, and a Euler characteristic of -4. It is shown to the right.

Each of its faces shares edges with only two other faces, four times each. All eight faces meet at each vertex, in the same cyclic order or its reverse.

Its Petrie polygons have eight edges, and its holes have two. It is the result of cantellating S³:{8,8}.

Its faces form antipodal pairs, its vertices form antipodal pairs, its edges form antipodal pairs.

Its rotational symmetry group has order 32.

Its dual is S³:{4,8}. It is self-Petrie dual.