A 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.

It is an embedding of the graph K_4,4 in S³.

Its Petrie polygons have eight edges, and its holes have four.

It is the result of cantellating S³:{8,8}.

It is different from this {8,4}, whose holes have two edges. However their diagrams are very similar, differing only in the lettering of the "sewing instructions".

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. It is a double cover of S²:{8,4}.