A Genus-3 Regular Map {4,8}

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

Each of its faces shares edges with four other faces. All eight faces meet at each vertex.

Its Petrie polygons have eight edges, its holes have four, its 2nd-order Petrie polygons have two, its 3rd-order holes have four, its 3rd-order Petrie polygons have eight, and its 4th-order holes have two.

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

It is a double cover of S²:{4,8}.

Its rotational symmetry group has order 32.

Its dual is S³:{8,4}.