The 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 only two other faces, twice each. All eight faces meet at each vertex, in the same cyclic order or its reverse.

Its Petrie polygons have eight edges, its holes have two, its 2nd-order Petrie polygons have four, 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, its vertices form antipodal pairs, its edges 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}.

A similar, but irregular, polyhedron

The figure shown and described below is not a regular map. Some faces border four different faces; some border three different faces, one of them twice.

This irregular map also 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 left.