The Genus-3 Regular map {8,3}

This regular map has 12 octagonal faces, of which three meet at each of its 32 vertices. It has 48 edges, and a Euler characteristic of -4. It is shown to the right.

Its Petrie polygons have six edges.

It is called the "Dyck map". As a graph, it is the Dyck graph

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

It is a double cover of S²:{8,3}.

It is the dual of S³:{3,8}. It is the Petrie dual of S¹{6,3}_(4,4).

Its rotational symmetry group has order 96.

If you prefer its diagram to be connected, the two octagonal components above can be pushed together to form a 14-gon, as shown to the right.

The diagram to the left shows the same regular map, drawn on a tunnelled cube with S4 symmetry.

A similar, but irregular, map

The figure shown and described below is not a regular map. Each face borders six other faces, two of them twice, so it is not edge-transitive.

This map was constructed from that for S³:{7,3} by merging adjacent pairs of faces and erasing their common edges. This has left angles within edges, which should be ignored.

Like the regular one shown above, has 12 octagonal faces, of which three meet at each of its 32 vertices. It has 48 edges, and a Euler characteristic of -4. However its Petrie polygons have 24 edges.

The diagram to the right has three faces in various shades of red, and of blue, and of yellow, and of green. If we regard the diagram as showing a cube with three tunnels bored through it, one tunnel-mouth in each face, we see that the colours clockwise around each tunnel-mouth are the six cyclic permutations of {red, green, yellow, blue}.

The diagram to the left is equivalent.