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 S^{2}:{8,3}.

It is the dual of S^{3}:{3,8}.
It is the Petrie dual of S^{1}{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.

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
S^{3}:{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.

Other regular maps on the genus-3 oriented surface.

Index to other pages on regular maps.

Some pages on groups

Copyright N.S.Wedd 2009