The Genus-2 Regular Map {8,8}

This genus-2 regular map, shown to the right, has one octagonal face, meeting itself eight times at the single vertex. It has four edges, and a Euler characteristic of -2.

It is self-dual. Its double cover is S3:{8,8}2. Its Petrie dual is the 4-hemihosohedron. It can be cantellated to produce S2:{8,4}.

Its rotational symmetry group is C8.

Faces share vertices with themselvesFaces share edges with themselvesVertices share edges with themselvesThe face shares all its vertices and all its edges with itself. The edges join a vertex to itself. Some readers may consider that this invalidates it as a regular map.

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

Antipodal Faces and Vertices

The face is antipodal to the vertex, and vice versa. The four edges form a single antipodal set. Rotating any one edge about its centre causes every other edge to remain where it is and rotate about its own centre: this is the involution of its rotational symmetry group.

Other regular maps on the genus-2 oriented surface.
Index to other pages on regular maps.
Some pages on groups

Copyright N.S.Wedd 2009