The Genus-2 Regular Map {10,5}

This genus-2 regular map, shown to the right, has one decagonal face, meeting five times at each of its two vertices. It has five edges, and a Euler characteristic of -2.

Its dual is {5,10}. Its double cover appears to be S³:{10,5}, which, however, is not a regular map (and therefore probaly not its double cover).

Its Petrie dual is the 5-hosohedron.

Its rotational symmetry group is D10.

The face shares all its vertices and all its edges with itself. Some readers may consider that this invalidates it as a regular map.

Its holes have ten edges. Its Petrie polygons have two edges.

Antipodal Faces and Vertices

Each vertex is antipodal to the other; the five edges form a single antipodal set. Rotating one edge about its centre causes every other edge to remain where it is and rotate about its centre: this is the central involution of its rotational symmetry group.