The Genus-3 Regular Map {3,7}

This genus-3 regular map has 56 triangular faces, of which seven meet at each of its 24 vertices. It has 84 edges, and a Euler characteristic of -4. It is shown to the right. The colouring is arbitary and irregular, it could have been done better.

Its Petrie polygons have 8 edges. A Petrie polygon is the route you traverse if you follow an edge of the polyhedron, taking the sharpest left turn and the sharpest right turn at alternate vertices.

Antipodal Faces

If we make one face red, then each face sharing an edge with a red face orange, then each face sharing an edge with a orange face yellow, then green, then cyan, then blue, then purple, we find that there is one purple face. This is shown in the diagram to the right.

Each vertex is antipodal to two other vertices. This can be seen by comparison with the dual map S³{7,3}, whose faces form antipodal threesomes.