The Genus-2 Map {10,3}

The figure shown and described below is not a regular map. See below.

This genus-2 map has three decagonal faces, all three meeting at each of its ten vertices. It has 15 edges, and a Euler characteristic of -2. It is shown to the right.

Its dual is {3,10}.

It appears to have a rotational symmetry group of order 30; but on examination it is found not to be symmetrical. To see this lack of symmetry, label the edges, arbitrarily, as shown to the left. Go round each of the decagons clockwise, noting alternate edge-labels. We find the six cycles (ABCDE) (GHIJK) (PQRST) (KHJGI) (PQRST) (ACEBD). The set of edges {P,Q,R,S,T} appears in the same cyclic order twice, the other two sets of edges appear in two different cyclic orders.

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