# Finding Regular Maps

These page lists the techniques I have used for finding the regular maps presented in these pages.

• For some genus, list all possible combinations of a,b,c,d,e such that {a,b} with c vertices, d faces and e edges might posibly exist. Choose one of these combinations. Try to draw it.
• When you have a regular map that is not self-dual, build its dual.
• When you have a regular map that is self-dual, cantellate it.
• When you have a regular map, find the size of its Petrie polygons, and try to build its Petrie dual. You can deduce from its Euler number what its genus might be (there are two possibilities if this number is even, one if it is odd).
• When you have a regular map, try to build a double cover of it. This will have twice the Euler number of the original.

You may be able to prove that there is no regular map for a particular manifold, number of edges per face, and number of edges per vertex; i.e. Schäfli formula M{g,h}. Then you can stop looking. But if you find a regular map for a particular M{g,h}, you may not have finished, there may be another one. Unless this is a manifold with Euler number 0, any others will have the same number of vertices etc. as the one you have found.

Some regular maps drawn on orientable 2-manifolds

Copyright N.S.Wedd 2010