Finding Regular Maps

Suppose you know all the statistics of some regular map, and you want to draw it. I know of no reliable method of findng a way to draw it. You just try one thing and another, until you happen to succeed.

But if you just want to draw some regular maps, and you don't mind which they are, you can use the method illustrated on this page. You start with a regular map (in this case, the cube), and you transform it in some way, and check to see if you have another regular map. Some possible transformations are listed at Relationships between pairs of Regular Maps.

 We start with the most familiar of all regular maps, the cube. cube Each vertex, each face, and each edge is antipodal to another such, so it must be the double cover of something. We find that it is the double cover of:  the hemicube. hemicube We form the full shuriken of the hemicube. It is C5:{8,4}. Unfortunately, C5:{8,4} is not a regular map – some of its Petrie polygons have three edges, and some have four. C5:{8,4} However, we don't give up at this point, we carry on, adding one major diagonal to each octagonal face so as to diagonalise C5:{8,4}. We obtain C5:{5,5}. C5:{5,5} is a regular map. C5:{5,5} Even better, it is self-dual. This means that we can rectify it, to give C5{5,4}. C5:{5,4} Then we can construct the dual of C5{5,4}. It is C5{4,5}. C5:{4,5} The last two are in a non-orientable surface. We can construct their double covers, S4:{5,4} and S4:{4,5}, in the orientable surface S4. S4:{5,4} S4:{4,5}