Some Questions

These questions are really addressed to myself. They may be easy, or stupid. If you can answer any of them, I will be pleased to hear from you, at nick@maproom.co.uk.

• Can there exist two different regular maps for the same genus (other than 1) and the same Schläfli formula?

  Yes. See C⁶{10,3}₅ and C⁶{10,3}₁₀.

• My definiton of a regular map allows for chiral regular maps. But every chiral regular map I have found is in S¹. Are there any in other closed manifolds?

  Here is a partial explanation. First. observe that nothing in a non-orientable manifold can be chiral. Now, most things (those with faces in antipodal pairs) in a manifold with an even Euler number are double covers of things in the manifold with half the Euler number. If we keep on halving, we end up with something in a manifold of Euler number odd or 0; but if odd it can't ever have been chiral. Exceptions are the tetrahedron, and S³{7,3}, whose faces are not in antipodal pairs (but they still aren't chiral).

• The page C² claims to show some regular maps, half-edge transitive, but with Petrie polygons of two different sizes. Is this plausible? If it is, might it happen for a regular map on an orientable surface?

  Probably my mistake.

• Often, a double cover of a regular map is itself a regular map. But not always. In what circumstances is it not?

  Probably my mistake.

• Consider a regular map M with rotational symmetry group G. Construct a double cover of M, and look at its rotational symmetry group H. H must be twice the size of G, and have G as a quotient. It may be that H is G×C2, or it may be that H has C2 as a non-trivial central subgroup. Can we tell which of these will be the case, from the way we built the double cover of M?
• I have defined the quality of a regular map in an arbitrary way. For each closed manifold that I have examined in sufficient detail, the highest quality of any regular map is a multiple of 3. Why?
• Three of the regular maps in S⁰ (the octahedron, dodecahedron and icosahedron) have stellated derivatives. It seems that a process analogous to stellation is possible for many regulary maps in other manifolds. Has this been studied?

  You can't sensibly stellate a polyhedron unless its dihedral angle exceeds a right angle. What, for a regular map, corresponds to "dihedral angle"?