Strictly face-regular spheres and tori

We say that a ( a, b), A)-map is aRt if every a-gonal face is adjacent exactly i times to a-gonal faces. It is said to be bRj if every h-gonal face is adjacent j times to fe-gonal faces. If the map is a cell-complex, then above i and j are just the numbers of a- and 6-gonal neighbors. 

An ( a, b], A)-map is said to be strictly face-regular map if it is aR[ and bRj for some i and j. It is weakly face-regular if it is aRt and/or bRj. In this chapter we will enumerate all strictly face-regular maps on sphere or plane. The classification on surfaces of higher genus is very difficult because there is an infinity of possibilities. 

We classify only the strictly face-regular spheres and strictly face-regular normal balanced planes. The plane case contains the toms case as a subcase. For the plane, the Euler formula does not hold, but the condition of normality, discussed thereafter, 

1 Classify maps, whose group acts transitively on vertices, i.e. Archimedean maps. The answer is known for the sphere and the plane. 

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