Automorphism group of r, q -polycycles

If an (r, ) poly cycle P is finite, then it has a single boundary and Aut(P) is a dihedral group consisting only of rotations and mirrors around this boundary. So its order divides 2r, 4, or 2q, depending on what Aut(P) fixes the center of an r-gon, the center of an edge, or a vertex. 

The number of chiral (i.e. with Aut(P) containing only rotations and translations) proper (5,3)-, (3,5)-polycycles is 12,208 (amongst, respectively, all 39,263.) 

Only r-gons and non-Platonic plane tilings r, q are isotoxal their respective automorphism groups are Crv and T 2, r, q). The group Aut([r, q] — f) is Crv in five Platonic cases none is isotoxal, isogonal, or isohedral polycycle, except of isohedral 3,3 — / = (3,3)-star. 

We call the set of q r-gons with a common vertex, the (r, q)-star of -gons. 

Theorem 6.2.1 Let P bean isohedral (r, q)-polycycle then it holds that  

---