Isohedral -polycycle

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. 

Using the algorithm of Theorem 6.2.2, in [DeStOOb], all elliptic isohedral polycycles were found (see Figure 6.1) 11 finite ones, 7 infinite ones with strip groups, and a (5,3)-polycycle with Aut P) = T (2,3, oo). 

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

Proofs Consider an isohedral (r, )-polycycle P and fix a face F in it The automorphism group of an r-gon is the dihedral group Cr the stabilizer Stab(F) of F in P is a subgroup of Crv. Since Crv is finite, we have a finite number of possibilities... 

If P is an isogonal (r, )-polycycle, then either no vertex belongs to the boundary and we have no boundary, or every vertex belongs to the boundary and P is outer-planar. If P is outerplanar, then we can consider it as an (r, q + l)-polycycle and so the outer dual is well defined and isohedral. ... 

Theorem 6.2.3 For any r, there exists isohedral (r, 4)-polycycle with exactly one boundary edge on each r-gon. 

The r-gon has r — 2 interior vertices. All those transformations fit together around those vertices. So, they yield an isohedral (r, 4)-polycycle. ... 

We now list some existence and classification results for isohedral (r, q)-polycycles obtained by using the previous formalism. 

Figure 6.1 All 19 isohedral elliptic (r, g)-polycycles only r-gons and Prismoo, APrisnico are isogonal (the remaining 9th elliptic isogonal polycycle is given on the right-hand side of Figure 6.3)...

Hgure 6.2 Conjectuially complete list of eight families of isohedral (r, g)-polycycles with a strip group of symmetry presented as decorated Prism (only 1st and 5th, for a = 0, are isogonal)... 

Table 6.1 Triples (jc, y, z) of numbers x of isohedral, y of isogonal, z of isohedral, and isogonal (r, q)-polycycles, different from r-gon and r, q],for r,q < 8...

We can consider -isotoxal, -isohedral, and -isogonal (r, [Pg.70]

An isohedral (6,3)-polycycle with non-Coxeter symmetry group... 

The hypothesis of simple connectedness of (r, )-polycycle radically simplifies the enumeration of the isohedral and isogonal ones. 

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