The major skeleton, elementary polycycles, and classification results

2 The major skeleton, elementary polycycles, and classification results 

The classification of all Frank-Kasper ( 5, b, 3)-spheres can be done using the elementary polycycles decomposition exposed in Chapter 7. If G is a Frank-Kasper ( 5, b), 3)-sphere, then we remove all its Agonal feces and obtain a (5, 3)gen-polycycle. This (5, 3)ge -polycycle is decomposed into elementary (5,3)gen-polycycles along bridges (see Chapter 7 for definition of those notions). In this chapter a bridge is an edge, where the two vertices are contained in different Agonal feces. 

Lemma 13.2.1 Take a Frank-Kasper ( 5, b), 3)-map that is not snub Prisms. Then the set of all bridges, together with edges incident to b-gonal faces, establish a partition of the set of 5-gonalfaces into O-elementary (5,3 fpolycycles. 

Our basic example is the graphite lattice sheet, i.e. the 3-valent tiling 6, 3 of the plane by 6-gons. At every vertex of this tiling, we can substitute a 0-elementary (5, 3)-polycycles, either Ei or C3. If we substitute only Ei, we obtain a ( 5, 12, 3)-plane that is 12R0. In order to obtain a ( 5, 13, 3)-plane, we need to substitute a part of the E, by some C3, such that every 6-gon is incident 

Furthermore, we can partition the set of vertices of the graphite lattice 6,3 into 6 sets Oi, such that every 6-gon contains exactly one vertex in the set Ot. So, by putting C3 into vertices of sets 01. 0 , we obtain a ( 5,12 + /, 3) plane that is (12 + 0 0- 

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