Frank-Kasper spheres and tori

We call the Frank-Kasper ( 5, , 3)-map any ( 5, b, 3)-map that is bRo (in chemistry and crystallography Frank-Kasper polyhedra are just four polyhedra, dual to all four ( 5,6, 3)-polyhedra that are 6Rq). Note that the only oriented ( a, b], 3)-maps with a 4 that is bRo are Prismb, Cube, Tetrahedron, and Bundle3. 

Recall that pb is the number of fc-gonal faces of a map and denote by xt with i = 0,1,2,3 the number of vertices contained exactly in i a-gonal faces. 

The above formula yield finiteness of the number of ( 5, b], 3)-polycycles for b 9 but we will see in the next section that more general finiteness results hold. 

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