Coxeter cell

Given a pair of dual period skeletal graphs, G and G" with a certain space group, the first step is to identify the Coxeter cell capable of filling space by repeated applications of the mirror symmetries of the space group (see Coxeter 1963). There are only seven possibilities for this cell, each of which is either a tetrahedron, a rectangular parallelepiped, or a right prism. 

At least one toroidal polyhex that is cell-complex exists for all numbers of vertices v > 14. The unique cell-complex toroidal fullerene at v = 14 is a realization of the Heawood graph. It is GC2,i(hexagon) in terms of Goldberg-Coxeter construction and is the dual of K7, which itself realizes the 7-color map on the torus. This map and its dual are shown in Figure 3.1. 

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