The Goldberg-Coxeter construction

The Goldberg-Coxeter construction takes a 3- or 4-valent plane graph Go, two integers k and l, and returns another 3- or 4-valent plane graph denoted by GC, /(Go)-This construction occurs in many contexts, whose (non-exhaustive) list (for the main case of G0 being Dodecahedron) is given below  

1 Every fullerene ( 5,6), 3) of symmetry I or Ih is of the form GC, / Dodecahedron) for some k and l, i.e. is parametrized by a pair of integers k, l 0. This result was proved by Goldberg in [Gol37] see some other proofs in [Cox71] and Theorem 2.2.2. 

2 The famous fullerene C Ih) (called buckminsterfullerene or soccer ball) has the skeleton of GC (Dodecahedron). GCpj(Dodecahedrori) constitute a particularly studied class of fullerenes (see [FoMa95, Diu03]). 

3 A certain class of virus capsides (protein shells of virions) have a spherical structure, that is modeled on dual GCk,i(Dodecahedron) (see [CaK162, Cox71, DDG98]). 

4 Geodesic domes, designed with the method of Buckminster Fuller, are based again on those two parameters k and l (see [Cox71]). 

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Triacon. He also called the Goldberg-Coxeter construction Breakdown of the initial plane graph Go. 

The Goldberg-Coxeter construction for 3- or 4-valent plane graphs can be seen, in algebraic tains, as the scalar multiplication by Eisenstein or Gaussian integers in the parameter space. More precisely, GC/y corresponds to multiplication by complex number k + l(o or k + li in the 3- or 4-valent case, respectively. 

Remark 2.2.4 The possible symmetries of ( 2,3), 6f spheres have not been determined yet. Also, the Goldberg-Coxeter construction has not been defined for 6-valent spheres, although we do not see an obstruction to it. Also it could be interesting to extend remark 2.2.3 on those spheres. 

Both of those cases correspond to the local configuration arising in the Goldberg-Coxeter construction (see Chapter 2). Moreover, the choice of a local configuration determines the whole structure completely, i.e. there is only one choice globally. 

Figure 1. The Goldberg/Coxeter construction of an icosahedral triangulation of the sphere. Twenty copies of the large equilateral triangle fit together to form the net of a master icosahedron, which yields an icosahedral fullerene on taking the dual. The Coxeter parameters in this particular example are a = 3 and b= i.e., to reach B from A, take 3 steps along the horizontal to the right, turn left through 60°, and take 1 step.

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. 

A method for counting the fullerenes within each symmetry can be based on a 60-year-old mathematical construction that has proved useful for classifying polyhe-dra, viruses, and geodesic domes. The original work by Goldberg, " as rediscovered by Caspar and Klug in the context of viruses, " implicit in the geodesic dome constructions of Buckminster Fuller, and reviewed by Coxeter, applies to structures of icosahedral (A or I) symmetry. 

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