Coxeter

H.S.M. Coxeter, Introduction to Geometry, Wiley, New York, 1961 Proc. Symp. Pure Math. Am. Math. Soc. 1 (1963) 53. 

Coxeter, H. (1991) Regular Complex Polytopes. New York Cambridge University Press. (Discusses the properties of polytopes, the 4-D analogs of polyhedra.)... 

Consider now 2-dimensional Coxeter groups (see, for example, [Cox73, Hum90]). By T (l, m, n) is denoted a Coxeter triangle group. It is defined abstractly as the group with generators a, b, c and relations ... 

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 ... 

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. 

Proof. The description of those spheres is given in [GrZa74] and it is, actually, a Goldbeig-Coxeter construction. ... 

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. 

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. 

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

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. 

Cox71] H.S.M. Coxeter, Virus macromolecules and geodesic domes, in A Spectrum of Mathematics, ed. by J. C. Butcher, 98-107, Oxford University Press/Auckland University Press, 1971. 

Cox73] H. S. M. Coxeter, Regular Polytopes, 3rd edition, Dover, 1973. 

Dre87] A. W. M. Dress, Presentations of discrete groups, acting on simply connected manifolds, in terms of parametrized systems of Coxeter matrices — a systematic approach, Advances in Mathematics 63-2 (1987) 196-212. 

DuDe03] M. Dutour and M. Deza, Goldbetg-Coxeter construction for 3- and 4-valent plane graphs, Electronic Journal of Combinatorics 11-1 (2004) R20. 

Hum90] J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, 1990. 

Private communication with Donald Coxeter, Prof. Emeritus, Math. Dept. Univ. Toronto, Toronto, Ontario, Canada M5S 3G3. 

An initial approach to fullerene enumeration was based on point-group symmetry (Fowler 1986 Fowler et al. 1988) and involved an extension of Coxeter s (1971) work on icosahedral tessellations of the sphere and of methods for the classification of virus structures (Caspar Klug 1962). This approach led to magic numbers in fullerene electronic structure (Fowler Steer 1987 Fowler 1990) and will be described briefly here. 

In the last two sections of this chapter, we impose specific conditions on sets of involutions. In Section 3.5, we look at constrained sets of involutions, and in the last of the six sections of this chapter, we look at constrained sets of involutions which satisfy the exchange condition. According to what we said in the preface of this monograph, such sets of involutions are called Coxeter sets. 

It might be worth mentioning that, via the group correspondence, most of the results about Coxeter sets which we compile in the last section are natural generalizations of well-known facts on Coxeter groups. 

We call L a Coxeter set if L is constrained and satisfies the exchange condition. 

We consider Theorem 3.6.4 and Theorem 3.6.6 to be the main results of this section. The equations given in these theorems are crucial for our approach to Coxeter sets. 

We are assuming that L is a Coxeter set. In particular, L is constrained. Under this hypothesis, one can modify the exchange condition, and we shall do this in the following lemma. 

In the last section of this chapter, in Section 8.7, we present identities about roots of unity in integral domains. The results will be useful in Section 12.4 where we shall investigate Coxeter sets of cardinality 2. 

Later in this chapter, in Section 10.6, we shall assume additionally that (L) has finite valency. It is one of the main goals of this chapter to show that (in this case) L is a Coxeter set (defined in Section 3.6) or a Moore set. (The definition of a Moore set will be given in Section 10.6.) This goal will be achieved in Theorem 10.6.6. 

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