Toroidal polyhex

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. 

Cash, G.G. (1998). A Simple Means of Computing the Kekule Structure Count for Toroidal Polyhex Fullerenes. J.Chem.lnf.Comput.Sci., 38, 58-61. 

For more specific cases where the surface network has a simple uniform pattern of rings, we specify the surface type and refer, for example, to a toroidal polyhex (also called a toroidal hexagonal system by John and Walther ) or a toroidal azulenoid, and so on. Others " -prefer to speak more generally of torusenes or toroidal forms of carbon or of graphitic carbon or graphite. 

In this way the adjacency of such a toroidal network can conveniently be represented as a planar rectangular polycyclic structure with opposite sides marked as being identical, i.e., in a labeling, the set of vertices along one side is the same, and in the same sequence, as those of the opposite side (or of a cyclic permutation of them). For toroidal polyhexes this is equivalent to the diagrammatic approach discussed elsewhere. Klein ° uses the Xtxmparity in discussing which identifications are possible. 

Figure 6. "Folding and gluing a polyhex sheet, first to a cylinder, and then to a toroidal polyhex. (The cylinder and the torus are reproduced by kind permission of Dr. Peter John. Technische UoiversitSt Ilmenau, from drawings published by John and WaIther(Ref. I2). ...

Toward a Characterization of the Planar Polyhex Precursor ofa Toroidal Polyhex... 

A polyhex precursor with h rings and , internal vertices, when all perimeter edges are fused by folding, generates a toroidal polyhex with 2h + 2- n,- fewer vertices. (This is because a planar polyhex has Ah + 2 - and a toroidal polyhex 2h vertices.)... 

In general there appears to be nothing unique about polyhex precursors or their derived tori. A given polyhex can be folded in different ways to give different toroidal polyhexes, and a given toroidal polyhex can be derived from more than one planar polyhex. 

In favorable and comparatively simple cases, such as the all-hexagon-faced toroidal polyhexes, and other graphs made by applying tiling rules, their identities are apparent from the methods of construction. Where this is not so, however, it can often be just as difficult with this, as with any other group of structures to decide whether two of them are isomorphic. 

Table 2. An Enumeration of Toroidal Polyhexes with up to 30 Hexagons (60 Vertices)"...

The closed but boundless network of hexagons embedded as an all-hexagon toroidal polyhex can be represented by an infinite planar lattice, on which the set of hexagons forms a parallelogram which repeats itself endlessly in two dimensions (i.e., it is doubly periodic). Figure 14 shows an example of a torus with nine hexagons (A-I). 

Figure 14. The infinitely-repeating-pattem representation of a toroidal polyhex on a planar lattice. This example has 9 hexagons (labeled A-1) and 18 vertices.

It was explained above that the three-element code used to characterize a toroidal polyhex is derived from a convenient special case of the matrix... 

An additional, fourth, rule was introduced to cater for a special case that arises from the automorphism of the C4 graph, when a toroidal polyhex consists of an even number of squares through which the torus is threaded, that is, they have the code (2-b-d) with = 0 or 1, and d is even. Such a case is the pair of toroidal polyhexes (2-0-4) and (2-1-4) in Figure 17. These are distinct, but are isomorphic. When the number of squares is odd, this difference does not arise. Its origin is the automorphism peculiar to the C4 graph that leaves alternate vertices fixed while permuting the others. We therefore write... 

Figure 17. Two pairs of (2-i-d) toroidal polyhex patterns (filled ones track the repeated occurrence of a marker hexagon). If these are carefully examined it can be seen that the patterns for (2-0-5) and (2-1-5), where d is odd, are superimposable (after reversing one of the images) and are therefore identical. When d is even, as in (2-0-4) and (2-1-4), they are not the same. They are, however, isomorphic (see text).

Adjacency Matrices and Their Eigenvalues for Toroidal Polyhexes... 

The smallest number that is divisible by the squares of three distinct prime numbers is 900 (equivalent to 2 x 3 x 5 ), and TPH(450-5-2) is an example of a toroidal polyhex that must be brought under a higher Case. 

Sachs and John devised another method in which K is evaluated as a determinant of a size not more than half the number of vertices, and in some special cases simple formulas can be derived for example, to a toroidal polyhex that can be represented as a folded parallelogram with a side having three hexagons, the following result can be applied ... 

As Aihara and Tamaribuchi recently pointed out, graphite can be classified as an infinite fully benzenoid. Some boundless toroidal polyhex surfaces also fall into this category, and examples are listed in Table 2. It was concluded that for a toroidal polyhex encoded (Section 8.9.1.1) as a-b-d, it will be fully benzenoid if and only if... 

Figure 23. A C50 toroidal polyhex, in fact TPH(30-8-1) (Section 8.9.1.1), that can be viewed as either fully benzenoid, with the vertices accounted for by ten hexagons (in this case 0, 3, 6.9, 12, 15, 18,21,24, and 27), or as fully naphthalenoid, with six hexagon-pairs (in this case 0-1, 5-6, 10-11, 15-16, 20-21, and 25-26). This concurrence does not occur among finite planar polyhexes. Note also that (again unlike the planar series) the "fuir -hexagon set is not unique. This example could equally well be described as fully benzenoid with the ten-hexagon set 1, 4, 7. 10, 13, 16, 19,22, 25, 28.

Molecular mechanics calculations have been carried out on several toroidal structures, including both a toroidal polyhex, and certain mixed systems containing 5-, 6-, and 7-ring sizes. Not surprisingly, suitable mixed systems are more favorable with regard to mechanical strain, and the energy of formation becomes comparable to values for medium-size fullerenes. 

To my knowledge, no electromagnetic radiation spectrum of interstellar space or of combustion processes has been shown to have peaks suggestive of toroidal structures. Nor has anyone suggested any obvious possible mechanism for their formation (although neither is it by any means certain yet how even spheres are formed ). On the other hand, there are encouraging signs, and in 1993 Kirby et al. wrote of toroidal polyhexes that... 

E. C. Kirby, Cylindrical and toroidal polyhex structures, Croat. Chem. Ada 66, 13-26 (1993). 

Before closing this subsection, we should also mention a simple algorithm that enumerates the isomers of a toroidal polyhex. Toroidal polyhexes are fullerenes embedded on the surface of a torus. The word fullerene is not appropriate here because only structures having six-membered rings (not five) are enumerated. This limitation greatly reduces the number of solutions. The number of isomers is formd to increase at only a modest rate that does not exceed 30% of the number of atoms. 

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