Affine Fullerene C60 in a GS-Quasigroup
It will be shown that the affine fullerene C60, which is defined as an affine image of buckminsterfullerene C60, can be obtained only by means of the golden section. The concept of the affine fullerene C60 will be constructed in a general GS-quasigroup using the statements about the relationships between affine regular pentagons and affine regular hexagons. The geometrical interpretation of all discovered relations in a general GS-quasigroup will be given in the GS-quasigroup .
The fullerenes are closed carbon-cage molecules containing only pentagonal and hexagonal rings.
is the first fullerene that was theoretically conceived and experimentally obtained. The geometrical structure of is a truncated icosahedron with a carbon atom at the corners of each hexagon and a bond along each edge (Figure 1). The sixty-carbon cluster with the geometry of a truncated icosahedron is named buckminsterfullerene [1, 2].
The affine regular icosahedron is defined as an affine image of a regular icosahedron. Let the affine regular icosahedron be given with the pairs of opposite vertices , ; , ; , ; , ; , ; , as in Figure 2. Let us divide each edge of this icosahedron into three equal parts and then omit two lateral parts. On each of the twenty faces of the icosahedron on the sides the three segments in their middle parts are left. Let us connect the adjacent ends of these segments, so that an affine regular hexagon is formed on each side of an icosahedron, and an affine regular pentagon appears in the neighborhood of each vertex of icosahedron (Figure 3).
The obtained polyhedron consists of twelve affine regular pentagons and twenty affine regular hexagons. It is an affine version of buckminsterfullerene which will be called affine fullerene . It is presented in Figure 3, from where it is obvious how the labels of vertices of that polyhedron are chosen, starting from the labels of vertices of the affine regular icosahedron.
We will prove that the complete affine fullerene can be presented only by means of the golden section. The concept of a GS-quasigroup will be used in this consideration.
A quasigroup is said to be a golden section quasigroup or shortly a GS-quasigroup  if it satisfies the (mutually equivalent) identities and the identity of idempotence GS-quasigroups are medial quasigroups; that is, the identity is valid .
As a consequence of the identity of mediality, the considered GS-quasigroup satisfies the identities of elasticity and left and right distributivity; that is, we have these identities: Further, the identities and equivalencies also hold.
Let be the set of points of the Euclidean plane. For any two different points we define if the point divides the pair in the ratio of the golden section. In , it is proved that is a GS-quasigroup. We will denote that quasigroup by because we have if and . The figures in this quasigroup can be used for illustration of “geometrical” concepts and relations in any GS-quasigroup.
3. Affine Regular Pentagons and Hexagons in GS-Quasigroups
From now on, let be any GS-quasigroup. The elements of the set are said to be points.
In each medial quasigroup, the concept of a parallelogram can be introduced by means of two auxiliary points. In , it is proved that the points are the vertices of a parallelogram denoted by , if and only if there are two points and such that and . It is also shown that if the statement holds, then the equalities and are equivalent. In a general GS-quasigroup, the notation of a parallelogram can be characterized by the equivalency (Figure 4).
In , some properties of the quaternary relation Par on the set are proved. We will mention only the property which will be used afterwards.
Lemma 1. From Par and Par there follows Par.
We will say that is the midpoint of the pair of points and we write if and only if holds. The statement holds if and only if .
The concept of the affine regular hexagon  in a GS-quasigroup is defined in the following way. We will say that is an affine regular hexagon with the vertices and the center and we write if the statements Par hold ,, where indexes are taken modulo 6 (Figure 5). The following statement can be proved .
Lemma 2. An affine regular hexagon is uniquely determined by any three consecutive vertices.
The points determine the figure which will be denoted by the symbol , “half” of the affine regular hexagon with the center (Figure 5).
The following results  will be very useful.
Lemma 3. Let , . If the statements , , and are valid, then there exists a unique point so that the statement is valid too. (The case for is illustrated in Figure 6.)
Lemma 3 implies the following statement.
Lemma 4. Let , . If the statements , , are valid, then there exist unique points so that the statement is valid, too.
The points successively are said to be the vertices of the golden section trapezoid  denoted by GST if the identity holds (Figure 7). It can be proved that the following equivalency holds. The following statement is also valid.
Lemma 5. Any of the three statements , , and is a consequence of the two remaining statements.
In , it is proved that any two of the five statements imply the remaining statements.
Lemma 6. An affine regular pentagon is uniquely determined by any three of its vertices.
Now, we are going to study the relationships between the previously defined geometrical concepts in a general GS-quasigroup.
Proof. (i) We have the equalities
and we have to prove the equivalency of the equalities
However, we get
and thus we get
and analogously . Because of that, it is necessary to prove the equivalency of the equality and the equality which is, according to (15), equivalent to . As we get
these should be equivalent equalities:
which is obvious.
(ii) Firstly, let us prove that from GST there follows .
We have the equalities Therefore we get so it is necessary to prove the equality Really, we have Now, we are going to prove that implies . According to (i), from the hypotheses of (ii), there follows the statement and, from this statement and the equality , according to Lemma 5, there follows the statement .
Lemma 8. If the statements , , and are valid, then the statement and equality are equivalent (Figure 9).
Proof. The assumptions of the lemma imply the statements Par, Par, Par, and Par, and then, according to Lemma 1, parallelograms Par, Par follow from the first and the third, and the second and the fourth parallelogram, respectively. Owing to these last statements, according to Lemma 7(ii), statements GST and are equivalent.
Lemma 9. With the assumption , the statement follows from the equalities , (Figure 10).
Proof. Supposing that a more precise statement is valid, so the statements Par and Par are valid. From the statements Par, there follows , which together with gives the equality , and this statement and the statement Par imply the equality .
Theorem 10. The statement follows from the statements , , , , , and (Figure 11).
Proof. It is sufficient to prove that the statement GST follows from the statements , , GST, GST, , and . Firstly, according to Lemma 8, the equality follows from the statements , , , and GST. Owing to the same lemma, the equality follows from the statements , , , and GST.
Now, all requirements of Lemma 9 are satisfied and the equality follows accordingly.
Finally, according to Lemma 8, the statements , , and and equality imply the statement GST,.
4. Construction of an Affine Fullerene C in a GS-Quasigroup
In this section, we are going to construct an affine fullerene in a general GS-quasigroup by means of the previously discovered statements about affine regular pentagons and hexagons in a general GS-quasigroup.
Theorem 11. An affine fullerene can be constructed in each GS-quasigroup.
Proof. For the sake of clarity, each step of the proof is precisely presented in figures in the GS-quasigroup and each sequence of the proof of the theorem can be followed on the Schlegel diagram (Figure 12).
Let us start with the four given points . The affine regular hexagons ARH and ARH can be constructed according to Lemma 2.
Owing to Lemma 6, affine regular pentagons ARP and ARP can be obtained. If we apply Lemma 2 again, we can get ARH and ARH.
According to Theorem 10, the existence of these obtained affine regular hexagons and affine regular pentagons around an affine regular hexagon will result in the existence of the affine regular pentagon ARP (Figure 13).
According to Lemma 2, the already obtained points uniquely determine and then, because of Lemma 4, the statements , , , and imply the statement ARH . In the same way, the statements , , ARH, and ARH imply and, analogously, the statements , , ARH, and ARH imply .
This consideration is presented in Figure 14.
These two precisely described procedures will also be used later. It will also be denoted which figure is used for the geometrical presentation of the obtained implications in the GS-quasigroup .
Now, according to Theorem 10, the statements , , , , , and imply ARP. The statements and can be obtained similarly (Figure 15).
Thanks to Lemma 4, the statements , ,, ARH, and ARH imply ARH . We can find the affine regular hexagons and ARH in the same way (Figure 16).
If we apply Theorem 10, we will discover three new affine regular pentagons (Figure 17). The statement ARP follows from ARH, ARH, ARP, ARP, , and ARH, and similarly we get ARP, and ARP.
Now, we are in a position to use Lemma 4 whose application gives the statements about three new affine regular hexagons (Figure 18). The statements ARH, ARH, ARH, and ARH imply ARH. Similarly, we can get the affine regular hexagons ARH and ARH.
If we apply Theorem 10, we will obtain the three new affine regular pentagons ARP, ARP, and ARP (Figure 19).
Finally, the application of Lemma 3 will allow us to close the complete structure. We have to prove that the points are the vertices of an affine regular hexagon.
By applying Lemma 3 we get that the statements HARH, HARH, HARH, and HARH imply HARH.
Analogously, we have that HARH, HARH, HARH, and HARH imply HARH, and HARH, HARH, HARH, and HARH imply HARH.
These obtained three halves of affine regular hexagons HARH, HARH, and HARH determine the affine regular hexagon ARH (Figure 19). This completes the proof of the theorem.
Thus, we have proved that any affine fullerene can be obtained only by applying the golden section.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the Ministry of Science, Education and Sports, Republic of Croatia, through Research Grant no. 037-0372785-2759.
P. W. Fowler and D. Manolopoulos, An Atlas of Fullerenes, Clarendon Press, Oxford, UK, 1995.