From the "Brazuca" ball to Octahedral Fullerenes: Their Construction and Classification
FFrom the “Brazuca” ball to Octahedral Fullerenes: TheirConstruction and Classification
Yuan-Jia Fan and Bih-Yaw Jin ∗ Department of Chemistry and Center for Emerging Material and Advanced Devices,National Taiwan University, Taipei 10617, Taiwan (Dated: November 19, 2018)
Abstract
A simple cut-and-patch method is presented for the construction and classification for fullerenesbelonging to the octahedral point groups, O or O h . In order to satisfy the symmetry requirementof the octahedral group, suitable numbers of four- and eight-member rings, in addition to thehexagons and pentagons, have to be introduced. An index consisting of four integers is introducedto specify an octahedral fullerenes. However, to specify an octahedral fullerene uniquely, we alsofound certain symmetry rules for these indices. Based on the transformation properties under thesymmetry operations that an octahedral fullerene belongs to, we can identify four structural typesof octahedral fullerenes. ∗ Electronic address: [email protected] a r X i v : . [ phy s i c s . a t m - c l u s ] J un . INTRODUCTION The 2014 FIFA World Cup championship in Brazil expected to attract the attentionof more than 3 billion people worldwide is the quadrennial soccer tournament since 1930.Whereas, the most popular design for the FIFA soccer ball, the Telstar, which was usedin the official logo for the 1970 World Cup, consists of twelve black pentagonal and 20white hexagonal panels, a truncated icosahedron belonging icosahedral point group[1, 2].Since then, a number of different designs have appeared, some with small variations suchas the Fevernova (2002) and the Teamgeist (2006) which still have icosahedral symmetrybut with low-symmetry tetrahedral patterns painted on the ball; some balls only show lowerpolyhedral patterns such as Jabulani (2012) with tetrahedral symmetry without icosahedralsymmetry superimposed. However, the soccer ball, “Brazuca” ball, used for the WorldCup this summer in Brazil has a new design based on octahedral symmetry. Basically, the“Brazuca” ball is composed of six bonded polyerethane panels with four-arm clover-shapedpanels that interlock like a jigsaw puzzle smoothly on a sphere[3, 4].In 1985, it was discovered that, in addition to diamond and graphite, carbon atoms canhave a third new allotrope consisting of 60-atom spherical molecules, C , sometimes nick-named molecular soccer ball because the shape of this molecule is identical to the standardsoccer ball, with 60 atoms located at 60 identical vertices[5]. More generally, this moleculebelongs to a family of sp -hybridized pure carbon systems now called fullerenes that containonly five- and six-membered rings. Since then, structures of fullerenes have been exten-sively studied experimentally and theoretically. Under this constraint, considerable efforthas been devoted to detailed enumerations of possible structures. For instance, a completelist of fullerenes with less than or equal to 60 carbon atoms and all fullerenes less than andequal to 100 carbon atoms that satisfy the isolated pentagon rule (IPR) is tabulated in themonograph by Fowler and Manolopolous[6]. Among all these fullerenes C N , N ≤ C , C s , C i , C m , C mv , C mh , S m , D n , D nd , D nh , T , T d , T h , I and I h , where m can be 2 or 3 and n can be 2, 3, 5 or 6. However, only twoout of three Platonic polyhedral groups, namely tetrahedral and icosahedral groups, seemsto be possible for fullerenes. So the question is, can we have fullerenes with octahedralsymmetry just like the “Brazuca” ball? If possible, what are the general construction andclassification rules for this family of octahedral fullerenes?2o answer this question, we start with the construction process of fullerenes with poly-hedral symmetries through a simple cut-and-patch procedure as shown in Figure 1. For in-stance, constructing a fullerene with icosahedral symmetry can be done by cutting 20 equiv-alent equilateral triangles from graphene and pasting them onto the triangular faces of anicosahedron. This will create twelve pentagons sitting at twelve vertices of the icosahedron[7–9]. Similar cut-and-patch procedure can be used to construct fullerenes with tetrahedral andoctahedral symmetries, too (Figure 1). However, the non-hexagons such as triangles andsquares will appear at the vertices of the template tetrahedron and octahedron, which arein contradiction to the definition of fullerenes. In the case of tetrahedral fullerenes, wecan replace the template tetrahedron with a truncated tetrahedron. This makes it possibleto the construction of tetrahedral fullerenes without triangles by a suitable cut-and-patchconstruction scheme[10]. But this technique is not applicable to octahedral fullerenes[11, 12].Albeit the appearance of squares in these caged octahedral fullerenes leads to energeticallyunstable molecules, one can still find in literatures that some studies have been carried out onthe geometric, topological[13], and electronic structures[14–17] of fullerenes with octahedralsymmetry by introducing squares on a template octahedron (Figure 1). In addition tothe pure carbon allotropes, the octahedral boron-nitride systems have also been vigorouslyinvestigated[18–22].In this paper, we present a general cut-and-patch construction and classification schemefor fullerenes with octahedral symmetry by systematically introducing some other non-hexagons such as octagons with a cantellated cube as the template. The octahedral fullerenespreviously considered in literatures are included as limiting cases in our general construc-tion scheme[13–17]. We also like to point out that the cut-and-patch method is a simpleand powerful method for building various kinds of fullerenes and graphitic structures. Forinstance, we have applied this method successfully to many other template polyhedral toriand concluded general structural rules of carbon nanotori[23, 24]. From there, structuralrelations for a whole family of topologically nontrivial fullerenes and graphitic structuressuch as carbon nanohelices, high-genus fullerenes, carbon Schwarzites and so on can bederived[25–28]. 3 a (a)(b) (c) (d)(e) (f) (g) FIG. 1: Goldberg polyhedra by the cut-and-patch construction. Here we cut a equilateral trianglewhich can be specified by an vector (2 ,
1) (also known as Goldberg vector) from graphene and thenpatch the triangle onto different platonic solids to construct fullerenes with different polyhedralsymmetries. The famous C , can be also constructed in this way using icosahedron as the templatewith Goldberg vector (1,1). II. REQUIREMENT OF OCTAHEDRAL FULLERENES
We start by briefly describing the icosahedral fullerenes that consist only of hexagonsand pentagons. The simplest icosahedral fullerene that satisfies IPR is C , which can alsobe viewed as a truncated icosahedron, one of the thirteen Archimedean solids if we ignorethe slight variation in bond lengths. In a truncated icosahedron, there are exactly twelvepentagons and twenty hexagons. This structure can be derived from a regular icosahedronby truncating the twelve vertices away appropriately. We will show that this is the only4ossibility if we want to construct an icosahedral fullerene with pentagons and hexagonsonly.Using the Euler’s polyhedron formula, V − E + F = 2 for a polyhedron with V vertices, E edges and F faces, and the condition 3 V = 2 E for trivalent carbon atoms in a fullerene,we can find easily the condition (cid:80) n (6 − n ) F n = 12, where F n is the number of n -gons. If weassign each face a topological charge 6 − n , the Euler’s polyhedron formula states that thesum of topological charges of a trivalent polyhedron must be twelve. Therefore, fullerenesthat contain only pentagons and hexagons must have twelve pentagons, i.e. F = 12, whilethere is no constraint on the number of hexagons, F , except the case with only one hexagon, F = 1, is forbidden. This conclusion is general and can be applied to any fullerene regardlessof its symmetry.An arbitrary icosahedral fullerene can be classified by its chiral vector ( h, k ), where h and k satisfy the inequality h ≥ k ≥ ∧ h >
0, according to the Goldberg construction [7].For instance, C corresponds to the fullerene with chiral vector (1 , × (cid:80) n (6 − n ) F n = 12, will require some n -gons where n > n -gons where n > (cid:80) n (6 − n ) F n = 12, is satisfied. The twofoldor the fourfold axes can also be chosen[11], but the resulting fullerenes are considerably moreenergetically unfavored because additional non-hexagons need to be introduced.5 a) (b) (c) OA BP P P a a P P A AP P O OP P B B
FIG. 2: Cut-and-patch procedure for constructing an octahedral fullerene. Points, P , P and P ,represent the high symmetry points of the octahedral symmetry respectively. (cid:52) OP A ( (cid:52) OP B ) isone-third of a regular triangle (the dotted triangle in (a)) and P ( P ) is the corresponding trianglecenter on graphene. Points O , A , and B become positions where twenty-four equivalent pentagonsare located at, while point P becomes the position for one of six equivalent octagons after theyare patched onto the cantellated cube. The two base vectors, −→ OA = ( i, j ) and −−→ OB = ( k, l ), canin general be any two vectors such that P does not coincide with an atom ( i.e. , i − j = 3 n ). { i, j, k, l } = { , , − , } in this example. (c) The 3D geometry of an octahedral fullerene specifiedaccording to its topological coordinates [9]. To illustrate this idea, we present a simple construction procedure using the cut-and-patch scheme as shown in Figure 2. We first cut the polygonal region as defined by thesolid thick line from graphene (Figure 2(a)) and then patch twenty-four replica of it on acantellated cube as shown in Figure 2(b). The points, O , A , and B in Figure 2(a) overlapwith vertices of the cantellated cube while P and P the centers of the triangle and thesquare faces, respectively. In this process, every four identical isosceles triangles, (cid:52) OP A ,cover one square face (Figure 2(b)). Since the angular deficit at P is 2 π − × π/ − π/ O is 2 π − ( π + 2 π/
3) = π/
3. Therefore a pentagon will be generated at O by this cut-and-patchprocess.We will define the area inside the solid thick line as shown in Figure 2(a) as the fun-damental polygon. Note that the two base vectors, −→ OA = ( i, j ) and −−→ OB = ( k, l ), in thefundamental polygon become the edges of the square and the regular triangle on the cantel-lated cube, respectively, as shown in Figure 2(b). For convenience, we refer to ( i, j ) as the6quare base vector and ( k, l ) the triangular base vector from now on. Using these two vectors,we can uniquely specify a scalene triangle with four integers { i, j, k, l } , which we will simplycall the indices of octahedral fullerenes later. In additional to this scalene triangle, we alsoneed to incorporate two extra triangles, (cid:52) OP A and (cid:52) OP B , corresponding to one-third ofthe regular triangles which share the same edges with the scalene triangle. The numbers ofcarbon atoms inside (cid:52) OP A , (cid:52) OP A , and (cid:52) OAB are ( i + ij + j ) /
3, ( k + kl + l ) /
3, and | il − jk | , respectively. After patching twenty-four fundamental polygons onto a cantellatedcube, we get an octahedral fullerene with 8( i + ij + j + k + kl + l ) + 24 | il − jk | carbonatoms.The octahedral fullerenes can be catagorized into two groups according to the sign of theangle θ formed by −→ OA and −−→ OB . Octahedral fullerenes with π > θ > α , { i, j, k, l } α , and octahedral fullerenes − π < θ < β , { i, j, k, l } β . This crite-rion is equivalent to determining the sign of il − jk , which stands for the signed area enclosedby the parallelogram spanned by the two base vectors up to a positive factor. Here we cantake one step further to include the degenerate cases, i.e. when (cid:52) OAB degenerates into aline, which can be considered as limiting cases when θ approaches to the boundaries of itsrange in each category. It is worthwhile to note that in general lim θ → + { i, j, k, l } α is inequiv-alent to lim θ → − { i, j, k, l } β and lim θ → π − { i, j, k, l } α is inequivalent to lim θ → π + { i, j, k, l } β , asshown in Figure 3. On the other hand, the category letter in the subscript can be omittedwhen there is no ambiguity. We will elaborate in later sections.Following the above cut-and-patch scheme, we can define a scalene triangle and thus thefundamental polygon, given the two base vectors ( i, j ) and ( k, l ) that satisfy the condition, i − j = 3 n . Each of these fundamental polygons uniquely defines an octahedral fullerene innon-degenerate case. When the two base vectors are parallel to each other, it is necessary tofurther specify the category explicitly. It is worthwhile to note that if the condition i − j = 3 n is not satisfied, P will coincide with a carbon atom, which is not allowed because this impliesthat the carbon atom is tetravalent. At first sight, one might think that there exists a one-to-one correspondence between an index, { i, j, k, l } X , and an octahedral fullerene. But thisis not true since it is possible that the octahedral fullerenes built from two different scalenetriangles are in fact identical. We will study this issue in details in the next section.Finally we can identify three limiting situations if one of the three sides of the scalenetriangle vanishes (see Fig. 4). 7 a) (b) O A BP P P O A BP P P a a (c) (d) a a a a a a O AB P P P O AP P P B FIG. 3: Four degenerate cases of octahedral fullerenes. (a) { , , , } α , (b) { , , , } β , (c) { , , − , − } α and (d) { , , − , − } β . We have { , , , } α (cid:54) = { , , , } β and { , , − , − } α (cid:54) = { , , − , − } β . However, T { , , , } α = { , , − , − } β and T { , , , } β = { , , − , − } α .The T transformation will be discussed in later sections.
1. The first limiting situation corresponds to a vanishing triangular base vector, ( k, l ) =(0 , { i, j, , } . Thus, the length of the triangular base vector −−→ OB vanishes and all triangles in the cantellated cube shrink to single points. And thetemplate polyhedron reaches the corresponding limit of the cantellation, namely thecube. Note also that three pentagons fuse to form a triangle at each corner of thecube, while the octagons remain at the centers of the faces of the cube. Thus, thereare eight triangles and six octagons in the resulting octahedral fullerene.2. The second limiting situation corresponds to a vanishing square base vector, ( i, j ) =(0 , { , , k, l } . In this limit, the length of the square base vector −−→ OB vanishes and eachsquare shrinks to a point. Thus, the template polyhedron reaches another limit of thecantellation, namely the octahedron. This case is identical to the Goldberg polyhedron8llustrated in Figure 1(c) and Figure 1(f) . Four pentagons and one octagon fuse toform a square at each corner of the octahedron. Therefore, we have six squares in atype II octahedral fullerene.3. The last limiting situation, denoted as type III, is when the length of the third sideof (cid:52) OAB , −→ AB , vanishes. In other words, −→ OA is equal to −−→ OB , i.e. ( i, j ) = ( k, l ). Onecan show that ( i, j ) = − ( k, l ) also corresponds to the same limiting case. { i, j, i, j } and { i, j, − i, − j } can be transformed to each other via additional symmetry trans-formations, T or T , which will be introduced in the next section. The indices forthis type are { i, j, i, j } or { i, j, − i, − j } and the template polyhedron in this limit is acuboctahedron. Two pentagons at A and B fuse to become one square, and there aresix octagons and twelve squares in total in this limiting case. Other collinear cases donot make the third side vanish though and pentagons will not fuse at all. In fact wecan use T or T introduced later to make these two base vectors nonparallel.When none of the sides of the scalene triangle vanishes, the corresponding octahedralfullerenes will be denoted as type IV. III. INDEX SYMMETRY
In the previous section, we showed that an octahedral fullerene can be constructed by cut-ting a fundamental polygon specified by a four-component index and its category, { i, j, k, l } X and patching twenty-four replica of this fundamental polygon onto a cantellated cube. Wealso pointed that this correspondence is not one-to-one, but many-to-one, since there aresome symmetry relationships in this indexing scheme. In other words, we mean that thereexist different indices { i, j, k, l } X that correspond to the same molecular structure. Thissection is devoted to find a systematic way to eliminate all such redundancies and fullycharacterize the nature of the index symmetry.In the limiting cases of octahedral fullerenes which belong to the types I to III, we onlyneed one independent two-component vector to specify their indices. It is obvious that theindex transformation arising from the geometric symmetry of graphene will lead to the sameoctahedral fullerene. For instance, a π/ O will transform the indexfrom { i, j, k, l } X to {− j, i + j, − l, k + l } X without altering the resulting octahedral fullerene.9 a)(d)(g) a a a (b) (c)(e) (f)(h) (i) a a a OP P AO P BP OP P A,B,P OP P A OP P AO P BP O P BP OP P A,B,P OP P A,B,P FIG. 4: Three limiting cases of octahedral fullerenes. (a)-(c) Type I octahedral fullerene with with { , , , } ; (d)-(f) Type II octahedral fullerene with with { , , , } ; (g)-(i) Type III octahedralfullerene with { , , , } . In this case points A , B , and P are coincident. Therefore these two indices correspond to the same molecular structure and should onlybe counted once. In fact, this applies to all twelve symmetry operations belonging to thepoint group C of graphene. Here, we ignore symmetry operation σ h that lies in the planeof graphene because it does not move any carbon atom at all. So, all indices that can berelated through these symmetry operations produce the same octahedral fullerene. Thisset of indices is called an orbit in group theory[29]. So to enumerate octahedral fullerene isequivalent to enumerate different orbits of all possible indices. Indices belonging to the sameorbit correspond to the same octahedral fullerene. In other words, only one out of the set of10ndices comprising an orbit is needed to represent an octahedral fullerene uniquely. In thesethree limiting situations, we can restrict the indices with the inequality, i ≥ j ≥ ∧ i > k ≥ l ≥ ∧ k > i ≥ j ≥ ∧ i > k = i and l = j ) for type IIIto remove all redundancies arising from the C symmetry operations.The situation for type IV octahedral fullerenes is more complicated. In addition tothe twelve symmetry operations from the point group C v , there are three more symmetryoperations, T , T , and T arising from different ways of dissecting each of the three differentkinds of faces of a cantellated cube into fundamental polygons. For each dissection scheme,different squares or regular triangles are drawn, and the square or triangular base vectorswill change respectively. Detailed description of these three symmetry operations will bedescribed later. These extra symmetry operations introduce redundancies which cannot beremoved by introducing inequalities of indices like the situations of types I to III.Although the redundancies produced by these three T -type symmetry operations cannotbe removed by such index restrictions, the parts of redundancies originating from the sixfoldrotational symmetry of graphene can be eliminated by introducing the canonical criterion, i > ∧ j ≥
0. This is because that these rotational operations commute with the three T -type operations, i.e. [ C n , T y ] = 0 , where y = 2, 3 or 4. Here, we do not impose therestriction, i ≥ j , to remove the redundancies produced by the six mirror symmetries M x .This will be discussed with the T symmetry in the next section. A. T symmetry The symmetry operation, T , comes from the two different ways to decompose a paral-lelogram as shown in Figure 5. The T operation stands for performing a local C operationwhich rotate one of base vectors by 180 ◦ . Thus the index { i, j, − k, − l } will generate thesame octahedral fullerene with { i, j, k, l } . We can define T explicitly with the followingmatrix notation T : ijkl X → i (cid:48) j (cid:48) k (cid:48) l (cid:48) X (cid:48) = − − ijkl X , where X (cid:54) = X (cid:48) . 11 a) (b) (c) a a O AB CP P P' P P' B BP' P' CCP' P' A AP P O OP P P P FIG. 5: The T symmetry operation illustrated with the example T { , , − , } α = { , , , − } β .If we choose {−→ OA, −−→ OB } α as the index, the corresponding fundamental polygon is OP AP BP .On the other hand, if we choose the index {−−→ BC, −−→ BO } β = {−→ OA, −−−→ OB } β , the fundamental poly-gon becomes BP (cid:48) CP OP (cid:48) . These two fundamental polygons essentially give the same octahedralfullerene with different ways of dissecting the parallelogram. Unlike usual matrix multiplications, we need to specify the category of the index beforeand after T transformation. Since il − jk stands for the signed area enclosed by the par-allelogram spanned by these two vectors up to a positive factor, it is clear that under thetransformations, T or M x , the signed area changes sign and hence the category. This is alsotrue in the degenerate case. Therefore, enumerating indices only in a single category canremove redundancies produced by T and M x , but not those produced by M x T = T M x . B. T symmetry The symmetry operation T involves different ways of dissecting the equilateral trianglesof the cantellated cube as shown in Figure 6. For instance, one possible choice of the two basevectors for the scalene triangle is {−→ OA, −−→ OB } α . However, there is another choice, {−→ OA, −→ OF } α ,which produce the same octahedral fullerene, but with a different way of dissecting thetriangles of the cantellated cube. The T transformation only changes the triangular basevectors.Unlike T and M x , the T transformation does not change the category. Moreover, for the T ,α transformation, which operates on octahedral fullerenes belonging to the category α , wealso need to impose an additional constraint on the domain i (cid:48) l (cid:48) − j (cid:48) k (cid:48) ≥ ⇒ − ik − jk − jl ≥ a) (b) (c) a a O AB CD E FP O OA AB B BC CP P D DE EF F
FIG. 6: An illustration of T symmetry. T transform the partition {−→ OA, −−→ OB } α = { , , − , } α to {−→ OA, −−→ OF } α = { , , − , } α , which can be also written as T ,α { , , − , } α = { , , − , } α .Similarly we have T − { , , − , } α = { , , − , } α . i + ij + j . The explicit form of T ,α can be written as T ,α : ijkl α → i (cid:48) j (cid:48) k (cid:48) l (cid:48) α = − − ijkl α . We may obtain T ,β easily by T ,β = M x T ,α M x and its domain by similar method. Theinverse of T transformation, namely T − , may be found by the usual matrix inversion, T − ,α : ijkl α → i (cid:48) j (cid:48) k (cid:48) l (cid:48) α = − − − − ijkl α Its domain can also be found by requiring that the category remains unchanged, i (cid:48) l (cid:48) − j (cid:48) k (cid:48) ≥ ⇒ ik + il + jk ≥ i + ij + j and so we have the identity, T − ,β = M x T − ,α M x . In addition, asshown in Figure 6, the T transformation always decrease | θ | by more than π/
3; while T − always increase | θ | by more than π/ C. T symmetry Similar to the symmetry operations T and T , the operation T involves different waysof assigning fundamental polygons on the cantellated cube as shown in Figure 7. In this13ase, we can see that two different fundamental polygons given by indices {−→ OA, −−→ OB } α and {−→ OF , −−→ OB } α are essentially equivalent in constructing an octahedral fullerene. The transfor-mation T does not change the category just like the transformation T . We can interchange T ,α and T ,β by sandwiching them between the mirror transformation M x . On the otherhand, in contrast to the transformation T , T changes the square base vector only. Therefore,both T and T will decrease | θ | by more than π/
3. In other words, the square base vectorwill be rotated by more than π/ i > ∧ j ≥ T ,α can be written as T ,α : ijkl α → i (cid:48) j (cid:48) k (cid:48) l (cid:48) α = − −
11 1 1 20 0 1 00 0 0 1 ijkl α . Again, an constraint on domain − ik − jk − jl ≥ k + kl + l is necessary to ensure that thecategory stays unchanged. The inverse T − ,α can be defined as follows T − ,α : ijkl α → i (cid:48) j (cid:48) k (cid:48) l (cid:48) α = − − − −
10 0 1 00 0 0 1 ijkl α , and the constraint on the domain is ik + il + jl ≥ k + kl + l . In summary, C n , T , T andtheir inverses do not change the categories, but M x and T do.Although T -type symmetry operations are defined for type IV octahedral fullerenes, theycan also be applied to three limiting cases. When T -type symmetry operations are appliedto type I and type II octahedral fullerenes, they reduce to the geometric rotation C n . Andwhen they are applied to type III octahedral fullerenes, we have following identities, T { i, j, i, j } X = { i, j, − i, − j } X (cid:48) ( X (cid:54) = X (cid:48) ) T − { i, j, i, j } X = { i, j, − i, − j } X C T − { i, j, i, j } X = { i, j, − i, − j } X . a) (b) a a O AB C DDEF GH P G ACOBF E H D P G ACOBF E H D (c) P P P P FIG. 7: An illustration of T symmetry. T transform the partition {−→ OA, −−→ OB } α = { , , − , } α to {−−→ OF , −−→ OB } α = {− , , − , } α , which can be also written as T ,α { , , − , } α = {− , , − , } α .Two points connected by a grey line should be patch into one point. Four shaded triangle in(a) will merge into square BCDE in (b) and four P points in (a) will become one P in (b)after patching. Note that since P always carries a topological charge, the vector −−→ OF does notcorrespond to ( − ,
2) in (a).
These formulae will give a torus-like orbit. The details for the enumeration of these orbitsare included in supporting information.
IV. CONCLUSION
In conclusion, we have developed a systematic cut-and-patch method to generate arbi-trary fullerenes belonging to the octahedral point group. A unique four-component vectorsatisfying certain constraints and symmetry rules can be used to specify these octahedralfullerenes. This work on the octahedral fullerenes fits in the final piece of the jigsaw puzzle ofall possible high symmetry caged fullerenes based on Platonic solids. Further investigationon the stability, elastic properties and electronic structures of these octahedral fullerenes andthe possibility of using them to build periodic carbon Schwarzites are currently undergoingin our group[28, 30].Finally, we also want to point out two observations: the “Brazuca” ball used in the WorldCup is close to a very round octahedral sphere, while the fullerenes discussed in this paperare still far from a round sphere. The explanation for the first observation is given in a moregeneral context by Delp and Thurston in a paper about the connection between clothing15esign and mathematics in the Bridges meeting three years ago.[31] The most importantfactor that makes it possible to wrap six clover-shaped panels used in the “Brazuca” arounda sphere smoothly is that the curved seams created by these interlocked 4-long-arms panelsare quite evenly distributed on the sphere. Readers interested in this problem should goto that paper for details. The observation on the shape of octahedral fullerenes is alsointeresting. All of the three-dimensional geometries shown in this paper are obtained throughtheir topological coordinates derived from the lowest three eigenvectors with single nodes bydiagonalizing the corresponding adjacency matrices[6]. Further investigations to rationalizehow the distribution of the non-hexagons affects the shapes of octahedral fullerenes in orderto obtain a round nanoscale “Brazuca” ball based on either elastic theory or quantumchemical calculations should be worth pursuing in the future.[6, 30, 32, 33] acknowledgements
The research was supported by the Ministry of Science and Technology, Taiwan. B.-Y.Jin thanks Center for Quantum Science and Engineering, and Center of Theoretical Sciencesof National Taiwan University for partial financial supports. We also wish to thank ChernChuang and Prof. Yuan-Chung Cheng for useful discussions and comments on this paper. [1] “Adidas Telstar.” Wikipedia: The Free Encyclopedia. Wikimedia Foundation, Inc. (Accessed17 Jul., 2014).[2] Kotschick, D. The Topology and Combinatorics of Soccer Balls,
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