## Fullerenes and Coordination Polyhedra versus Half-Cubes Embeddings (1997)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Deza97fullerenesand,

author = {Antoine Deza and Michel Deza and Viatcheslav Grishukhin},

title = {Fullerenes and Coordination Polyhedra versus Half-Cubes Embeddings},

year = {1997}

}

### OpenURL

### Abstract

A fullerene F n is a 3-regular (or cubic) polyhedral carbon molecule for which the n vertices - the carbons atoms - are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes F n for n ! 60 and of all preferable fullerenes C n for n ! 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onion-like metallic clusters and geodesic domes. Quasi-embeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells. Contents 1 Introduction and Basic Properties 2 1...

### Citations

269 | Convex Polytopes
- GRUNBAUM
- 1967
(Show Context)
Citation Context ...arbons atoms - are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons and the 3 2 n edges correspond to carbon-carbon bonds. Fullerenes F n can be constructed for all even ns20 except n = 22 (see =-=[37]-=- p. 271 and [51]). For a given n, different arrangements of facets are possible; for example, there are 1; 1; 1; 2; 3; 40; 271 and 1812 fullerenes F n for n = 20; 24; 26; 28; 30; 40; 50 and 60 respect... |

104 |
Physical principles in the construction of regular viruses
- Ireland, CASPAR, et al.
- 1962
(Show Context)
Citation Context ...erenes Call icosadeltahedron the duals - C 20T (I h ) or C 20T (I) - of an icosahedral fullerene. Those icosadeltahedra were introduced by Goldberg [35] in 1937 and (independently) by Caspar and Klug =-=[7]-=- in 1962 as capsids (protein coats) of some viruses. The first icosadeltahedron is the icosahedron F 20 (I h ); and, for (a; b) = (1; 1), we have the dual buckminsterfullerene C 60 (I h ) which is als... |

68 |
Distance preserving subgraphs of hypercubes
- Djoković
- 1973
(Show Context)
Citation Context ... G is an isometric subgraph of a half-cube ) dG is an ` 1 -graph ) dG is hypermetric ) dG is (2k + 1)-gonal ) dG is (2k \Gamma 1)-gonal ) dG is 3-gonal (= metric). We also recall that Djokovi' c, see =-=[19]-=-, gave a characterization of ` 1 -graphs with scales= 1 which can be reformulated (see Avis [1]) in the following way: a graph is an isometric subgraph of a hypercube if and only if it is bipartite an... |

58 |
An Atlas of Fullerenes
- Fowler, Manolopoulos
- 1995
(Show Context)
Citation Context ...edral group of order 120). While C 60 (I h ) is a 2 soccer ball, C 70 (D 5h ) and C 84 (D 2d ) respectively resemble to a rugby and a baseball ball. The general references are Fowler and Manolopoulos =-=[28]-=- for fullerenes and Gr unbaum [37] for polyhedra. References for so-called point groups is [59], and [29, 37] for operations on polyhedra such as pentakis, leapfrog, truncation and dual. In this paper... |

36 |
and Wim Sweldens. Spherical wavelets: Efficiently representing functions on the sphere
- Schröder
- 1995
(Show Context)
Citation Context ...elements) technique, in computeraided geometric design, graphical rendering, solid modeling and finite element analysis. In particular, this is used for spherical wavelets (see Schr oder and Sweldens =-=[60, 61]-=-) leading to efficient algorithms in computer graphics. For example, Schr oder and Sweldens use the dual 8-chamfered dodecahedron (20 \Delta 2 16 = 1310720 facets!) for a spherical texture processing ... |

30 | and Wim Sweldens. Spherical Wavelets: Texture Processing
- Schröder
- 1995
(Show Context)
Citation Context ...elements) technique, in computeraided geometric design, graphical rendering, solid modeling and finite element analysis. In particular, this is used for spherical wavelets (see Schr oder and Sweldens =-=[60, 61]-=-) leading to efficient algorithms in computer graphics. For example, Schr oder and Sweldens use the dual 8-chamfered dodecahedron (20 \Delta 2 16 = 1310720 facets!) for a spherical texture processing ... |

26 |
Geometry of cuts and metrics. Algorithms and Combinatorics 15 Springer-Verlag
- Deza, Laurent
- 1997
(Show Context)
Citation Context ...if: jfi 2 f1; : : : ; mg : x i 6= y i gj = 2. If moreover, the embedding is ` 1 -rigid, as for F 20 (I h ) and C 80 (I h ), then we have an essentially unique encoding (see the end of Section 1.1 and =-=[15]-=- for details). Such labeling of vertices can also be useful for the nomenclature of fullerenes [24] and for the calculation of molecular parameters depending only on the graphical distances such as th... |

26 | A class of multi-symmetric polyhedra - Goldberg - 1939 |

21 | McLaren’s improve snub cube and other new spherical design in three dimensions, Discrete and Computational Geometry 15
- Hardin, Sloane
- 1996
(Show Context)
Citation Context ... isolated hexagons. In chemistry, their duals are called Frank-Kasper polyhedra (see [30]). (iii) The vertex sets of tetrahedral F 28 , F 68 , C 116 and, perhaps, C 164 and C 188 appear in Table 1 of =-=[39]-=- as putative best spherical t-designs with m points with (m; t) = (16; 5), (36; 8), (60; 10), (84; 12) and (96; 13). (iv) In the Mosseri-Sadoc [53, 57] models similar to the one presented in Section 4... |

19 | Analyzing and visualizing sequence and distance data using
- DRESS, HUSON, et al.
- 1996
(Show Context)
Citation Context ...i(S) defined for a given subset S of V n = f1; 2; : : : ; ng by ffi(S) ij = 1 if 3 exactly one of i, j is in S and 0 otherwise, for 1si ! jsn (cut metrics are also sometimes called split metrics, see =-=[20]-=-). In other words, a graph G (or any metric) is ` 1 -embeddable if the path metric dG is a non-negative linear combination of cuts: dG = X SaeVn a S ffi(S) with a Ss0 for all S. If the embedding is es... |

19 |
On scale embedding of graphs into hypercubes
- Shpectorov
- 1993
(Show Context)
Citation Context ...it is bipartite and 5-gonal. A graph G is an ` 1 -graph if and only if G is an isometric subgraph of a Cartesian product of cocktail-party graphs (i.e. dual hypercube graph) and half-cube graphs, see =-=[62]-=- (or [12] for another proof). As for complexity results: while recognizing a metric as ` 1 -embeddable (that is, isometrically embeddable into an ` m 1 ) is NP-complete, see Karzanov [42] and Avis and... |

18 |
The cut cone, L 1 embeddability, complexity, and multicommodity flows
- Avis, Deza
- 1991
(Show Context)
Citation Context ...12] for another proof). As for complexity results: while recognizing a metric as ` 1 -embeddable (that is, isometrically embeddable into an ` m 1 ) is NP-complete, see Karzanov [42] and Avis and Deza =-=[2]-=-, recognizing a graph as ` 1 -embeddable is polynomial, see Deza and Shpectorov [16]. See also [15] for a general study of ` 1 -metrics. 4 1.2 Fullerenes and the isoperimetric problem for polyhedra 1.... |

18 |
Metrics and undirected cuts
- Karzanov
- 1985
(Show Context)
Citation Context ... graphs, see [62] (or [12] for another proof). As for complexity results: while recognizing a metric as ` 1 -embeddable (that is, isometrically embeddable into an ` m 1 ) is NP-complete, see Karzanov =-=[42]-=- and Avis and Deza [2], recognizing a graph as ` 1 -embeddable is polynomial, see Deza and Shpectorov [16]. See also [15] for a general study of ` 1 -metrics. 4 1.2 Fullerenes and the isoperimetric pr... |

18 |
Induction and analogy in mathematics
- Pólya
- 1954
(Show Context)
Citation Context ...cets which contains the largest volume V . In other words, minimize the Steinitz number S 3 V 2 or, equivalently, maximize the isoperimetric quotient IQ = 36 V 2 S 3 - a term introduced by P' olya in =-=[56]-=- (Chap. 10, problem 43). See Florian [26] for a survey and recall the classical isoperimetric inequality for solids: 36 V 2 S 3s1, with equality only for the sphere. The polyhedron Pm with m facets wi... |

17 |
Zur Morphologie der Polyeder
- Eberhard
- 1891
(Show Context)
Citation Context ...], we have the following dual fullerenes: F 4 , F 6 , F 8 , F 10 , F 16 , F 18 , F 20 , ~ F 22 , F 24 , F 26 , F 28 (T d ), F 30 (D 5h ), C 60 (I h ) and C 80 (D 5h ). Kuijlaar and Saff (see [47] and =-=[20,21]-=- therein) did extensive search and indicated that for 32sns200, the Voronoi domains correspond to fullerenes with only handful of exceptions. Case k = 2. Among the spherical Voronoi domains of the con... |

16 | Hypermetric spaces and the Hamming cone
- Avis
- 1981
(Show Context)
Citation Context ... + 1)-gonal ) dG is (2k \Gamma 1)-gonal ) dG is 3-gonal (= metric). We also recall that Djokovi' c, see [19], gave a characterization of ` 1 -graphs with scales= 1 which can be reformulated (see Avis =-=[1]-=-) in the following way: a graph is an isometric subgraph of a hypercube if and only if it is bipartite and 5-gonal. A graph G is an ` 1 -graph if and only if G is an isometric subgraph of a Cartesian ... |

14 | A constructive enumeration of fullerenes - Brinkmann, Dress - 1997 |

11 |
macromolecules and geodesic domes, in A spectrum of mathematics
- Coxeter
- 1971
(Show Context)
Citation Context ...edral fullerenes Any icosahedral fullerene F n (I h ) and F n (I) comes by folding of triangular net and satisfies n = 20T with triangulation number T = a 2 +ab+ b 2 for positive integers (a; b), see =-=[8, 35]-=-. All icosahedral fullerenes being preferable except F 20 (I h ), we denote them C 20T (I h ) or C 20T (I) by abuse of notation. Moreover, n = 60i if (a \Gamma b) is divisible by 3, and n = 60i + 20 o... |

8 |
M.I.Shtogrin, Isometric embedding of semi-regular polyhedra, partitions and their duals into hypercubes and cubic lattices
- Deza
- 1996
(Show Context)
Citation Context ...l structure of some virus capsids (see Section 3.3). So, this paper can be considered as an investigation of ` 1 -embeddability of supposed best polyhedral approximation of spheres following [13] and =-=[17]-=- which are devoted to ` 1 -embedding of, respectively, highly regular polyhedra and infinite polyhedra (plane partitions and lattices). 8 2 Half-cube Embeddings of Fullerenes We expected the number of... |

8 |
Systematics of bonding in non-icosahedral carbon clusters, Theor
- Fowler, Cremona, et al.
- 1988
(Show Context)
Citation Context ... of 5 pentagons surrounding a hexagon, we get the i-layered F 24 (D 6d ): F 12(i+2) (D 6h ) for odd i and F 12(i+2) (D 6d ) for even i, see Fig. 2.12. Those two families were introduced in Table 5 of =-=[27]-=- as dihedral fullerenes with two caps. file=F40.ps,height=5.0cm Figure 2.12: F 36 (D 6h ) = 1-layered F 24 (D 6d ) and F 40 (D 5d ) = 2-layered dodecahedron Proposition 2.3 For any integer is1: (i) Th... |

8 | Leapfrog Transformation and polyhedra of Clar Type
- PW, Pisanski
- 1994
(Show Context)
Citation Context ...ly resemble to a rugby and a baseball ball. The general references are Fowler and Manolopoulos [28] for fullerenes and Gr unbaum [37] for polyhedra. References for so-called point groups is [59], and =-=[29, 37]-=- for operations on polyhedra such as pentakis, leapfrog, truncation and dual. In this paper we focus on the following metric question. Can we embed into a hypercube the graph formed by the vertices an... |

8 |
Geometry of Spatial Forms
- Gasson
- 1983
(Show Context)
Citation Context ...20 facets!) for a spherical texture processing of topography/bathymetry data of Earth (see [61]). This spherical triangulation was also considered, for example, by Gasson (see Sections 7.5 and 7.6 of =-=[32]-=-). file=heche4octa.ps,height=5.4cm Figure 4.33: Dual 4-chamfered cube 3 2048 file=heche4dode.ps,height=7.4cm Figure 4.34: Dual 4-chamfered dodecahedron C 5120 (I h ) (Test planetarium, Iena 1922) 31 4... |

8 |
Distributing many points on a sphere, Mathematical Intelligencer 19(1
- Saaf, Kuijlaars
- 1997
(Show Context)
Citation Context ...ndson [23], we have the following dual fullerenes: F 4 , F 6 , F 8 , F 10 , F 16 , F 18 , F 20 , ~ F 22 , F 24 , F 26 , F 28 (T d ), F 30 (D 5h ), C 60 (I h ) and C 80 (D 5h ). Kuijlaar and Saff (see =-=[47]-=- and [20,21] therein) did extensive search and indicated that for 32sns200, the Voronoi domains correspond to fullerenes with only handful of exceptions. Case k = 2. Among the spherical Voronoi domain... |

8 |
Convex Polyhedra with Regular Faces, Consultants Bureau
- Zalgaller
- 1969
(Show Context)
Citation Context ...] for description of clathrate crystal structures of some ice or silicate compounds. Another space co-filler of IR 3 is F 12 (item VIII of Fig. 1.4); this dual of the bisdisphenoid (see Fig. 4.29 and =-=[4, 71]-=-) embeds into 1 2 H 8 . We also recall that the right angled hexagonal barrel F 24 tiles the hyperbolic space H 3 - it is the fundamental polyhedron of a compact hyperbolic manifold called the L obell... |

7 | Hypermetric graphs
- Deza, Grishukhin
- 1993
(Show Context)
Citation Context ...artite and 5-gonal. A graph G is an ` 1 -graph if and only if G is an isometric subgraph of a Cartesian product of cocktail-party graphs (i.e. dual hypercube graph) and half-cube graphs, see [62] (or =-=[12]-=- for another proof). As for complexity results: while recognizing a metric as ` 1 -embeddable (that is, isometrically embeddable into an ` m 1 ) is NP-complete, see Karzanov [42] and Avis and Deza [2]... |

5 |
Graph Invariants for Fullerenes
- Balaban, Liu, et al.
(Show Context)
Citation Context ...vertices can also be useful for the nomenclature of fullerenes [24] and for the calculation of molecular parameters depending only on the graphical distances such as the Wiener, J , and other indices =-=[3]-=-. The path metric dG associated with a connected graph G(V; E) is the integer valued metric on the vertices of G defined by setting dG (v 1 ; v 2 ) equal to the length of the shortest path in G joinin... |

5 |
and V.P.Grishukhin, A clin d'oeil on l 1 -embeddable planar graphs, Discrete
- Chepoi
- 1997
(Show Context)
Citation Context ...p to scalesinto a hypercube, we have: (i)s= 1 , G is an isometric subgraph of a hypercube, we write: G ! Hm (ii)s= 1 or 2 , G is an isometric subgraph of a half-cube, we write: G ! 1 2 Hm . Lemma 1.1 =-=[9]-=- The scalesof a planar ` 1 -graph is either 1 or 2. All ` 1 -metrics on n vertices (that is, all ` 1 -metric spaces (d; V n ) with jV n j = n) form a i n 2 j -dimensional pointed cone called the cut c... |

5 |
l 1 --rigid graphs
- Deza, Laurent
- 1994
(Show Context)
Citation Context ...uts: dG = X SaeVn a S ffi(S) with a Ss0 for all S. If the embedding is essentially unique, that is, if the above linear combination is unique up to an isomorphism, the graph is called ` 1 -rigid (see =-=[14]-=-). Restated in terms of cone, a metric is ` 1 -rigid if and only if it belongs to a simplex face of the cut cone. For example, see Lemma 2.1, the embeddings of F 20 into 1 2 H 10 and of its dual F 20 ... |

5 | Recognition of l 1 --graphs with complexity O(nm); or football in a hypercube, Europ - Deza, Shpectorov |

5 |
The isoperimetric problem for polyhedra
- Goldberg
- 1935
(Show Context)
Citation Context ...ses as given in Fig. 1.1 wheres= 1+ p 5 2 denotes the golden ratio and the upper bound IQ m of Fejes T' othGoldbergsis the one given in Theorem 37 of [26]. This bound is sharper than the one given in =-=[34]-=-: IQ(Pm )s(m\Gamma2) 2 m(m\Gamma1) . m Polyhedron P IQ(P ) Upper bound IQ m 4 Tetrahedron 6 p 3 ' 0:302 6 p 3 6 Cube 6 ' 0:524 6 8 Octahedron 3 p 3 ' 0:605 ' 0:637 12 Dodecahedron 7=2 3\Delta5 5=4 ' 0... |

5 |
Space Structures: Their Harmony and Counterpoint
- Loeb
- 1991
(Show Context)
Citation Context ... a (2,0)-leapfrog transform and we have Cham 1 (F n (S g )) = C 4n (S g ) where S g is the symmetry group of F n . We also recall in Fig. 3.24 the well-known edge truncation given in Chap. 10 of Loeb =-=[49]-=-. file=loeb.ps,width=7.0cm Figure 3.24: Usual edge truncation Proposition 3.5 With the definition of alternating circuits given in the proof of Proposition 3.3, two necessary conditions for the ` 1 -e... |

4 |
Regular-faced convex polyhedra
- Berman
- 1971
(Show Context)
Citation Context ...] for description of clathrate crystal structures of some ice or silicate compounds. Another space co-filler of IR 3 is F 12 (item VIII of Fig. 1.4); this dual of the bisdisphenoid (see Fig. 4.29 and =-=[4, 71]-=-) embeds into 1 2 H 8 . We also recall that the right angled hexagonal barrel F 24 tiles the hyperbolic space H 3 - it is the fundamental polyhedron of a compact hyperbolic manifold called the L obell... |

3 |
Polyoxometalates: A class of compounds with remarkable topology
- Friedrichs, Dress, et al.
- 1993
(Show Context)
Citation Context ...s the rhombic dodecahedron. In addition to some frequent coordination polyhedra, the ` 1 -status of some metallopolyhedra (convex hull of atoms of metallic cluster) or metal and oxygen polyhedra (see =-=[10, 45, 67, 70]-=-) are given in Fig. 4.29. The names and the number (as regular-faced polyhedra) are taken from the list of such 112 polyhedra given in [4, 71]. For example, see Fig. 4.28 where the embedding of the rh... |

3 |
A zoo of l 1 --embeddable polytopal graphs
- Deza, Grishukhin
- 1997
(Show Context)
Citation Context ...F 16 , are ` 1 -embeddable. One can check (see Table 13.3.1 in [37]) that all simple polyhedra with only 3, 4 or 5-gonal facets are those 8 items and 3 ` 1 -polyhedra from Figs. 7 (a), (b) and (c) of =-=[13]-=-. Small fullerenes also appear in the following forms: (i) F 20 , ~ F 22 (C 2v ), F 24 , F 26 and F 28 (D 2 ) as skeletons of central soap bubbles (see Fig. 9.13 in [25]) (ii) F 20 , F 24 , F 26 and F... |

3 |
private communication
- Dillencourt
- 1991
(Show Context)
Citation Context ...dges). Square fullerenes 3 n can be constructed for all even ns8 except n = 10 (see [37] p. 271). There are 1; 0; 1; 1; 1; 1; 3; 1; 3; 3; 3; 2; 8; 3; 7; 7; 7; 5; 14 square fullerenes 3 2k for 4sks22 (=-=[18]-=-). The cube and the hexagonal prism are the unique square fullerenes 3 8 and 3 12 . Besides the (a; 0)-cube for a ? 0, examples of preferable polyhedra, that is, without pair of 4-gons sharing a commo... |

3 |
1619), Harmonices mundi 1619
- Kepler
(Show Context)
Citation Context ... Icosahedron 4 15 p 3 ' 0:829 ' 0:851 Figure 1.1: Isoperimetric quotient of Platonic solids file=kepler.ps,width=6.1cm Figure 1.2: The five Platonic solids (adapted from a drawing by Kepler) 5 Kepler =-=[44]-=- interpreted a high (resp. low) IQ as wetness (resp. dryness) and as a justification for why the icosahedron and the tetrahedron symbolizes respectively the water and the fire, see Fig. 1.2. 1.2.2 Ful... |

3 |
Properties of planar graphs with uniform vertex and face structure
- Malkevitch
- 1970
(Show Context)
Citation Context ...re arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons and the 3 2 n edges correspond to carbon-carbon bonds. Fullerenes F n can be constructed for all even ns20 except n = 22 (see [37] p. 271 and =-=[51]-=-). For a given n, different arrangements of facets are possible; for example, there are 1; 1; 1; 2; 3; 40; 271 and 1812 fullerenes F n for n = 20; 24; 26; 28; 30; 40; 50 and 60 respectively (see, for ... |

3 |
A generalized leapfrog for fullerene structures
- Sah
- 1994
(Show Context)
Citation Context ...) C 80 (I h ) corresponds to (a; b) = (2; 0) (see Figs. 3.13 and 3.14, or item XLII of Fig. 1.4). In other words, an icosahedral fullerene is the (a,b)-generalized leapfrog of a dodecahedron (see Sah =-=[58]-=-) or the f5+; 3g a;b (see Coxeter [8]). file=C60.ps,width=5cm Figure 3.13: The C 60 (I h ) or (1; 1)-dodecahedron file=C80.ps,width=5cm Figure 3.14: Chamfered dodecahedron C 80 (I h ) (2; 0)-dodecahed... |

2 |
Magic numbers for sphere packings: experimental verification in free xenon clusters
- Echt, Sattler, et al.
- 1981
(Show Context)
Citation Context ...hedral and cuboctahedral metallic clusters The evidence for fl-clusters was also observed as maxima for concentration profiles of cluster spectra in inert gas, and called there Mackay icosahedra. See =-=[22]-=- for xenon 3; 4 and 5-clusters and see [31] for argon 3; 4; 5 and 6-clusters. 32 4.3.2 Other metallic clusters Metallic clusters can also be realized by tetrahedral, octahedral or cubic shells (see fo... |

2 |
A.: Magic numbers in polygonal and polyhedral clusters. Inorganic Chemistry 24
- Theo, Sloane
- 1985
(Show Context)
Citation Context ... 3; 4 and 5-clusters and see [31] for argon 3; 4; 5 and 6-clusters. 32 4.3.2 Other metallic clusters Metallic clusters can also be realized by tetrahedral, octahedral or cubic shells (see for example =-=[66]-=-). The first 5 metallic clusters given in Fig. 4.37 are made of only one shell. The last one is made of 2 non-consecutive octahedral shells: the dual (1; 0) and (3; 0)-cubes 3 8 and 3 72 . Contrary to... |

1 |
N.: Closo boron hybrides with 13 to 24 boron atoms. Organic Chemistry 16
- Brown, Lipscomb
- 1977
(Show Context)
Citation Context ...on of Escher, which can also be seen as a small stellated dodecahedron with pentagram faces. Remark 3.1 Another interesting family of dual fullerenes is the hypothetical boron cages (for example, see =-=[6]-=-). The first one, B 12 = F 20 (I h ), is known experimentally: [B 12 H 12 ] 2\Gamma . Next cases are: B 14 = F 24 (D 6d ), B 15 = F 26 (D 3h ), B 16 = F 28 (T d ), B 17 = all three F 30 , B 18 = three... |

1 |
Dublanchet A.: On structure of icosahedral viruses
- Deza
(Show Context)
Citation Context ...en if less commonly, considered. Remark that the majority of viruses of Fig. 3.19 are with symmetry I h and especially of type (a; 0). And among the remaining with symmetry I, only one is dextro (see =-=[11]-=- for a presentation of icosahedral viruses). The discovery that the majority of human virulent viruses are icosahedral recalls Plato's belief that an excess of water - which element was supposed to be... |

1 |
R.: The distrubution of point charges on the surface of a sphere. Acta Crystallographica A 48
- Edmundson
- 1992
(Show Context)
Citation Context ...qn 1 d k pq for ks1 max X 1p!qn 1 d k pq for ks\Gamma1 Case k = 1. This case corresponds to the Coulombic potential (the Thomson problem). Among the configurations minimizing V n;1 found by Edmundson =-=[23]-=-, we have the following dual fullerenes: F 4 , F 6 , F 8 , F 10 , F 16 , F 18 , F 20 , ~ F 22 , F 24 , F 26 , F 28 (T d ), F 30 (D 5h ), C 60 (I h ) and C 80 (D 5h ). Kuijlaar and Saff (see [47] and [... |

1 |
A canonical assignment of locant numbers to fisular compounds - especially fullerenes - based on graph theoretical principles
- Elk
- 1995
(Show Context)
Citation Context ... (I h ) and C 80 (I h ), then we have an essentially unique encoding (see the end of Section 1.1 and [15] for details). Such labeling of vertices can also be useful for the nomenclature of fullerenes =-=[24]-=- and for the calculation of molecular parameters depending only on the graphical distances such as the Wiener, J , and other indices [3]. The path metric dG associated with a connected graph G(V; E) i... |

1 |
Complex allow structures regarded as sphere packings 1. Definitions and basic principle. Acta Crystallographica 11
- Frank, Kasper
- 1958
(Show Context)
Citation Context ...f central soap bubbles (see Fig. 9.13 in [25]) (ii) F 20 , F 24 , F 26 and F 28 (T d ) are the only fullerenes with isolated hexagons. In chemistry, their duals are called Frank-Kasper polyhedra (see =-=[30]-=-). (iii) The vertex sets of tetrahedral F 28 , F 68 , C 116 and, perhaps, C 164 and C 188 appear in Table 1 of [39] as putative best spherical t-designs with m points with (m; t) = (16; 5), (36; 8), (... |

1 |
Magic numbers for argon and nitrogen cluster ions
- Friedman, Beuhler
- 1983
(Show Context)
Citation Context ...The evidence for fl-clusters was also observed as maxima for concentration profiles of cluster spectra in inert gas, and called there Mackay icosahedra. See [22] for xenon 3; 4 and 5-clusters and see =-=[31]-=- for argon 3; 4; 5 and 6-clusters. 32 4.3.2 Other metallic clusters Metallic clusters can also be realized by tetrahedral, octahedral or cubic shells (see for example [66]). The first 5 metallic clust... |

1 |
W.: Search for largest polyhedra
- Grace
- 1963
(Show Context)
Citation Context ...facets and gave the following conjecture. Conjecture 1.3 [Goldberg 1935] Among polyhedra with m facets, m 6= 11; 13, a medial polyhedron F 2(m\Gamma2) reaches the highest IQ. We also recall the Grace =-=[36]-=- and Steiner [63] conjectures. Conjecture 1.4 [Grace 1963] Among all polyhedra with n vertices lying on the unit sphere, n 6= 18; 22, a dual medial polyhedronsF n reaches the highest volume. Conjectur... |

1 |
P.: Khimiya klusterof
- Gubin
- 1987
(Show Context)
Citation Context ...dral and cuboctahedral metallic clusters Besides virus capsids and geodesic domes, the dual (a; 0)-dodecahedra also occur as levels (concentric onion skins) in some large metallic clusters (see Gubin =-=[38]-=- p. 241). Those flclusters are made up of fl successive shells of atoms surrounding a central atom, each shell being either a dual (a; 0)-dodecahedron or a dual (a; 0)-rhombic dodecahedron. Since an i... |

1 |
C.: Huckel treatment
- Huang, Tang
- 1966
(Show Context)
Citation Context ... Next cases with tetrahedral or higher symmetry are B 28 = F 52 (T ), B 32 = C 60 (I h ) and B 92 = C 180 (I h ). The last two are fairly stable in terms of eigenvalues of the Huckel Hamiltonian (see =-=[41]-=-). Both dual quasi-medial polyhedra ~ F 18 (C 2v ) and ~ F 22 (C 2v ) - see Fig. 1.3 - are also putative boron cages for B 11 and B 13 . 14 file=dualC80.ps,width=6.7cm C 60 (I h ) file=escher.ps,width... |

1 |
B.: Topological aspects of chemically significant polyhedra
- King
- 1991
(Show Context)
Citation Context ... in Fig. 1.3 are almost medial and we call them quasimedialspolyhedra with 18 and 22 vertices denoted by ~ F 18 (C 2v ) and ~ F 22 (C 2v ). While ~ F 18 (C 2v ) is the edge-coalesced icosahedron (see =-=[45]-=-) given in Fig. 4.30, ~ F 22 (C 2v ) is the one-edge truncated dodecahedron mentioned p. 274 in [57]. This polyhedron has 10 pentagonal facets which is the maximum among all simple polyhedra with 22 v... |