The icosahedral symmetry group H3 of order 120 and its dihedral subgroup H2 of order 10 are used for exact geometric construction of polytopes that are known to exist in nature. The branching rule for the H3 orbit of the fullerene C60 to the subgroup H2 yields a union of eight orbits of H2: four of them are regular pentagons and four are regular decagons. By inserting into the branching rule one, two, three or n additional decagonal orbits of H2, one builds the polytopes C70, C80, C90 and nanotubes in general. A minute difference should be taken into account depending on whether an even or odd number of H2 decagons are inserted. Vertices of all the structures are given in exact coordinates relative to a non-orthogonal basis naturally appropriate for the icosahedral group, as well as relative to an orthonormal basis. Twisted fullerenes are defined. Their surface consists of 12 regular pentagons and 20 hexagons that have three and three edges of equal length. There is an uncountable number of different twisted fullerenes, all with precise icosahedral symmetry. Two examples of the twisted C60 are described.
Polynomials in this paper are defined starting from a compact semisimple Lie group. A known classification of maximal, semisimple subgroups of simple Lie groups is used to select the cases to be considered here. A general method is presented and all the cases of rank ≤ 3 are explicitly studied. We derive the polynomials of simple Lie groups B 3 and C 3 as they are not available elsewhere. The results point to far reaching Lie theoretical connections to the theory of multivariable orthogonal polynomials.
Abstract.Properties of four infinite families of special functions of two real variables, based on the compact simple Lie group G 2 , are compared and described. Two of the four families (called here C-and S-functions) are well known. New results of the paper are in description of two new families of G 2 -functions not found in the literature. They are denoted as S L -and S S -functions.It is shown that all four families have analogous useful properties. In particular, they are orthogonal when integrated over a finite region F of the Euclidean space. They are also orthogonal as discrete functions when their values are sampled at the lattice points F M ⊂ F and added up with appropriate weight function. The weight functions are determined for the new families. Products of ten types among the four families of functions, namelyand S L S L , are completely decomposable into the finite sum of the functions belonging to just one of the families. Uncommon arithmetic properties of the functions are pointed out and questions about numerous other properties are brought forward.
This paper completes the series of three independent articles [Bodner et al. (2013). Acta Cryst. A69, 583-591, (2014), PLOS ONE, 10.1371/journal.pone.0084079] describing the breaking of icosahedral symmetry to subgroups generated by reflections in three-dimensional Euclidean space {\bb R}^3 as a mechanism of generating higher fullerenes from C60. The icosahedral symmetry of C60 can be seen as the junction of 17 orbits of a symmetric subgroup of order 4 of the icosahedral group of order 120. This subgroup is noted by A1 × A1, because it is isomorphic to the Weyl group of the semi-simple Lie algebra A1 × A1. Thirteen of the A1 × A1 orbits are rectangles and four are line segments. The orbits form a stack of parallel layers centered on the axis of C60 passing through the centers of two opposite edges between two hexagons on the surface of C60. These two edges are the only two line segment layers to appear on the surface shell. Among the 24 convex polytopes with shell formed by hexagons and 12 pentagons, having 84 vertices [Fowler & Manolopoulos (1992). Nature (London), 355, 428-430; Fowler & Manolopoulos (2007). An Atlas of Fullerenes. Dover Publications Inc.; Zhang et al. (1993). J. Chem. Phys. 98, 3095-3102], there are only two that can be identified with breaking of the H3 symmetry to A1 × A1. The remaining ones are just convex shells formed by regular hexagons and 12 pentagons without the involvement of the icosahedral symmetry.
Exact icosahedral symmetry of C 60 is viewed as the union of 12 orbits of the symmetric subgroup of order 6 of the icosahedral group of order 120. Here, this subgroup is denoted by A 2 because it is isomorphic to the Weyl group of the simple Lie algebra A 2 . Eight of the A 2 orbits are hexagons and four are triangles. Only two of the hexagons appear as part of the C 60 surface shell. The orbits form a stack of parallel layers centered on the axis of C 60 passing through the centers of two opposite hexagons on the surface of C 60 . By inserting into the middle of the stack two A 2 orbits of six points each and two A 2 orbits of three points each, one can match the structure of C 78 . Repeating the insertion, one gets C 96 ; multiple such insertions generate nanotubes of any desired length. Five different polytopes with 78 carbon-like vertices are described; only two of them can be augmented to nanotubes.
Abstract.This paper considers Platonic solids/polytopes in the real Euclidean space R n of dimension 3 ≤ n < ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 ≤ d ≤ n−1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of types An, Bn, Cn, F 4 and of non-crystallographic Coxeter groups H 3 , H 4 .Our method consists in recursively decorating the appropriate CoxeterDynkin diagram. Each recursion step provides the essential information about faces of a specific dimension. If, at each recursion step, all of the faces are in the same Coxeter group orbit, i.e. are identical, the solid is called Platonic.
We describe the existence and structure of large fullerenes in terms of symmetry breaking of the molecule. Specifically, we describe the existence of in terms of breaking of the icosahedral symmetry of by the insertion into its middle of an additional decagon. The surface of is formed by 12 regular pentagons and 25 regular hexagons. All 105 edges of are of the same length. It should be noted that the structure of the molecules is described in exact coordinates relative to the non-orthogonal icosahedral bases. This symmetry breaking process can be readily applied, and could account for and describe other larger cage cluster fullerene molecules, as well as more complex higher structures such as nanotubes.
Icosahedral symmetry is ubiquitous in nature, and understanding possible deformations of structures exhibiting it can be critical in determining fundamental properties. In this work we present a framework for generating and representing deformations of such structures while the icosahedral symmetry is preserved. This is done by viewing the points of an orbit of the icosahedral group as vertices of an icosahedral polytope. Contraction of the orbit is defined as a continuous variation of the coordinates of the dominant point -which specifies the orbit in an appropriate basis -toward smaller positive values. Exact icosahedral symmetry is maintained at any stage of the contraction. All icosahedral orbits or polytopes can be built by successive contractions. This definition of contraction is general and can be applied to orbits of any finite reflection group. PACS Nos.: 61.05.-a, 61.18.-j, 61.48.-c.Résumé : La symétrie icosaédrique est omniprésente dans la nature et la compréhension des déformations possibles des structures est critique pour déterminer les propriétés fondamentales. Nous présentons ici un cadre de travail pour générer et représenter les déformations de ce type de structures, tout en préservant la symétrie icosaédrique. Ceci est fait en prenant des points d'une orbite du groupe icosaèdre comme des sommets d'un polytope icosaédrique. La contraction de l'orbite est définie par des variations continues des coordonnées du point dominant -ceci positionne l'orbite dans une base appropriée -vers de plus petites valeurs positives. Nous maintenons la symétrie icosaédrique à toutes les étapes de la contraction. Toutes les orbites/polytopes icosaédriques peuvent être construites par contractions successives. Cette définition de la contraction est générale et peut être appliquée aux orbites de n'importe quel groupe de réflexion fini. [Traduit par la Rédaction]
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