C70, C80, C90and carbon nanotubes by breaking of the icosahedral symmetry of C60

Bodner, Mark; Patera, J.; Szajewska, Marzena

doi:10.1107/s0108767313021375

Cited by 17 publications

(37 citation statements)

References 17 publications

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“…In recent years evidence has been obtained that there exist in nature molecules with imperfect symmetries (Bodner et al, 2013(Bodner et al, , 2014Fowler & Manolopoulos, 2007;McKenzie et al, 1992) of the WðgÞ type, not necessarily Platonic. They could be considered as WðgÞ symmetries broken to the symmetry of a subgroup.…”

Section: Discussionmentioning

confidence: 99%

Faces of Platonic solids in all dimensions

Szajewska

2014

Acta Cryst Sect A

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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.

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Section: Discussionmentioning

confidence: 99%

Faces of Platonic solids in all dimensions

Szajewska

2014

Acta Cryst Sect A

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“…The polytopes of interest, or, more precisely, their vertices, are all formed starting from a single ("seed") point and applying to it the reflections (2.2) for as long as new points (vertices) are being generated [15,16]. We say that such vertices belong to one orbit of H 3 , fixed by our choice of the seed point.…”

Section: Preliminariesmentioning

confidence: 99%

Polytope contractions within icosahedral symmetry

2014

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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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“…from the structure published in Bodner et al (2013) and the known structures of the C 80 fullerene isomers, suggesting that this structure is not realized in nature.…”

Section: Electronic Reprintmentioning

confidence: 99%

Viruses and fullerenes – symmetry as a common thread?

Dechant¹,

Wardman²,

Keef³

et al. 2014

Acta Cryst Sect A

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Reidun (2014) 'Viruses and fullerenes -symmetry as a common thread ?', Acta crystallographica A., 70 (2). pp. 162-167. Further information on publisher's website:http://dx.doi.org/10.1107/S2053273313034220Publisher's copyright statement:Additional information:Deposited paper was the cover article for the March issue of this journal. Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Author(s) of this paper may load this reprint on their own web site or institutional repository provided that this cover page is retained. Republication of this article or its storage in electronic databases other than as specified above is not permitted without prior permission in writing from the IUCr.For further information see http://journals.iucr.org/services/authorrights.html Acta Crystallographica Section A: Foundations and Advances publishes articles reporting f undamental advances in all areas of crystallography in the broadest sense. This includes metacrystals such as photonic or phononic crystals, i.e. structures on the mesoor macroscale that can be studied with crystallographic methods. The central themes are, on the one hand, experimental and theoretical studies of the properties and arrangements of atoms, ions and molecules in condensed matter, periodic, quasiperiodic or amorphous, ideal or real, and, on the other, the theoretical and experimental aspects of the various methods to determine these properties and arrangements. In the case of metacrystals, the focus is on the methods for their creation and on the structure-property relationships for their interaction with classical waves. The principle of affine symmetry is applied here to the nested fullerene cages (carbon onions) that arise in the context of carbon chemistry. Previous work on affine extensions of the icosahedral group has revealed a new organizational principle in virus structure and assembly. This group-theoretic framework is adapted here to the physical requirements dictated by carbon chemistry, and it is shown that mathematical models for carbon onions can be derived within this affine symmetry approach. This suggests the applicability of affine symmetry in a wider context in nature, as well as offering a novel perspective on the geometric principles underpinning carbon chemistry.

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C₇₀, C₈₀, C₉₀and carbon nanotubes by breaking of the icosahedral symmetry of C₆₀

Cited by 17 publications

References 17 publications

Faces of Platonic solids in all dimensions

Faces of Platonic solids in all dimensions

Polytope contractions within icosahedral symmetry

Viruses and fullerenes – symmetry as a common thread?

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