Icosahedral Symmetry

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  • Icosahedral Symmetry breaking: C60 to C84, C108 and to related nanotubes
    Acta Crystallographica Section A Foundations and Advances, 2015
    Co-Authors: Mark Bodner, Jiri Patera, Emmanuel Bourret, Marzena Szajewska
    Abstract:

    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.

  • Icosahedral Symmetry breaking: C(60) to C(84), C(108) and to related nanotubes.
    Acta crystallographica. Section A Foundations and advances, 2015
    Co-Authors: Mark Bodner, Jiri Patera, Emmanuel Bourret, Marzena Szajewska
    Abstract:

    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.

  • Icosahedral Symmetry breaking: C60to C78, C96and to related nanotubes
    Acta Crystallographica Section A Foundations and Advances, 2014
    Co-Authors: Mark Bodner, Jiri Patera, Emmanuel Bourret, Marzena Szajewska
    Abstract:

    Exact Icosahedral Symmetry of C60is 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 byA2because it is isomorphic to the Weyl group of the simple Lie algebraA2. Eight of theA2orbits are hexagons and four are triangles. Only two of the hexagons appear as part of the C60surface shell. The orbits form a stack of parallel layers centered on the axis of C60passing through the centers of two opposite hexagons on the surface of C60. By inserting into the middle of the stack twoA2orbits of six points each and twoA2orbits of three points each, one can match the structure of C78. Repeating the insertion, one gets C96; 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.

  • Breaking of Icosahedral Symmetry: C60 to C70.
    PloS one, 2014
    Co-Authors: Mark Bodner, Jiri Patera, Marzena Szajewska
    Abstract:

    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.

  • Polytope contractions within Icosahedral Symmetry
    Canadian Journal of Physics, 2014
    Co-Authors: Mark Bodner, Jiri Patera, Marzena Szajewska
    Abstract:

    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.

K.h. Kuo - One of the best experts on this subject based on the ideXlab platform.

  • crystalline phases displaying pseudo Icosahedral Symmetry in ag42in42re16 re gd tb dy ho er tm yb and lu
    Journal of Alloys and Compounds, 2004
    Co-Authors: Juanfang Ruan, K.h. Kuo, J.q. Guo, An Pang Tsai
    Abstract:

    A bcc phase (a = 1.510-1.519 nm), possibly isostructural with Cd6RE, exists in Ag42In42RE16 alloys, where RE = Gd, Tb, Dy, Ho, Er, Tm, and Lu. In Ag42In42Yb16, however, a C-centered monoclinic phase (a = 2.60 nm, b = 1.48 nm, c = 2.63 nm, beta similar to 108degrees) coexists with the Icosahedral quasicrystal (IQC). All these crystalline phases show local pseudo-Icosahedral Symmetry. (C) 2003 Elsevier B.V. All rights reserved.

  • Crystalline phases displaying pseudo-Icosahedral Symmetry in Ag42In42RE16 (RE = Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu)
    Journal of Alloys and Compounds, 2003
    Co-Authors: Juanfang Ruan, K.h. Kuo, J.q. Guo, An Pang Tsai
    Abstract:

    A bcc phase (a = 1.510-1.519 nm), possibly isostructural with Cd6RE, exists in Ag42In42RE16 alloys, where RE = Gd, Tb, Dy, Ho, Er, Tm, and Lu. In Ag42In42Yb16, however, a C-centered monoclinic phase (a = 2.60 nm, b = 1.48 nm, c = 2.63 nm, beta similar to 108degrees) coexists with the Icosahedral quasicrystal (IQC). All these crystalline phases show local pseudo-Icosahedral Symmetry. (C) 2003 Elsevier B.V. All rights reserved.

  • Hierarchic multishell structures with Icosahedral Symmetry
    Journal of Alloys and Compounds, 2002
    Co-Authors: K.h. Kuo, Degang Deng
    Abstract:

    Abstract Icosahedral quasicrystals are generally divided into the Mackay icosahedron (MI) type and the Frank–Kasper (F–K) type based on the multishell Icosahedral structural motifs in their related crystalline phases. Their different hierarchic Icosahedral structures have been discussed separately in the past. It is shown here that they both are faulted Icosahedral shell structure of Mackay with the f.c.c. ABCA… sequence [Acta Crystallogr. 15 (1962) 916]: a h.c.p. fault in the ABCB sequence yields the MI type whereas an ABA sequence the F-K type hierarchic structure. This can also result from the different connections of the core and outer icosahedra in the hierarchic Icosahedral structures: inversely oriented icosahedra yield the MI type while similarly oriented icosahedra yield the F-K type hierarchic structure.

Juanfang Ruan - One of the best experts on this subject based on the ideXlab platform.

Robijn Bruinsma - One of the best experts on this subject based on the ideXlab platform.

  • Icosahedral packing of RNA viral genomes.
    Physical review letters, 2005
    Co-Authors: Joseph Rudnick, Robijn Bruinsma
    Abstract:

    Many spherelike RNA viruses package a portion of their genome in a manner that mirrors the Icosahedral Symmetry of the protein container, or capsid. Graph-theoretical constraints forbid exact realization of Icosahedral Symmetry. This paper explores the consequences of graph-theoretical constraints on quasi-Icosahedral genome structures. A key result is the prediction that the genome organization is a Hamiltonian path or cycle and that the associated assembly scenario of such single-stranded spherelike RNA viruses resembles that of cylindrical RNA viruses, such as tobacco mosaic viruses.

  • Origin of Icosahedral Symmetry in viruses
    Proceedings of the National Academy of Sciences of the United States of America, 2004
    Co-Authors: Roya Zandi, Robijn Bruinsma, William M. Gelbart, David Reguera, Joseph Rudnick
    Abstract:

    With few exceptions, the shells (capsids) of sphere-like viruses have the Symmetry of an icosahedron and are composed of coat proteins (subunits) assembled in special motifs, the T-number structures. Although the synthesis of artificial protein cages is a rapidly developing area of materials science, the design criteria for self-assembled shells that can reproduce the remarkable properties of viral capsids are only beginning to be understood. We present here a minimal model for equilibrium capsid structure, introducing an explicit interaction between protein multimers (capsomers). Using Monte Carlo simulation we show that the model reproduces the main structures of viruses in vivo (T-number icosahedra) and important nonIcosahedral structures (with octahedral and cubic Symmetry) observed in vitro. Our model can also predict capsid strength and shed light on genome release mechanisms.

  • Viral self-assembly as a thermodynamic process.
    Physical review letters, 2003
    Co-Authors: Robijn Bruinsma, William M. Gelbart, David Reguera, Joseph Rudnick, Roya Zandi
    Abstract:

    The protein shells, or capsids, of nearly all spherelike viruses adopt Icosahedral Symmetry. In the present Letter, we propose a statistical thermodynamic model for viral self-assembly. We find that Icosahedral Symmetry is not expected for viral capsids constructed from structurally identical protein subunits and that this Symmetry requires (at least) two internal "switching" configurations of the protein. Our results indicate that Icosahedral Symmetry is not a generic consequence of free energy minimization but requires optimization of internal structural parameters of the capsid proteins.

  • Icosahedral packing of RNA viral genomes
    arXiv: Soft Condensed Matter, 2003
    Co-Authors: Joseph Rudnick, Robijn Bruinsma
    Abstract:

    Recent studies reveal that certain viruses package a portion of their genome in a manner that mirrors the Icosahedral Symmetry of the protein container, or capsid. Graph theoretical constraints forbid exact realization of Icosahedral Symmetry. This paper proposes a model for the determination of quasi-Icosahedral genome structures and discusses the connection between genomic structure and viral assembly kinetics.