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Bodner M.,11 Academy Drive | Bourret E.,University of Montréal | Patera J.,11 Academy Drive | Patera J.,University of Montréal | And 3 more authors.
Acta Crystallographica Section A: Foundations and Advances | Year: 2014

Exact icosahedral symmetry of C60 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 A2 because it is isomorphic to the Weyl group of the simple Lie algebra A2. Eight of the A2 orbits are hexagons and four are triangles. Only two of the hexagons appear as part of the C60 surface shell. The orbits form a stack of parallel layers centered on the axis of C60 passing through the centers of two opposite hexagons on the surface of C60. By inserting into the middle of the stack two A2 orbits of six points each and two A2 orbits 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. © 2014 International Union of Crystallography.


Bodner M.,11 Academy Drive | Patera J.,11 Academy Drive | Patera J.,University of Montréal | Szajewska M.,University of Montréal | Szajewska M.,University of Bialystok
Acta Crystallographica Section A: Foundations of Crystallography | Year: 2013

The icosahedral symmetry group H 3 of order 120 and its dihedral subgroup H 2 of order 10 are used for exact geometric construction of polytopes that are known to exist in nature. The branching rule for the H 3 orbit of the fullerene C60 to the subgroup H 2 yields a union of eight orbits of H 2: 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 H 2, 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 H 2 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. © 2013 International Union of Crystallography Printed in Singapore - all rights reserved.


Bodner M.,11 Academy Drive | Bourret E.,University of Montréal | Patera J.,11 Academy Drive | Patera J.,University of Montréal | And 3 more authors.
Acta Crystallographica Section A: Foundations and Advances | Year: 2015

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 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 A 1 × A 1, because it is isomorphic to the Weyl group of the semi-simple Lie algebra A 1 × A 1. Thirteen of the A 1 × A 1 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 H 3 symmetry to A 1 × A 1. The remaining ones are just convex shells formed by regular hexagons and 12 pentagons without the involvement of the icosahedral symmetry. © 2015 International Union of Crystallography.


Bodner M.,11 Academy Drive | Chadzitaskos G.,Czech Technical University | Patera J.,11 Academy Drive | Patera J.,University of Montréal | And 2 more authors.
Acta Polytechnica | Year: 2013

About 60 years ago, I. Shmushkevich presented a simple ingenious method for computing the relative probabilities of channels involving the same interacting multiplets of particles, without the need to compute the Clebsch-Gordan coefficients. The basic idea of Shmushkevich is "isotopic non-polarization" of the states before the interaction and after it. Hence his underlying Lie group was SU(2). We extend this idea to any simple Lie group. This paper determines the relative probabilities of various channels of scattering and decay processes following from the invariance of the interactions with respect to a compact simple a Lie group. Aiming at the probabilities rather than at the Clebsch-Gordan coefficients makes the task easier, and simultaneous consideration of all possible channels for given multiplets involved in the process, makes the task possible. The probability of states with multiplicities greater than 1 is averaged over. Examples with symmetry groups O(5), F(4), and E(8) are shown. © Czech Technical University in Prague, 2013.


PubMed | 11 Academy Drive
Type: Journal Article | Journal: Acta crystallographica. Section A, Foundations of crystallography | Year: 2013

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.


PubMed | 11 Academy Drive and University of Montréal
Type: Journal Article | Journal: Acta crystallographica. Section A, Foundations and advances | Year: 2015

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.

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