Icosahedral symmetry breaking: C60to C84, C108and to related nanotubes

2015 ◽  
Vol 71 (3) ◽  
pp. 297-300 ◽  
Author(s):  
Mark Bodner ◽  
Emmanuel Bourret ◽  
Jiri Patera ◽  
Marzena Szajewska
Keyword(s):  
Convex Polytopes ◽  
Three Dimensional ◽  
Chem Phys ◽  
Dimensional Euclidean Space ◽  
Icosahedral Symmetry ◽  
Acta Cryst ◽  
Line Segments ◽  
Symmetric Subgroup ◽  
Higher Fullerenes ◽  
Surface Shell

This paper completes the series of three independent articles [Bodneret 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 C60can 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 byA1×A1, because it is isomorphic to the Weyl group of the semi-simple Lie algebraA1×A1. Thirteen of theA1×A1orbits are rectangles and four are line segments. The orbits form a stack of parallel layers centered on the axis of C60passing 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.; Zhanget al. (1993).J. Chem. Phys.98, 3095–3102], there are only two that can be identified with breaking of theH3symmetry toA1×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

2014 ◽  
Vol 70 (6) ◽  
pp. 650-655 ◽  
Author(s):  
Mark Bodner ◽  
Emmanuel Bourret ◽  
Jiri Patera ◽  
Marzena Szajewska
Keyword(s):  
Symmetry Breaking ◽  
Weyl Group ◽  
Lie Algebra ◽  
Icosahedral Symmetry ◽  
Simple Lie Algebra ◽  
Icosahedral Group ◽  
Symmetric Subgroup ◽  
Surface Shell

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.

scholarly journals Crystal structure of a new monoclinic polymorph ofN-(4-methylphenyl)-3-nitropyridin-2-amine

2014 ◽  
Vol 70 (8) ◽  
pp. 58-61
Author(s):  
Aina Mardia Akhmad Aznan ◽  
Zanariah Abdullah ◽  
Vannajan Sanghiran Lee ◽  
Edward R. T. Tiekink
Keyword(s):  
Three Dimensional ◽  
Basis Set ◽  
Dihedral Angles ◽  
Asymmetric Unit ◽  
Acta Cryst ◽  
Common Features ◽  
Compound C ◽  
B3lyp Level ◽  
The Common ◽  
Independent Molecules

The title compound, C12H11N3O2, is a second monoclinic polymorph (P21, withZ′ = 4) of the previously reported monoclinic (P21/c, withZ′ = 2) form [Akhmad Aznanet al.(2010).Acta Cryst.E66, o2400]. Four independent molecules comprise the asymmetric unit, which have the common features of asyndisposition of the pyridine N atom and the toluene ring, and an intramolecular amine–nitro N—H...O hydrogen bond. The differences between molecules relate to the dihedral angles between the rings which range from 2.92 (19) to 26.24 (19)°. The geometry-optimized structure [B3LYP level of theory and 6–311 g+(d,p) basis set] has the same features except that the entire molecule is planar. In the crystal, the three-dimensional architecture is consolidated by a combination of C—H...O, C—H...π, nitro-N—O...π and π–π interactions [inter-centroid distances = 3.649 (2)–3.916 (2) Å].

scholarly journals Crystal Structure of an Aquabirnavirus Particle: Insights into Antigenic Diversity and Virulence Determinism

2009 ◽  
Vol 84 (4) ◽  
pp. 1792-1799 ◽  
Author(s):  
Fasséli Coulibaly ◽  
Christophe Chevalier ◽  
Bernard Delmas ◽  
Félix A. Rey
Keyword(s):  
Crystal Structure ◽  
Three Dimensional ◽  
Salient Feature ◽  
Infectious Pancreatic Necrosis Virus ◽  
Antigenic Determinants ◽  
Binding Motif ◽  
Icosahedral Symmetry ◽  
Fold Axis ◽  
Cell Internalization ◽  
Pancreatic Necrosis Virus

ABSTRACT Infectious pancreatic necrosis virus (IPNV), a pathogen of salmon and trout, imposes a severe toll on the aquaculture and sea farming industries. IPNV belongs to the Aquabirnavirus genus in the Birnaviridae family of bisegmented double-stranded RNA viruses. The virions are nonenveloped with a T=13l icosahedral capsid made by the coat protein VP2, the three-dimensional (3D) organization of which is known in detail for the family prototype, the infectious bursal disease virus (IBDV) of poultry. A salient feature of the birnavirus architecture is the presence of 260 trimeric spikes formed by VP2, projecting radially from the capsid. The spikes carry the principal antigenic sites as well as virulence and cell adaptation determinants. We report here the 3.4-Å resolution crystal structure of a subviral particle (SVP) of IPNV, containing 20 VP2 trimers organized with icosahedral symmetry. We show that, as expected, the SVPs have a very similar organization to the IBDV counterparts, with VP2 exhibiting the same overall 3D fold. However, the spikes are significantly different, displaying a more compact organization with tighter packing about the molecular 3-fold axis. Amino acids controlling virulence and cell culture adaptation cluster differently at the top of the spike, i.e., in a central bowl in IBDV and at the periphery in IPNV. In contrast, the spike base features an exposed groove, conserved across birnavirus genera, which contains an integrin-binding motif. Thus, in addition to revealing the viral antigenic determinants, the structure suggests that birnaviruses interact with different receptors for attachment and for cell internalization during entry.

scholarly journals On a Question of Bourgain about Geometric Incidences

2008 ◽  
Vol 17 (4) ◽  
pp. 619-625 ◽  
Author(s):  
JÓZSEF SOLYMOSI ◽  
CSABA D. TÓTH
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Geometric Incidences

Given a set of s points and a set of n2 lines in three-dimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, we show that s = Ω(n11/4). This is the first non-trivial answer to a question recently posed by Jean Bourgain.

scholarly journals Orthogonal Isomorphic Representations Of Free Groups

1956 ◽  
Vol 8 ◽  
pp. 256-262 ◽  
Author(s):  
J. De Groot
Keyword(s):  
Euclidean Space ◽  
Free Groups ◽  
Three Dimensional ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Orthogonal Matrices ◽  
Abelian Subgroups ◽  
Proper Orthogonal

1. Introduction. We consider the group of proper orthogonal transformations (rotations) in three-dimensional Euclidean space, represented by real orthogonal matrices (aik) (i, k = 1,2,3) with determinant + 1 . It is known that this rotation group contains free (non-abelian) subgroups; in fact Hausdorff (5) showed how to find two rotations P and Q generating a group with only two non-trivial relationsP2 = Q3 = I.

Gröbner basis and resultant method for the forward displacement of 3-DoF planar parallel manipulators in seven-dimensional kinematic space

Robotica ◽  
2015 ◽  
Vol 34 (11) ◽  
pp. 2610-2628 ◽  
Author(s):  
Davood Naderi ◽  
Mehdi Tale-Masouleh ◽  
Payam Varshovi-Jaghargh
Keyword(s):  
Euclidean Space ◽  
Gröbner Basis ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Parallel Robots ◽  
Dimensional Euclidean Space ◽  
Grobner Basis ◽  
Kinematic Space ◽  
Forward Kinematic Problem ◽  
Forward Kinematic

SUMMARYIn this paper, the forward kinematic analysis of 3-degree-of-freedom planar parallel robots with identical limb structures is presented. The proposed algorithm is based on Study's kinematic mapping (E. Study, “von den Bewegungen und Umlegungen,” Math. Ann.39, 441–565 (1891)), resultant method, and the Gröbner basis in seven-dimensional kinematic space. The obtained solution in seven-dimensional kinematic space of the forward kinematic problem is mapped into three-dimensional Euclidean space. An alternative solution of the forward kinematic problem is obtained using resultant method in three-dimensional Euclidean space, and the result is compared with the obtained mapping result from seven-dimensional kinematic space. Both approaches lead to the same maximum number of solutions: 2, 6, 6, 6, 2, 2, 2, 6, 2, and 2 for the forward kinematic problem of planar parallel robots; 3-RPR, 3-RPR, 3-RRR, 3-RRR, 3-RRP, 3-RPP, 3-RPP, 3-PRR, 3-PRR, and 3-PRP, respectively.

Integral criteria for closed surfaces with constant mean curvature

2007 ◽  
Vol 463 (2081) ◽  
pp. 1199-1210
Author(s):  
Luca Guzzardi ◽  
Epifanio G Virga
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Second Variation ◽  
Closed Surfaces ◽  
Area Functional ◽  
Symmetric Surface ◽  
The Second Variation ◽  
Integral Identities

We propose three integral criteria that must be satisfied by all closed surfaces with constant mean curvature immersed in the three-dimensional Euclidean space. These criteria are integral identities that follow from requiring the second variation of the area functional to be invariant under rigid displacements. We obtain from them a new proof of the old result by Delaunay, to the effect that the sphere is the only closed axis-symmetric surface.

scholarly journals Bis(μ2-4-nitrophenolato)bis(4-nitrophenolato)di-μ3-oxido-octaphenyltetratin chloroform sesquisolvate [+ solvate]: a tetranuclear stannoxane

IUCrData ◽  
2019 ◽  
Vol 4 (8) ◽  
Author(s):  
Patrick Butler
Keyword(s):  
Hydrogen Bonds ◽  
Electron Density ◽  
Three Dimensional ◽  
The Other ◽  
Dimensional Structure ◽  
Coordination Geometry ◽  
Asymmetric Unit ◽  
Acta Cryst ◽  
Complex Molecules ◽  
Three Dimensional Structure

The title tetranuclear stannoxane, [Sn4(C6H5)8(C6H4NO3)4O2]·1.5CHCl3·solvent, crystallized with two independent complex molecules, A and B, in the asymmetric unit together with 1.5 molecules of chloroform. There is also a region of disordered electron density, which was corrected for using the SQUEEZE routine [Spek (2015). Acta Cryst. C71, 9–18]. The oxo-tin core of each complex is in a planar `ladder' arrangement and each Sn atom is fivefold SnO3C2 coordinated, with one tin centre having an almost perfect square-pyramidal coordination geometry, while the other three Sn centres have distorted shapes. In the crystal, the complex molecules are arranged in layers, composed of A or B complexes, lying parallel to the bc plane. The complex molecules are linked by a number of C—H...O hydrogen bonds within the layers and between the layers, forming a supramolecular three-dimensional structure.

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