scholarly journals Icosahedral Polyhedra from D6 Lattice and Danzer’s ABCK Tiling

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1983
Author(s):  
Abeer Al-Siyabi ◽  
Nazife Ozdes Koca ◽  
Mehmet Koca
Keyword(s):  
Maximal Subgroup ◽  
Group Action ◽  
Point Group ◽  
Fibonacci Sequence ◽  
Fundamental Region ◽  
Icosahedral Symmetry ◽  
Root Lattice ◽  
Linear Combinations ◽  
Icosahedral Group ◽  
3D Space

It is well known that the point group of the root lattice D6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H3, its roots, and weights are determined in terms of those of D6. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D6 determined by a pair of integers (m1, m2) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of H3, and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers, which are linear combinations of the integers (m1, m2) with coefficients from the Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the <ABCK> octahedral tilings in 3D space with icosahedral symmetry H3, and those related transformations in 6D space with D6 symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in D6. The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron”, and “C-polyhedron” generated by the icosahedral group have been discussed.

scholarly journals Dodecahedral structures with Mosseri–Sadoc tiles

2021 ◽  
Vol 77 (2) ◽  
Author(s):  
Nazife Ozdes Koca ◽  
Ramazan Koc ◽  
Mehmet Koca ◽  
Abeer Al-Siyabi
Keyword(s):  
Euclidean Space ◽  
Golden Ratio ◽  
Root Lattice ◽  
Third Order ◽  
Face To Face ◽  
Icosahedral Group ◽  
3D Space ◽  
Edge Matching ◽  
Inflation Factor

The 3D facets of the Delone cells of the root lattice D 6 which tile the 6D Euclidean space in an alternating order are projected into 3D space. They are classified into six Mosseri–Sadoc tetrahedral tiles of edge lengths 1 and golden ratio τ = (1 + 51/2)/2 with faces normal to the fivefold and threefold axes. The icosahedron, dodecahedron and icosidodecahedron whose vertices are obtained from the fundamental weights of the icosahedral group are dissected in terms of six tetrahedra. A set of four tiles are composed from six fundamental tiles, the faces of which are normal to the fivefold axes of the icosahedral group. It is shown that the 3D Euclidean space can be tiled face-to-face with maximal face coverage by the composite tiles with an inflation factor τ generated by an inflation matrix. It is noted that dodecahedra with edge lengths of 1 and τ naturally occur already in the second and third order of the inflations. The 3D patches displaying fivefold, threefold and twofold symmetries are obtained in the inflated dodecahedral structures with edge lengths τ n with n ≥ 3. The planar tiling of the faces of the composite tiles follows the edge-to-edge matching of the Robinson triangles.

scholarly journals Exploring the Interplay between Virology and Molecular Crystallography

2008 ◽  
Vol 9 (3-4) ◽  
pp. 167-173 ◽  
Author(s):  
Aloysio Janner
Keyword(s):  
Point Group ◽  
Morphological Characterization ◽  
Three Dimensions ◽  
Icosahedral Symmetry ◽  
Coat Proteins ◽  
Viral Capsids ◽  
Icosahedral Group ◽  
Higher Dimensional ◽  
Icosahedral Viruses ◽  
The One

Polyhedra with icosahedral symmetry and vertices labelled by rational indices of points of a six-dimensional lattice left invariant by the icosahedral group allow a morphological characterization of icosahedral viruses which includes the Caspar–Klug classification as a special case. Scaling transformations relating the indexed polyhedron enclosing the surface with the one delimiting the central cavity lead to models of viral capsids observed in nature. Similar scaling relations can be obtained from projected images in three dimensions of higher-dimensional crystallographic point groups having the icosahedral group as a subgroup. This crystallographic approach can be extended to axial-symmetric clusters of coat proteins around icosahedral axes of the capsid. One then gets enclosing forms with vertices at points of lattices left invariant by the corresponding point group and having additional crystallographic properties also observed in natural crystals, but not explained by the known crystallographic laws.

An Aspect of Icosahedral Symmetry

2002 ◽  
Vol 45 (4) ◽  
pp. 686-696 ◽  
Author(s):  
Jan Rauschning ◽  
Peter Slodowy
Keyword(s):  
Symmetric Group ◽  
Moduli Space ◽  
Irreducible Module ◽  
Projective Line ◽  
Icosahedral Symmetry ◽  
3 Dimensional ◽  
Icosahedral Group ◽  
Regular Icosahedron

AbstractWe embed the moduli space Q of 5 points on the projective line S5-equivariantly into (V), where V is the 6-dimensional irreducible module of the symmetric group S5. This module splits with respect to the icosahedral group A5 into the two standard 3-dimensional representations. The resulting linear projections of Q relate the action of A5 on Q to those on the regular icosahedron.

scholarly journals Explicit expressions for totally symmetric spherical functions and symmetry-dependent properties of multipoles

2014 ◽  
Vol 470 (2171) ◽  
pp. 20140435
Author(s):  
Boris Zapol ◽  
Peter Zapol
Keyword(s):  
Point Group ◽  
Product Structure ◽  
Spherical Functions ◽  
Matrix Elements ◽  
Linear Combinations ◽  
Selection Rules ◽  
Charge Distributions ◽  
Symmetry Operators ◽  
Point Groups ◽  
Symmetry Operations

Closed expressions for matrix elements 〈 lm' | A (G)| lm 〉, where | lm 〉 are spherical functions and A (G) is the average of all symmetry operators of point group G, are derived for all point groups (PGs) and then used to obtain linear combinations of spherical functions that are totally symmetric under all symmetry operations of G. In the derivation, we exploit the product structure of the groups. The obtained expressions are used to explore properties of multipoles of symmetric charge distributions. We produce complete lists of selection rules for multipoles Q l and their moments Q lm , as well as of numbers of independent moments in a multipole, for any l and m and for all PGs. Periodicities and other trends in these properties are revealed.

scholarly journals 4D Pyritohedral Symmetry

2016 ◽  
Vol 21 (2) ◽  
pp. 150
Author(s):  
Nazife O. Koca ◽  
Amal J.H. Al Qanobi ◽  
Mehmet Koca
Keyword(s):  
Euclidean Space ◽  
Maximal Subgroup ◽  
Symmetry Group ◽  
Coxeter Groups ◽  
Dimensional Euclidean Space ◽  
Root Lattice

We describe an extension of the pyritohedral symmetry in 3D to 4-dimensional Euclidean space and construct the group elements of the 4D pyritohedral group of order 576 in terms of quaternions. It turns out that it is a maximal subgroup of both the rank-4 Coxeter groups W (F4) and W (H4), implying that it is a group relevant to the crystallographic as well as quasicrystallographic structures in 4-dimensions. We derive the vertices of the 24 pseudoicosahedra, 24 tetrahedra and the 96 triangular pyramids forming the facets of the pseudo snub 24-cell. It turns out that the relevant lattice is the root lattice of W (D4). The vertices of the dual polytope of the pseudo snub 24-cell consists of the union of three sets: 24-cell, another 24-cell and a new pseudo snub 24-cell. We also derive a new representation for the symmetry group of the pseudo snub 24-cell and the corresponding vertices of the polytopes.  

Angular reconstitution of ice embedded macromolecules with arbitrary point group symmetry

1994 ◽  
Vol 52 ◽  
pp. 90-91
Author(s):  
Marin van Heel ◽  
Michael Schatz ◽  
Prakash Dube ◽  
Elena V. Orlova
Keyword(s):  
Point Group ◽  
3D Structure ◽  
Three Dimensional ◽  
Group Symmetry ◽  
Specimen Preparation ◽  
Biological Macromolecules ◽  
Icosahedral Symmetry ◽  
Single Shot ◽  
Preparation Technique ◽  
Tilt Series

Electron microscopy of individual non-crystallized (large) macromolecules is a very rapid technique for probing the three-dimensional (3D) structure of biological macromolecules. Since there is no need for extensive crystallization experiments, specimen preparation can be simple and fast. In particular the vitreous-ice embedding specimen preparation technique, in which the macromolecules are kept in a waterlike environment, has proved very suited for this purpose. One may extract three-dimensional information from the data - without ever collecting tilt series in the microscope - by exploiting thedifferent ("random") orientations the macromolecules have with respect to the grid. Collecting tilt series can be cumbersome and requires multiple exposure of the sensitive molecules. The only techniquewhich was available for such single-shot 3D microscopy was the common-lines technique for virusses with icosahedral symmetry [Crowther (1971), Fuller (1987)]. The other technique available for asymmetric non-crystalline specimens is the random conical tilt technique [Radermacher (1986)]. However, this technique is a tilt series technique requiring two exposures per specimen area. Moreover, it requiresthe molecules to be preferentially oriented with respect to the plane of the support film which - in turn - requires strong (and thus unfavourable) interactions between molecule and support.

Application of Over-Sample Policy and Rotated Angle Invariability of Radial Sampling Points in the Isaf Reconstruction Algorithm

2011 ◽  
Vol 216 ◽  
pp. 485-489
Author(s):  
Gong Ming Wang ◽  
Fa Zhang ◽  
Qi Chu ◽  
Zhi Yong Liu
Keyword(s):  
High Resolution ◽  
Reconstruction Algorithm ◽  
Icosahedral Symmetry ◽  
Running Speed ◽  
Radial Sampling ◽  
3D Space ◽  
B Virus ◽  
Sampling Points ◽  
Order Of Magnitude ◽  
Symmetry Adapted Functions

ISAF (icosahedral symmetry-adapted functions) algorithm is the new high-resolution algorithm of icosahedral macromolecules. But its running speed is very slow because of the time-consuming operations of mapping sampling points into 3D space. In this paper, a new sampling method is proposed to improve the running speed of this stage. First of all, the angle corresponding to one pixel arc in the maximum Fourier ring is taken as the sampling angle and the same angle sampling is applied in every rings. After that, the sampling points in radius one ring are mapped into 3D space. Finally, the 3D spatial positions of radial sampling points in other rings can be deduced quickly according to the rotate angle invariability of radial sampling points. The HBV (Hepatitis B Virus) Cryo-electron microscopy images are used for validating this strategy. At the high resolution of 6.64 angstrom, the local speedup in the stage of mapping sampling points reaches to 50, and the overall speedup can be improved in an order of magnitude. The overall speedup increases with the increasement number of EM images and the improvement of target resolution.

scholarly journals Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Tianlan Chen ◽  
Muhammad Nadeem Bari ◽  
Muhammad Aslam Malik ◽  
Hafiz Muhammad Afzal Siddiqui ◽  
Jia-Bao Liu
Keyword(s):  
Group Action ◽  
Modular Group ◽  
Equivalence Classes ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Icosahedral Group ◽  
Real Quadratic

Reduced numbers play an important role in the study of modular group action on the PSL2,ℤ-subset of Qm/Q. For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in PSL2,ℤ-orbits of real quadratic fields. In particular, we classify PSL2,ℤ-orbits of Qm/Q=∪k∈NQ∗k2m containing G-circuits of length 6 and determine that the number of equivalence classes of G-circuits of length 6 is ten. We also employ the icosahedral group to explore cyclically equivalence classes of circuits and similar G-circuits of length 6 corresponding to each of these ten circuits. This study also helps us in classifying reduced numbers lying in the PSL2,ℤ-orbits.

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