Concepts of symmetry in quasicrystals: root lattice D6, icosahedral group, etc.

1992 ◽  
Author(s):  
P. Kramer ◽  
Z. Papadopolos ◽  
D. Zeidler
Keyword(s):  
Root Lattice ◽  
Icosahedral Group

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 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.

C60and the icosahedral group Ih: Simplification methods and group-theoretical applications

1999 ◽  
Vol 69 (3) ◽  
pp. 363-394
Author(s):  
Weiler R. Hurren ◽  
Dorian M. Hatch
Keyword(s):  
Icosahedral Group

Square-root-like higher-order topological states in three-dimensional sonic crystals

2021 ◽  
Author(s):  
Zhiguo Geng ◽  
Huanzhao Lv ◽  
Zhan Xiong ◽  
Yu-Gui Peng ◽  
Zhaojiang Chen ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Surface States ◽  
Higher Order ◽  
Square Root ◽  
Root Lattice ◽  
Third Order ◽  
Topological Materials ◽  
Topological States ◽  
Sonic Crystal ◽  
Sonic Crystals

Abstract The square-root descendants of higher-order topological insulators were proposed recently, whose topological property is inherited from the squared Hamiltonian. Here we present a three-dimensional (3D) square-root-like sonic crystal by stacking the 2D square-root lattice in the normal (z) direction. With the nontrivial intralayer couplings, the opened degeneracy at the K-H direction induces the emergence of multiple acoustic localized modes, i.e., the extended 2D surface states and 1D hinge states, which originate from the square-root nature of the system. The square-root-like higher order topological states can be tunable and designed by optionally removing the cavities at the boundaries. We further propose a third-order topological corner state in the 3D sonic crystal by introducing the staggered interlayer couplings on each square-root layer, which leads to a nontrivial bulk polarization in the z direction. Our work sheds light on the high-dimensional square-root topological materials, and have the potentials in designing advanced functional devices with sound trapping and acoustic sensing.

scholarly journals Analysis of an SU(8) model with a spin-1 2 field directly coupled to a gauged Rarita–Schwinger spin-3 2 field

2019 ◽  
Vol 34 (33) ◽  
pp. 1950230
Author(s):  
Stephen L. Adler
Keyword(s):  
Representation Formula ◽  
Root Lattice ◽  
Anomaly Cancellation ◽  
Text Field ◽  
Text Model ◽  
Gauge Anomaly ◽  
Chiral Representation

In earlier work we analyzed an abelianized model in which a gauged Rarita–Schwinger spin-[Formula: see text] field is directly coupled to a spin-[Formula: see text] field. Here, we extend this analysis to the gauged [Formula: see text] model for which the abelianized model was a simplified substitute. We calculate the gauge anomaly, show that anomaly cancellation requires adding an additional left chiral representation [Formula: see text] spin-[Formula: see text] fermion to the original fermion complement of the [Formula: see text] model, and give options for restoring boson–fermion balance. We conclude with a summary of attractive features of the reformulated [Formula: see text] model, including a possible connection to the [Formula: see text] root lattice.

The polytopes of the H 3 group with 60 vertices and their orbit decompositions

2019 ◽  
Vol 75 (3) ◽  
pp. 541-550
Author(s):  
Emmanuel Bourret ◽  
Zofia Grabowiecka
Keyword(s):  
Coxeter Group ◽  
Dynkin Diagram ◽  
Geometrical Structure ◽  
Three Dimensions ◽  
Icosahedral Group ◽  
Finite Coxeter Group

The goal of this article is to compare the geometrical structure of polytopes with 60 vertices, generated by the finite Coxeter group H 3, i.e. an icosahedral group in three dimensions. The method of decorating a Coxeter–Dynkin diagram is used to easily read the structure of the reflection-generated polytopes. The decomposition of the vertices of the polytopes into a sum of orbits of subgroups of H 3 is given and presented as a `pancake structure'.

Tensor algebra for the icosahedral group

1987 ◽  
Vol 10 (4) ◽  
pp. 1-33 ◽  
Author(s):  
S. Cesare ◽  
V. Del Duca
Keyword(s):  
Tensor Algebra ◽  
Icosahedral Group
Sign in / Sign up

Export Citation Format

Share Document