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.

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.

A geometric approach to manipulator path planning in 3D space in the presence of obstacles

Robotica ◽  
1992 ◽  
Vol 10 (4) ◽  
pp. 321-328
Author(s):  
Manja Kirćanski ◽  
Olga Timčenko
Keyword(s):  
Euclidean Space ◽  
Path Planning ◽  
Real Time ◽  
Control Level ◽  
Geometric Approach ◽  
Geometric Method ◽  
Reachable Workspace ◽  
3D Space ◽  
Simulation Results ◽  
Polyhedral Obstacles

SUMMARYThe paper presents a geometric method for collision-free manipulator path planning in 3D Euclidean space with polyhedral obstacles. It ensures that none of the links nor the manipulator tip collide with the objects. The method is computationally very cheap and it does not require intensive off-line preprocessing. Hence, it is real-time applicable if the information about obstacles positions and shapes is obtained from a higher control level. The trajectories generated lie within the reachable workspace. The method is implemented on a VAX 11/750 computer and the simulation results are included.

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.  

scholarly journals On the Theory of a Non-Periodic Quasilattice Associated with the Icosahedral Group

1985 ◽  
Vol 40 (8) ◽  
pp. 775-788
Author(s):  
P. Kramer
Keyword(s):  
Euclidean Space ◽  
General Theory ◽  
Structure Factor ◽  
Point Symmetry ◽  
Space Filling ◽  
Icosahedral Group ◽  
Local Properties ◽  
Global And Local ◽  
Periodic Space

A quasilattice in the euclidean space E3 is generated by dualization of a hexagrid. The construction is based on a general theory of periodic and non-periodic space filling by projection from hypergrids. Global and local properties of the quasilattice are discussed, including the structure factor, the form and packing of the cells, and the point symmetry. The quasilattice is expected to give a description of quasicrystals without periodic order which have recently been found in experiments.

scholarly journals 4-Point Minimal Pick & Place Trajectory Design in Robotics

2019 ◽  
Vol 8 (9S) ◽  
pp. 801-806
Keyword(s):  
Euclidean Space ◽  
Motion Planning ◽  
Motion Control ◽  
Shortest Path ◽  
Mathematical Formulation ◽  
Theoretical Background ◽  
Dimensional Euclidean Space ◽  
Trajectory Design ◽  
3 Dimensional ◽  
3D Space

Motion planning in robotics plays a very important role in the movement of objects from the source to the destination. Robots are classified according to motion control as PNP, PTP and CP robots. Hence, there are 3 basic types of trajectory motions + 1 trajectory which is the shortest path between two points in the 3D space. This type of motion or trajectory is exhibited by PNP robots. In this paper, we develop the theoretical background along with the mathematical formulation relating to the design & development of a 4-point minimal pick & place trajectory from the source to the destination during the transportation of an object in the 3 dimensional Euclidean space R3.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

Face-to-Face

1992 ◽  
Vol 1 (2) ◽  
pp. 11-12 ◽  
Author(s):  
James Jerger
Keyword(s):  
Face To Face
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