scholarly journals Quasicrystal Tilings in Three Dimensions and Their Empires

Crystals ◽  
2018 ◽  
Vol 8 (10) ◽  
pp. 370 ◽  
Author(s):  
Dugan Hammock ◽  
Fang Fang ◽  
Klee Irwin
Keyword(s):  
Projection Method ◽  
Voronoi Cell ◽  
Three Dimensions ◽  
New Method ◽  
Dimensional Lattice ◽  
Cell Complexes ◽  
Cubic Lattices ◽  
Higher Dimensional ◽  
Icosahedral Quasicrystals ◽  
Quasiperiodic Tilings

The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal’s vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and Voronoi cell complexes of the lattice being projected. We describe a new method for calculating empires (forced tiles) which also borrows from the dualisation formalism and which generalizes to tilings generated projections of non-cubic lattices. These techniques were used to compute the vertex configurations, frequencies and empires of icosahedral quasicrystals obtained as a projections of the D 6 and Z 6 lattices to R 3 and we present our analyses. We discuss the implications of this new generalization.

Indexing approximants of icosahedral quasicrystals

1999 ◽  
Vol 55 (6) ◽  
pp. 975-983 ◽  
Author(s):  
M. Quiquandon ◽  
A. Katz ◽  
F. Puyraimond ◽  
D. Gratias
Keyword(s):  
Unit Cell ◽  
Dimensional Lattice ◽  
Geometrical Properties ◽  
Actual Model ◽  
Icosahedral Quasicrystals ◽  
Low Dimensional

It is well known that the crystallography of approximants is directly related to that of the parent quasicrystal, once its unit-cell vectors are identified as parallel projections of certain N-dimensional lattice nodes {\bf A}^{i}. Derived here are explicit simple relations for calculating the shear matrices {\boldvarepsilon} and the related crystallographic properties of the corresponding approximants, including diffraction indexing and the determination of the lattice in perpendicular space. Applied to low-dimensional approximants, the derivation shows that the systematic `accidental' extinction rules observed in the pentagonal phases are generic extinctions that are due to the geometrical properties of the projected 1D lattice and are independent of the actual model of the quasicrystal.

scholarly journals Visualising higher-dimensional space-time and space-scale objects as projections to ℝ3

2017 ◽  
Vol 3 ◽  
pp. e123 ◽  
Author(s):  
Ken Arroyo Ohori ◽  
Hugo Ledoux ◽  
Jantien Stoter
Keyword(s):  
Dimensional Space ◽  
Space Time ◽  
Three Dimensions ◽  
Time And Space ◽  
3D Objects ◽  
Levels Of Detail ◽  
Higher Dimensional Space Time ◽  
Space Scale ◽  
Higher Dimensional ◽  
Dimensional Space Time

Objects of more than three dimensions can be used to model geographic phenomena that occur in space, time and scale. For instance, a single 4D object can be used to represent the changes in a 3D object’s shape across time or all its optimal representations at various levels of detail. In this paper, we look at how such higher-dimensional space-time and space-scale objects can be visualised as projections from ℝ4to ℝ3. We present three projections that we believe are particularly intuitive for this purpose: (i) a simple ‘long axis’ projection that puts 3D objects side by side; (ii) the well-known orthographic and perspective projections; and (iii) a projection to a 3-sphere (S3) followed by a stereographic projection to ℝ3, which results in an inwards-outwards fourth axis. Our focus is in using these projections from ℝ4to ℝ3, but they are formulated from ℝnto ℝn−1so as to be easily extensible and to incorporate other non-spatial characteristics. We present a prototype interactive visualiser that applies these projections from 4D to 3D in real-time using the programmable pipeline and compute shaders of the Metal graphics API.

Low-dimensional lattices. VI. Voronoi reduction of three-dimensional lattices

1992 ◽  
Vol 436 (1896) ◽  
pp. 55-68 ◽  
Keyword(s):  
Simple Formula ◽  
Three Dimensional ◽  
Voronoi Cell ◽  
Dimensional Lattice ◽  
Usual Treatment ◽  
Low Dimensional

The aim of this paper is to describe how the Voronoi cell of a lattice changes as that lattice is continuously varied. The usual treatment is simplified by the introduction of new parameters called the vonorms and conorms of the lattice. The present paper deals with dimensions n ≼ 3; a sequel will treat four-dimensional lattices. An elegant algorithm is given for the Voronoi reduction of a three-dimensional lattice, leading to a new proof of Voronoi’s theorem that every lattice of dimension n ≼ 3 is of the first kind, and of Fedorov’s classification of the three-dimensional lattices into five types. There is a very simple formula for the determinant of a three-dimensional lattice in terms of its conorms.

Existence of traveling wave solutions in nonlocal delayed higher-dimensional lattice systems with quasi-monotone nonlinearities

2021 ◽  
Vol 71 (6) ◽  
pp. 1459-1470
Author(s):  
Kun Li ◽  
Yanli He
Keyword(s):  
Traveling Wave ◽  
Solution Method ◽  
Traveling Wave Solutions ◽  
Lower Solution ◽  
Cooperative System ◽  
Lattice Systems ◽  
Dimensional Lattice ◽  
Wave Solutions ◽  
Higher Dimensional ◽  
Lower Solution Method

Abstract In this paper, we are concerned with the existence of traveling wave solutions in nonlocal delayed higher-dimensional lattice systems with quasi-monotone nonlinearities. By using the upper and lower solution method and Schauder’s fixed point theorem, we establish the existence of traveling wave solutions. To illustrate our results, the existence of traveling wave solutions for a nonlocal delayed higher-dimensional lattice cooperative system with two species are considered.

Incremental Hyperplane Partitioning for Classification

2013 ◽  
Vol 4 (2) ◽  
pp. 67-79 ◽  
Author(s):  
Tao Yang ◽  
Sheng-Uei Guan ◽  
Jinghao Song ◽  
Binge Zheng ◽  
Mengying Cao ◽  
...  
Keyword(s):  
Genetic Algorithm ◽  
Linear Equations ◽  
Curse Of Dimensionality ◽  
Experimental Results ◽  
New Method ◽  
Complex Pattern ◽  
Classification Problems ◽  
Encoding Scheme ◽  
Incremental Approach ◽  
Higher Dimensional

The authors propose an incremental hyperplane partitioning approach to classification. Hyperplanes that are close to the classification boundaries of a given problem are searched using an incremental approach based upon Genetic Algorithm (GA). A new method - Incremental Linear Encoding based Genetic Algorithm (ILEGA) is proposed to tackle the difficulty of classification problems caused by the complex pattern relationship and curse of dimensionality. The authors solve classification problems through a simple and flexible chromosome encoding scheme, where the partitioning rules are encoded by linear equations rather than If-Then rules. Moreover, an incremental approach combined with output portioning and pattern reduction is applied to cope with the curse of dimensionality. The algorithm is tested with six datasets. The experimental results show that ILEGA outperform in both lower- and higher-dimensional problems compared with the original GA.

Reduction-by-Projection Method for Linear Programming Problems

2014 ◽  
Vol 672-674 ◽  
pp. 1968-1971
Author(s):  
Xue Tong ◽  
Jun Qiang Wei
Keyword(s):  
Linear Programming ◽  
Linear Systems ◽  
Integer Linear Programming ◽  
Projection Method ◽  
Simplex Method ◽  
New Method ◽  
Linear Programming Problems

This paper defines the projection of algebic systems, and studies the projecting algorithm for linear systems. As its application, a new method is given to solve linear programming problems, which is called reduction-by-projection method. For many problems, especially when the problems have many constraint conditions in comparison with the number of their variables, the method needs less computation than simplex method and others. The great advantage of the method is shown when solving the integer linear programming problems.

Some Helical Structures Generated by Reflexions

1985 ◽  
Vol 38 (3) ◽  
pp. 299 ◽  
Author(s):  
AC Hurley
Keyword(s):  
Dimensional Space ◽  
Helical Structure ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Helical Structures ◽  
Higher Dimensional ◽  
Translation Component ◽  
Explicit Formulae ◽  
Different Dimensions ◽  
Early Discussion

There has recently been a revival of interest in the helical structure built up as a column of face-sharing tetrahedra, because of possible applications in structural crystallography (Nelson 1983). This structure and its analogues in spaces of different dimensions are investigated here. It is shown that the only crystallographic cases are the structures in one- and two-dimensional space. For three and higher dimensional space the structures are all non-crystallographic. For the physically important case of three dimensions, this result is implicit in an early discussion by Coxeter (1969). Results obtained here include explicit formulae for the positions of all vertices of the simplexes for dimensions n = 1-4 and a demonstration that, for arbitrary n, the ratio of the translation component of the screw to the edge of the simplex is {6/ n(n+ I)(n+ 2)}1/2

Primary Functions on a Curve and its Applications

2014 ◽  
Vol 926-930 ◽  
pp. 3149-3152
Author(s):  
Bi Cai Xu
Keyword(s):  
Dimensional Space ◽  
New Method ◽  
Definite Integral ◽  
Primary Function ◽  
Higher Dimensional ◽  
Curvilinear Integral ◽  
Finite Increment

It is very simple to calculate the definite integral and some special curvilinearintegral. Based on this method, the author put forward the concept of primary functions on a curve and sets up a new method on curvilinear integral with primary function. Finally the author gave a generalization of finite increment theorem in higher dimensional space.

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