Investigation of nodal domains in a chaotic three-dimensional microwave rough billiard with the translational symmetry

2008 ◽  
Vol 372 (11) ◽  
pp. 1851-1855 ◽  
Author(s):  
Nazar Savytskyy ◽  
Oleg Tymoshchuk ◽  
Oleh Hul ◽  
Szymon Bauch ◽  
Leszek Sirko
Keyword(s):  
Three Dimensional ◽  
Translational Symmetry ◽  
Nodal Domains

Joint Normal Partitions and Hierarchical Filling of N-Dimensional Spaces

2021 ◽  
pp. 261-272
Keyword(s):  
Three Dimensional ◽  
Translational Symmetry ◽  
Higher Dimension

The structures arising in spaces of various dimensions with simultaneous normal partitioning of spaces and their hierarchical fillings are considered. The conditions for the appearance of translational symmetry in these structures are investigated. It is shown that simultaneous hierarchical filling and normal tiling in three-dimensional spaces do not lead to the formation of translational symmetry. Such consistent transformations lead to many elements of translational symmetry in spaces of higher dimension. The higher the dimension of space, the more complex the emerging structure and the more symmetry the elements.

Symmetry

2004 ◽  
Author(s):  
Robert E. Newnham
Keyword(s):  
Single Crystals ◽  
Point Group ◽  
Three Dimensional ◽  
Unit Cell ◽  
Translational Symmetry ◽  
Identity Operator ◽  
Space Groups ◽  
Rotation Axes ◽  
Point Groups ◽  
And Inversion

All single crystals possess translational symmetry, and most possess other symmetry elements as well. In this chapter we describe the 32 crystallographic point groups used for single crystals. The seven Curie groups used for textured polycrystalline materials are enumerated in the next chapter. We live in a three-dimensional world which means that there are basically four kinds of geometric symmetry operations relating one part of this world to another. The four primary types of symmetry are translation, rotation, reflection, and inversion. As pictured in Fig. 3.1, these symmetry operators operate on a point with coordinates Z1, Z2, Z3 and carry it to a new position. By definition, all crystals have a unit cell that is repeated many times in space, a point Z1, Z2, Z3 is repeated over and over again as one unit cell is translated to the next. A mirror plane perpendicular to one of the principal axes is a two-dimensional symmetry element that reverses the sign of one coordinate. Rotation axes are one-dimensional symmetry elements that change two coordinates, while an inversion center is a zero-dimensional point that changes all three coordinates. In developing an understanding of the macroscopic properties of crystals, we recognize that the scale of physical property measurements is much larger than the unit cell dimensions. It is for this reason that we are not concerned about translational symmetry and work with the 32 point group symmetries rather than the 230 space groups. This greatly simplifies the structure–property relationships in crystal physics. Aside from the identity operator 1, there are only four types of rotational symmetry consistent with the translation symmetry common to all crystals. Fig. 3.2 shows why. Parallelograms, equilateral triangles, squares, and hexagons will pack together to fill space but, pentagons symmetry axes are found in crystals. This is the starting point for generating the 32 crystal classes. When taken in combination with mirror planes and inversion centers, these four types of rotation axes are capable of forming 32 self-consistent three-dimensional symmetry patterns around a point. These are the so-called 32 crystal classes or crystallographic point groups.

Quasicrystals Perspectives and Potential Applications

MRS Bulletin ◽  
1997 ◽  
Vol 22 (11) ◽  
pp. 34-39 ◽  
Author(s):  
Daniel J. Sordelet ◽  
Jean Marie Dubois
Keyword(s):  
Window Glass ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Unit Cell ◽  
Translational Symmetry ◽  
Integer Number ◽  
Unit Cells ◽  
Potential Applications ◽  
Incommensurate Structures ◽  
Three Dimensional Space

For decades scientists have accepted the premise that solid matter can only order in two ways: amorphous (or glassy) like window glass or crystalline with atoms arranged according to translational symmetry. The science of crystallography, now two centuries old, was able to relate in a simple and efficient way all atomic positions within a crystal to a frame of reference in which a single unit cell was defined. Positions within the crystal could all be deduced from the restricted number of positions in the unit cell by translations along vectors formed by a combination of integer numbers of unit vectors of the reference frame. Of course disorder, which is always present in solids, could be understood as some form of disturbance with respect to this rule of construction. Also amorphous solids were naturally referred to as a full breakdown of translational symmetry yet preserving most of the short-range order around atoms. Incommensurate structures, or more simply modulated crystals, could be understood as the overlap of various ordering potentials not necessarily with commensurate periodicities.For so many years, no exception to the canonical rule of crystallography was discovered. Any crystal could be completely described using one unit cell and its set of three basis vectors. In 1848 the French crystallographer Bravais demonstrated that only 14 different ways of arranging atoms exist in three-dimensional space according to translational symmetry. This led to the well-known cubic, hexagonal, tetragonal, and associated structures. Furthermore the dihedral angle between pairs of faces of the unit cell cannot assume just any number since an integer number of unit cells must completely fill space around an edge.

Higher Dimensions of Clusters of Intermetallic Compounds

2019 ◽  
Vol 4 (1) ◽  
pp. 8-25 ◽  
Author(s):  
Gennadiy V Zhizhin
Keyword(s):  
Diffraction Pattern ◽  
Intermetallic Compounds ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Translational Symmetry ◽  
Higher Dimension ◽  
Internal Geometry ◽  
Metallic Nanoclusters

The author has previously proven that diffraction pattern of intermetallic compounds (quasicrystals) have translational symmetry in the space of higher dimension. In this paper, it is proved that the metallic nanoclusters also have a higher dimension. The internal geometry of clusters was investigated. General expressions for calculating the dimension of clusters is obtained, from which it follows that the dimension of metallic nanoclusters increases linearly with increasing number of cluster shells. The dimensions of many experimentally known metallic nanoclusters are determined. It is shown that these clusters, which are usually considered to be three - dimensional, have a higher dimension. The Euler-Poincaré equation was used, the internal geometry of clusters was investigated.

scholarly journals Categorizing Three-Dimensional Symmetry Using Reflection, Rotoinversion, and Translation Symmetry

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1132 ◽  
Author(s):  
Baclig ◽  
Westover ◽  
Adeeb
Keyword(s):  
Standard Deviation ◽  
Simple Procedure ◽  
Three Dimensional ◽  
Translational Symmetry ◽  
Contour Map ◽  
Fundamental Symmetry ◽  
Dimensional Measurements ◽  
Translation Symmetry ◽  
Arbitrary Plane ◽  
Magnitude Difference

Symmetry is a property that has been widely examined clinically as a measurement of health and aesthetic appeal. Many current techniques that assess geometric symmetry rely on interpretation from a trained operator or produce two-dimensional measurements that cannot express the three-dimensional character of an object. In this article, we propose a comprehensive markerless method that describes an object’s symmetry using three types of fundamental symmetry, reflection, rotoinversion a combination of reflection and rotation and translation a process of reflection and rigid movement. This is done by mirroring an object over an arbitrary plane and aligning the mirrored image with the original object in a position that minimizes deviation between both objects. Each object’s symmetry can be displayed in two ways, numerically, with a best plane of symmetry or “Psym”, a fixed point and the mirrored objects rotation and magnitude of translation in relation to the original object, and visually, through a 3D deviation contour map. Three examples were made: Model 1 showed reflection symmetry and resulted in a standard deviation of 0.002 mm, Model 2 expressed rotoinversion symmetry and produced a standard deviation of 0.003 mm and Model 3 expressed translational symmetry which resulted in a translation magnitude difference of 0.015% with respect to model height. This simple procedure accurately recognizes reflection, rotoinversion and translation symmetry, takes minimal time and expertise and has the ability to expand previous case specific methods to a global application of symmetry analysis.

STRUCTURAL EVOLUTIONS AND PROPERTIES OF GERMANIUM CLUSTERS DURING RAPID COOLING PROCESSES

2013 ◽  
Vol 27 (32) ◽  
pp. 1350241
Author(s):  
T. H. GAO ◽  
W. J. YAN ◽  
X. T. GUO ◽  
X. M. QIN ◽  
Q. XIE
Keyword(s):  
Crystal Structure ◽  
Molecular Dynamics ◽  
Three Dimensional ◽  
Translational Symmetry ◽  
Transition Barrier ◽  
Germanium Cluster ◽  
Simulation Results ◽  
Dynamics Simulations ◽  
Germanium Clusters ◽  
Good Agreement

In this paper, structural evolutions of germanium cluster are studied by molecular dynamics simulations during quenching processes. Three-dimensional atomic configurations of germanium cluster are established. Our simulation results are in good agreement with the experimental ones. The structural properties of germanium are described in detail by means of several structural analysis methods. It is obtained that the 〈2, 3, 0, 0 〉 and 〈4, 0, 0, 0 〉 polyhedra play different roles in the course of liquid-to-amorphous transition. 〈4, 0, 0, 0〉 tend to be gathered together to form single crystal regions. However, 〈2, 3, 0, 0 〉 has five neighboring atoms that destroy the translational symmetry of the crystal structure, and enhances the transition barrier to crystals. Consequently, it is difficult for 〈4, 0, 0, 0 〉 to form crystal germanium at the cooling rate of 1.0 × 1010 °C/s.

Higher Dimensions of Clusters of Intermetallic Compounds

2021 ◽  
pp. 31-57
Keyword(s):  
Intermetallic Compounds ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Translational Symmetry ◽  
Higher Dimension ◽  
Internal Geometry ◽  
Metallic Nanoclusters ◽  
Diffraction Patterns

The author has previously proved that diffraction patterns of intermetallic compounds (quasicrystals) have translational symmetry in the space of higher dimension. In this chapter, it is proved that the metallic nanoclusters also have a higher dimension. The internal geometry of clusters was investigated. General expressions for calculating the dimension of clusters are obtained from which it follows that the dimension of metallic nanoclusters increases linearly with increasing number of cluster shells. The dimensions of many experimentally known metallic nanoclusters are determined. It is shown that these clusters, which are usually considered to be three-dimensional, have a higher dimension. The Euler-Poincaré equation was used, and the internal geometry of clusters was investigated.

scholarly journals Crystal structure of (E)-N′-(3,4-difluorobenzylidene)-4-methylbenzenesulfonohydrazide

2015 ◽  
Vol 71 (10) ◽  
pp. o761-o761
Author(s):  
Yeming Wang ◽  
Hong Yan
Keyword(s):  
Crystal Structure ◽  
Hydrogen Bonds ◽  
Dihedral Angle ◽  
Torsion Angle ◽  
Title Compound ◽  
Three Dimensional ◽  
Translational Symmetry ◽  
Aromatic Rings ◽  
Compound C ◽  
Dimensional Network

In the title compound, C14H12F2N2O2S, the dihedral angle between the aromatic rings is 70.23 (8)° and the S—N—N=C torsion angle is 172.11 (11)°. In the crystal, N—H...O hydrogen bonds link the molecules into [100]C(4) chains, with adjacent molecules in the chain related by translational symmetry. The chains are linked by weak C—H...F and C—H...O interactions, thereby forming a three-dimensional network.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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