Crystallography, geometry and physics in higher dimensions. IX. Counting and geometry of the 32 crystal families of five-dimensional space

1991 ◽  
Vol 47 (3) ◽  
pp. 233-238 ◽  
Author(s):  
R. Veysseyre ◽  
T. Phan ◽  
D. Weigel
Keyword(s):  
Dimensional Space ◽  
Higher Dimensions

scholarly journals Rational Trigonometry in Higher Dimensions and a Diagonal Rule for $2$-planes in Four-dimensional Space

KoG ◽  
2017 ◽  
pp. 47-54
Author(s):  
Norman Wildberger
Keyword(s):  
Dimensional Space ◽  
Higher Dimensions ◽  
Natural Generalization ◽  
Principal Angles ◽  
Rational Trigonometry ◽  
Pythagoras Theorem ◽  
The Cross

We extend rational trigonometry to higher dimensions by introducing rational invariants between $k$-subspaces of $n$-dimensional space to give an alternative to the canonical or principal angles studied by Jordan and many others, and their angular variants. We study in particular the cross, spread and det-cross of $2$-subspaces of four-dimensional space, and show that Pythagoras theorem, or the Diagonal Rule, has a natural generalization forsuch $2$-subspaces.

The Osculatory Packing of a Three Dimensional Sphere

1973 ◽  
Vol 25 (2) ◽  
pp. 303-322 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Specific Knowledge ◽  
Precise Information ◽  
Practical Applications ◽  
Physical Problems ◽  
Three Dimensional Space

Packings by unequal spheres in three dimensional space have interested many authors. This is to some extent due to the practical applications of such investigations to engineering and physical problems (see, for example, [16; 17; 31]). There are a few general results known concerning complete packings by spheres in N-dimensional Euclidean space, due mainly to Larman [20; 21]. For osculatory packings, although there is a great deal of specific knowledge about the two-dimensional situation, the results for higher dimensions, such as [4], rely on general methods which do not give particularly precise information.

NATURE OF THE SINGULARITIES IN HIGHER-DIMENSIONAL HUSAIN SPACE–TIME

2006 ◽  
Vol 15 (09) ◽  
pp. 1359-1371 ◽  
Author(s):  
K. D. PATIL ◽  
S. S. ZADE
Keyword(s):  
Gravitational Collapse ◽  
Dimensional Space ◽  
Higher Dimensions ◽  
Cosmic Censorship ◽  
Space Time ◽  
Spherically Symmetric ◽  
Naked Singularities ◽  
Higher Dimensional ◽  
Dimensional Space Time ◽  
Cosmic Censorship Hypothesis

We generalize the earlier studies on the spherically symmetric gravitational collapse in four-dimensional space–time to higher dimensions. It is found that the central singularities may be naked in higher dimensions but depend sensitively on the choices of the parameters. These naked singularities are found to be gravitationally strong that violate the cosmic censorship hypothesis.

Crystallography, geometry and physics in higher dimensions. V. Polar and mono-incommensurate point groups in the four-dimensional space %E4

1989 ◽  
Vol 45 (2) ◽  
pp. 187-193 ◽  
Author(s):  
R. Veysseyre ◽  
D. Weigel
Keyword(s):  
Euclidean Space ◽  
Dimensional Space ◽  
Physical Space ◽  
Higher Dimensions ◽  
Dimensional Euclidean Space ◽  
Acta Cryst ◽  
Magnetic Structures ◽  
Convenient Means ◽  
Incommensurate Structures ◽  
Point Groups

The crystallographic point groups of the four-dimensional Euclidean space {\bb E}4are a convenient means of studying some crystallized solids of physical space, for instance the groups of magnetic structures and the groups of mono-incommensurate structures, as is demonstrated by a simple example. The concept of polar crystallographic point groups defined here in {\bb E}4, and also in {\bb E}nenables the list and the WPV notation {geometric symbol of Weigel, Phan & Veysseyre [Acta Cryst.(1987), A43, 294-304]} of these special structures to be stated in a more precise way. This paper is especially concerned with the mono-incommensurate structures while a discussion on magnetic structures will be published later.

Free Particle, Wave Packet and Dynamics, Quantum Dots, Defects and Traps

2021 ◽  
pp. 32-56
Author(s):  
Duncan G. Steel
Keyword(s):  
Wave Packet ◽  
Dimensional Space ◽  
Free Particle ◽  
Plane Waves ◽  
Higher Dimensions ◽  
Destructive Interference ◽  
Piecewise Constant ◽  
Wave Package ◽  
Quantum Behavior ◽  
The Waves

This chapter continues with a study of the time independent Schrödinger equation and seeks to contrast the quantum behavior of a free particle with that of a particle localized in a potential quantum well. A free particle can exist over all space or can be localized in a wave package. The wave packet is a coherent superposition of the plane waves that make up the wave function that localizes the particle because of constructive and destructive interference. The wave packet spreads out in time because the waves leading to constructive interference get out of phase. In Chapter 2, the particle was localized by a quadratic potential energy. Here, the potentials are described as piecewise constant. The approach is based on assuming a one-dimensional space, x, which is relevant to many problems in the laboratory. The solution is easily generalized to higher dimensions (x-y or x-y-z), but the physics remains the same. The objective is to understand the shape of the eigenfunctions in space and to be able to relate this to the probability density of locating the particle, as well as understanding the relevance of these systems to today’s technology.

The Orthocentric Simplex in Space of Three and Higher Dimensions.

1935 ◽  
Vol 19 (233) ◽  
pp. 102-108
Author(s):  
H. Lob
Keyword(s):  
Dimensional Space ◽  
Flat Space ◽  
Higher Dimensions ◽  
Higher Order ◽  
Orthocentric Simplex

By a simplex is meant the figure in general flat space of which the triangle and tetrahedron are examples. In space of n dimensions the simplex has (n + 1) vertices and (n + 1) faces, each face consisting of a prime, i.e. a space of (n - 1) dimensions, containing n of the vertices and hence itself a simplex in (n - 1)-dimensional space. There is a general likeness between the triangle and simplexes of higher order, but this is marred by the fact that, in general, a simplex does not possess an orthocentre.

scholarly journals Un modelo para visualizar objetos en 4D con el Mathematica

2018 ◽  
pp. 51-58
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Two Dimensional ◽  
3D Objects ◽  
El Sistema ◽  
Definition Of ◽  
Three Dimensional Space

Un modelo para visualizar objetos en 4D con el Mathematica A model to visualize objects in 4D with Mathematica Ricardo Velezmoro y Robert Ipanaqué Universidad Nacional de Piura, Urb. Miraflores s/n, Castilla, Piura, Perú.  DOI: https://doi.org/10.33017/RevECIPeru2014.0008/ Resumen Una variedad de técnicas de gráficos por computadora han permitido la visualización de objetos, que existen en dimensiones más altas, en una pantalla 2D. En este artículo se propone un nuevo modelo a partir de la extensión de una técnica útil en la visualización de objetos en 3D en una pantalla 2D para realizar algo similar con objetos en 4D. Dicha técnica se basa en la definición de una inmersión, en primera instancia, del espacio tridimensional en el espacio bidimensional que luego se toma como referencia para definir otra inmersión, que constituye el modelo propuesto en este artículo, del espacio tetra dimensional en el espacio tridimensional. En teoría la visualización de objetos en 4D en una pantalla 2D se consigue mediante la composición de las dos inmersiones mencionadas, pero en la práctica se aprovechan los comandos incorporados en el sistema de cálculo simbólico Mathematica para tal fin. Descriptores: objetos 4D, modelo, inmersión Abstract A variety of computer graphics techniques have enabled the display of objects, which exist in higher dimensions, on a 2D screen. In this paper a new model from the extension of a technique useful in visualizing 3D objects on a 2D screen to make something similar with 4D objects is proposed. This technique is based on the definition of a immersion, in the first instance, from the three-dimensional space in two-dimensional space which is then taken as a reference to define another immersion, which is the model proposed in this paper, from the fourdimensional space in three dimensional space. Theoretically the visualization of objects in 4D on a 2D screen is achieved by the composition of the two immersions mentioned, but in practice the incorporated commands into the computer algebra system Mathematica for this purpose are utilized. Keywords: objects 4D, model, immersion.

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