Crystal families and systems in higher dimensions, and geometrical symbols of their point groups. I. Crystal families in five-dimensional space with two-, three-, four- and sixfold symmetries

2008 ◽  
Vol 64 (6) ◽  
pp. 675-686 ◽  
Author(s):  
R. Veysseyre ◽  
D. Weigel ◽  
Th. Phan
Keyword(s):  
Dimensional Space ◽  
Higher Dimensions ◽  
Point Groups

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.

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.

Choosing the best parameters for method of deformed stars in n-dimensional space

2021 ◽  
pp. 24-28
Author(s):  
Maryna Antonevych ◽  
Anna Didyk ◽  
Nataliia Tmienova ◽  
Vitaliy Snytyuk
Keyword(s):  
Genetic Algorithm ◽  
Evolutionary Algorithms ◽  
Differential Evolution ◽  
Classical Method ◽  
Dimensional Space ◽  
New Method ◽  
Evolutionary Strategy ◽  
Point Groups ◽  
Best Parameters

This paper is devoted to the problem of optimization of a function in -dimensional space, which, in general case, is polyextreme and undifferentiated. The new method of deformed stars in n-dimensional space was proposed. It is built on the ideas and principles of the evolutionary paradigm. Method of deformed stars is based on the assumption of using potential solutions groups. There by it allows to increase the rate of the accuracy and the convergence of the achieved result. Populations of potential solutions are used to optimize the multivariable function. In contrast to the classical method of deformed stars, we obtained a method that solves problems in -dimensional space, where the population of solutions consists of 3-, 4-, and 5-point groups. The advantages of the developed method over genetic algorithm, differential evolution and evolutionary strategy as the most typical evolutionary algorithms are shown. Also, experiments were performed to investigate the best configuration of method of deformed stars parameters.

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