This paper mainly consists of the classification of all crystallographic point groups of n-dimensional space with n ≤ 6 into different isomorphism classes. An isomorphism class is defined by a type of finite mathemat...This paper mainly consists of the classification of all crystallographic point groups of n-dimensional space with n ≤ 6 into different isomorphism classes. An isomorphism class is defined by a type of finite mathematic group;for instance, the different types of mathematic groups have been well defined and studied by Coxeter. This classification may be used in the investigation of several domains of crystallography such as the study of the incommensurate phases, the quasi crystals … Indeed, each mathematic substitution group characterizes an isomorphism class of crystallographic point groups (spaces E2 or E3), of point groups of super crystals (spaces E4 or E5), and of molecular symmetry groups (spaces E2 or E3). This mathematic group gives interesting information about: 1) the incommensurate phase structures and their phase transitions according to the Landau’s theory in their super spaces E4, E5, E6, ···;2) the molecular symmetry group of chemisorbed molecules in space E2 (paragraph 2) or of the molecular crystal or solution in view of studying the molecule structure or its rotations or vibrationsin space E3;3) the geometric polyhedron symmetry groups as the regular rhombohedron in space E3, the rhombotope in space E4 or the rhombotope in space E5. Then, thanks to the isomorphism classes, we shall give properties of some crystal families that we have not published up to now. This formalism may be used to study crystal families in n-dimensional space with n > 6.展开更多
In two previous papers, we explained the classification of all crystallographic point groups of n-dimensional space with n ≤ 6 into different isomorphism classes and we describe some crystal families. This paper main...In two previous papers, we explained the classification of all crystallographic point groups of n-dimensional space with n ≤ 6 into different isomorphism classes and we describe some crystal families. This paper mainly consists in the study of three crystal families of space E5, the (di-iso hexagons)-al, the hypercube 5 dim and the (hypercube 4 dim)-al crystal families. For each studied family, we explain their name, we describe their cell and we list their point groups which are classified into isomorphism classes. Then we give a WPV symbol to each group. (WPV means Weigel Phan Veysseyre). Our method is based on the description of the cell of the holohedry of each crystal family and of the results given by the Software established by one of us. The advantage to classify the point groups in isomorphism classes is to give their mathematical structures and to compare their WPV symbols. So the study of all crystal families of space E5 is completed. Some crystal families of space E5 can be used to describe di incommensurate structures and quasi crystals.展开更多
In the paper N0II, we describe some isomorphism classes and we apply their properties to the study of five crystal families of space E5. The names of these families are the following ones (monoclinic di iso squares)-a...In the paper N0II, we describe some isomorphism classes and we apply their properties to the study of five crystal families of space E5. The names of these families are the following ones (monoclinic di iso squares)-al, decadic-al, (monoclinic di iso hexagons)-al, (rhombotopic cosa=-1/4)-al and rhombotopic cosa=-1/5. The meaning of these names will be given in Paragraphs 5 and 6 with some geometric properties of their cell.展开更多
文摘This paper mainly consists of the classification of all crystallographic point groups of n-dimensional space with n ≤ 6 into different isomorphism classes. An isomorphism class is defined by a type of finite mathematic group;for instance, the different types of mathematic groups have been well defined and studied by Coxeter. This classification may be used in the investigation of several domains of crystallography such as the study of the incommensurate phases, the quasi crystals … Indeed, each mathematic substitution group characterizes an isomorphism class of crystallographic point groups (spaces E2 or E3), of point groups of super crystals (spaces E4 or E5), and of molecular symmetry groups (spaces E2 or E3). This mathematic group gives interesting information about: 1) the incommensurate phase structures and their phase transitions according to the Landau’s theory in their super spaces E4, E5, E6, ···;2) the molecular symmetry group of chemisorbed molecules in space E2 (paragraph 2) or of the molecular crystal or solution in view of studying the molecule structure or its rotations or vibrationsin space E3;3) the geometric polyhedron symmetry groups as the regular rhombohedron in space E3, the rhombotope in space E4 or the rhombotope in space E5. Then, thanks to the isomorphism classes, we shall give properties of some crystal families that we have not published up to now. This formalism may be used to study crystal families in n-dimensional space with n > 6.
文摘In two previous papers, we explained the classification of all crystallographic point groups of n-dimensional space with n ≤ 6 into different isomorphism classes and we describe some crystal families. This paper mainly consists in the study of three crystal families of space E5, the (di-iso hexagons)-al, the hypercube 5 dim and the (hypercube 4 dim)-al crystal families. For each studied family, we explain their name, we describe their cell and we list their point groups which are classified into isomorphism classes. Then we give a WPV symbol to each group. (WPV means Weigel Phan Veysseyre). Our method is based on the description of the cell of the holohedry of each crystal family and of the results given by the Software established by one of us. The advantage to classify the point groups in isomorphism classes is to give their mathematical structures and to compare their WPV symbols. So the study of all crystal families of space E5 is completed. Some crystal families of space E5 can be used to describe di incommensurate structures and quasi crystals.
文摘In the paper N0II, we describe some isomorphism classes and we apply their properties to the study of five crystal families of space E5. The names of these families are the following ones (monoclinic di iso squares)-al, decadic-al, (monoclinic di iso hexagons)-al, (rhombotopic cosa=-1/4)-al and rhombotopic cosa=-1/5. The meaning of these names will be given in Paragraphs 5 and 6 with some geometric properties of their cell.
