摘要
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.
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.
