R. Veysseyre

Crystallographic point groups of five-dimensional space. 1. Their elements and their subgroups

The purpose of this work is to introduce a method with a view to obtaining the crystallographic point groups of five-dimensional space, i.e. the subgroups of the holohedries of these space crystal families. This paper is specifically devoted to numerical analysis, whereas the following ones deal with some applications to crystallography. These results have been obtained through a collaboration between two teams: H. Veysseyre (Institut Supérieur des Matériaux) for the numerical analysis, R. Veysseyre, D. Weigel and Th. Phan (Ecole Centrale Paris) for the crystallographic part.

Crystal families and systems in higher dimensions, and geometrical symbols of their point groups. II. Cubic families in five- and n-dimensional spaces.

This paper is devoted to the study of the crystal families with cubic symmetries and to the mathematical construction of all their point-symmetry groups. The mono cubic crystal families of n-dimensional space (E(n)) are defined and a list of these families is given for spaces E4, E5, E6 and E7 with the Weigel-Phan-Veysseyre (WPV) symbols of their holohedries. The cubic and iso cubic crystal point-symmetry groups of space E(n) are also defined together with their properties and their WPV symbols. Some examples of these point groups are given. The 16 point groups of the three isomorphic mono cubic crystal families, the cubic-al family of space E4 (No. 19), the cube oblique (or cube parallelogram) family of space E5 (No. XVIII) and the triclinic cubic family of space E6 (No. 21) are listed. All the WPV symbols of the point-symmetry groups of all the mono cubic crystal families of space E5, i.e. the cube rectangle family (No. XXII), the cube square family (No. XXVI) and the cube hexagon family (No. XXVII), are given together with an explanation of the mathematical construction of these point-symmetry groups. All the di cubic crystal families of spaces E6, E7 and E8 are predicted and the symbols of their holohedries are given. Finally, some tri cubic crystal families of spaces E9, E10 and E11 are listed.

Crystallography, geometry and physics in higher dimensions. VII. The WPV symbols of the 38 cyclic crystallographic point symmetry groups in the five-dimensional space %E5

This paper is the first of a series of three devoted to crystallography in the five-dimensional space \bb E5. The 38 types of point symmetry operations (PSO for short) are described i.e. 19 types of PSO+s or rotations and 19 types of PSO-s or improper rotations; each of them generates a cyclic point group. A WPV (Weigel, Phan, Veysseyre) symbol is given both to the PSOs and to the cyclic groups. There is a generalization of the well known symbols of \bb E3. For instance, \bar 6 is the symbol of a point group of \bb E3 (and \bb E4), and \bar 6 has application in \bb E5 (and \bb E6); but new symbols such as \bar {\bar 6}, \bar {\bar 6} \bar {\bar 6} are also required.

Crystallographic point groups of five-dimensional space. 2. Their geometrical symbols.

Our previous paper emphasized a method for obtaining the crystallographic point groups of five-dimensional space, i.e. the subgroups of the crystal family holohedries. Moreover, it recalled the names of the crystal families and the symbols of their holohedries. These results being obtained, this paper gives a geometrical symbol to each of these point groups described as Weigel-Phan-Veysseyre symbols (WPV symbols). In most cases, these symbols make it possible to reconstitute all the elements of the groups. The point symmetry operation symbols, which are the basis of the Hermann-Mauguin symbols (HM symbols) as well as of the WPV symbols, that have been defined from the cyclic groups generated by the five-dimensional point symmetry operations are recalled. The basic principles of the WPV system of crystallographic point-group symbols are explained and a list of 196 symbols of five-dimensional space out of 955 is given. All the information given by the WPV symbol of a point group is detailed and analysed through some examples and the study of the (hexagon oblique)-al crystal family. Finally, the polar point groups of five-dimensional space are specified.

Crystallography, Geometry and Physics in Higher Dimensions. II. Point Symmetry of Holohedries of the Two Hypercubic Crystal Systems in Four-Dimensional Space

The interpretation of physical properties of incommensurate modulated crystals leads to the use of their point groups and their total character tables in their superspaces. Examples are chosen of point groups of holohedries of the two hypercubic crystal systems - primitive and body-centred- in four-dimensional space. A geometrical presentation is given of the point group - including its character table - of the primitive hypercubic crystal system, as it is useful for the prediction and simplification of tensorial physical properties of the corresponding crystals. Through geometrical considerations, the exceptional splitting of the hypercubic family of \bb E4 into two crystal systems is easily proved. Finally, the two different relations - according to the parity of n - existing between the point group of the primitive hypercube of \bb En and its subgroup of rotations are explained.

Crystallography, Geometry and Physics in Higher Dimensions. I. Point-Symmetry Operations

Physical phenomena such as incommensurate phases or diffraction enhancement of symmetry are interpreted by using symmetry groups in four, five or six dimensions. This first paper concerns the point-symmetry operations (PSO) in these Euclidean superspaces. Elementary, non-elementary, degenerate and non-degenerate PSOs are defined and their geometrical supports and geometrical symbols are specified. A geometrical description is thus given of nineteen types of PSO which are either the crystallographic rotations of the four-dimensional space or the crystallographic rotations and improper rotations of the five-dimensional space or the improper crystallographic rotations of the six-dimensional space. These PSOs are elements of crystallographic point groups of these spaces and the physical application to polar point groups is given.

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

This paper mainly consists in counting the crystal families of five-dimensional space and then in giving a geometrical name to each of them. All crystal cells of E 5 are obtained as orthogonal products of cells belonging to spaces of dimension less than five. Thanks to this geometrical approach, many general results can easily be found as WPV symbols of the holohedries (Weigel, Phan, Veysseyre symbols), the quadratic form associated with each lattice, the subgroups of these holohedries. The number of crystal families counted here is obviously the same as the number given by Plesken [Match (1981), No. 10, pp. 97-119] by a quite different method.

Crystallography, geometry and physics in higher dimensions. XI. A new geometrical method for systematic construction of the n-dimensional crystal families: reducible and irreducible crystal families

A geometrical method is described for counting and constructing all crystal families of Euclidean spaces of dimension n from the different well known crystal families of spaces E1, E2, E3 and the geometrically Z-irreducible families of each space. The definition of the geometrically Z-irreducible (gZ-irr.) or geometrically Z-reducible (gZ-red.) families is connected to the properties of the character table of the holohedry of these families. Indeed, a crystal family of space En is said to be gZ-irr, if the n translation operators corresponding to a basis of a primitive Bravais cell belong to the same irreducible representation with integer entries of its holohedry. In the opposite case, the family is said to be gZ-red. This method enables a name to be given to the crystal families. This name is connected to the geometrical construction, except for the families considered as irreducible. As far as possible, it also recalls the name of the crystal families of spaces E1, E2and E3. Moreover, the WPV (Weigel–Phan–Veysseyre) symbols of the holohedries can be defined thanks to the properties of the crystal cells. Finally, all the point-symmetry operations of these groups and subgroups can be listed.

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

The aim of this paper and of the following one [Weigel, Phan & Veysseyre (2008). Acta Cryst. A64, 687-697] is to complete the list of the Weigel-Phan-Veysseyre (WPV) symbols of the point groups of space E5 that was started in previous papers and in two reports of an IUCr Subcommittee on the Nomenclature of n-Dimensional Crystallography. In this paper, some crystal families of space E5 are studied. The cells of these are right hyperprisms with as a basis either two squares, or two hexagons, or a square and a hexagon. If the basis is made up of two squares, the two families are the (monoclinic di squares)-al family (No. XVI) and the (di squares)-al family (No. XIX). If the basis is made up of two hexagons, the two families are the (monoclinic di hexagons)-al family (No. XVII) and the (di hexagons)-al family (No. XXI). If the basis is made up of one square and one hexagon, the family is the (square hexagon)-al family (No. XX). In order to link space E5 to spaces E2, E3 and E4, some results published in previous papers are recalled. In fact, most of the symbols of the point groups of space E5 can be deduced from the symbols of the four, six and 23 crystal families of spaces E2, E3 and E4, respectively.

Crystallography, geometry and physics in higher dimensions. X, Super point groups in five-dimensional space for the di-incommensurate structures

This paper, the third of a series devoted to crystallography in five-dimensional space E5, deals with the di-incommensurate structures. Physical considerations on the vectors of modulation have enabled the definition and listing of the di-incommensurate point symmetry operations, the di-incommensurate point symmetry groups and the di-incommensurate crystal families of the space E5.