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.
Laboratoire Structures, Propriétés et Modélisation des Solides UMR 8580, Ecole Centrale de Paris, F-92295 Châtenay-Malabry CEDEX, France.
R
R
Veysseyre
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D
Weigel
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T
Phan
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H
Veysseyre