NUMBER AND SYMBOLS OF CRYSTALLOGRAPHIC POINT GROUPS OF FIVE-DIMENSIONAL SPACE. R. Veysseyre¹, H. Veysseyre², D. Weigel¹, T. Phan¹, ¹Laboratoire de Chimie Physique du Solide et Département de Mathématiques Ecole Centrale Paris 92295 Châtenay Malabry France, ²Institut Supérieur des Matériaux (ISMCM) 3 rue Fernand Hainaut 93407 Saint-Ouen France

This is an addition and conclusion to the paper introduced during the IUCr XVI congress (Beijing 1993). A classification of the crystal families into geometrically Z reducible and geometrically Z irreducible families has been suggested as a result of both geometrical properties of their cells and different types of point symmetry operations [1], [2], [3]. For example, there are 29 gZ-red. and 3 gZ-irred. crystal families in space E⁵. Each crystal family holohedry has been given a WPV (Weigel, Phan, Veysseyre) symbol [4] [5], generalizing the Hermann-Mauguin symbols. The family indexing has been achieved according to the decreasing number of the cell geometrical parameters.

By means of computer, all the subgroups of the different holohedries i.e. the crystallographic point groups of space E⁵ has been listed.

The crystal family study having proved dissatisfactory as regards the working out of a logical link between all the point groups, it was necessary to go into all possible centerings of space E⁵, hence to define "centered families" in connection with the previously defined families; the family VII splits into the the triclinic-hexagon R centered family (holohedry , mm) and the triclinic-hexagon family (holohedry ).

The definition of the WPV symbols has been achieved out of either algebraic properties of groups (direct product or semi-direct product) or the cell geometry (generator research belonging to supplementary orthogonal subspaces). These symbols should permit to list all point elements.Results about these point groups have been given showing the number of point groups of given orders or belonging to each family.

[1] T. Phan Thesis Paris VI 1989

[2] R. Veysseyre, D. Weigel and T. Phan Acta Cryst 1993 A49, 481-486

[3] Plesken W. Match 1981 ndeg. 10 97-119

[4] D. Weigel, T. Phan and R. Veysseyre Acta Cryst. 1987 A 43, 294-304

[5] R. Veysseyre Thesis Paris VI 1987