CONSTRUCTION OF THE 4, 5, 6,..DIM CRYSTALLOGRAPHIC POINT GROUPS OUT OF A CONSISTENT SYSTEM OF SYMBOLS
Dominique Weigel, Renée Veysseyre and Thérese Phan
Laboratoire de Chimie Physique du Solide
(Unité de Recherche CNRS 453) et Département de Mathématiques,
Ecole Centrale de Paris, Grande Voie des Vignes, 92295
Chatenay-Malabry CEDEX, France
weigel@cps.ecp.fr
Keywords : n dim. Crystallography, Geometrical (g)Z Reducible and Irreducible Crystal Families, Point Groups, Generators, Geometrical Supports of the Elements.
The definition of the reducibility (and the irreducibility) of the Crystal Families are given in references (1) and (2).
The Symbols of that System immediately enable us to find :
- the crystal family,
- the construction of its holohedry point group,
- a set of generators of the point group and its order,
- the geometrical supports of its elements.
Examples :
4 dim. 4_2m.4 gZ-Reducible Family idi (orthogonal) squaresî whose the holohedry is 4mm^4mm, of order 64 (8x8). Point group 4_2m acts in xyz space and 4 acts in zt space, orhogonal to xy space; its order is 32 (8x4) and the point group symbol is the product of two group symbols : 4_2m and 4, from which one may obtain sets of generators and geometrical supports of its elements.
4 dim. 4_3m.[10] gZ Irreducible Family : iRhombotope (cos a = -1/4)i; this cell is bounded by ten 3 dim. regular tetrahedra; it is the generalization of the rhombohedron (cos a = -1/3). Cyclic group [10] is generated by a double rotation with angles 2p/10 in two orthogonal planes. Point group 4_3m.[10] is the holohedry of this family, its order is 240 (24x10).
5 dim. (4_2m.4)^m gZ Reducible Family idi squares-alî (right hyper prism based on di (orthogonal) squares. Its order is 64 (32x2)
6 dim. m3_m^4_3m gZ Reducible Family idi (orthogonal) cubesî whose the holohedry is m3_m^m3_m, of order 2304 (48x48). Group m3_m acts in xyz space while 3_3m acts in tuv space, orthogonal to xyz space. The point group order is 1152 (48x24).