CRYSTALLOGRAPHY IN 2-DIMENSIONAL METRIC SPACES

Crystallography in two-dimensional Euclidean, Galilean and Minkowskian space is defined and treated by means of a formalism which does not require an a priori specification of the character (definite, singular or indefinite) of the metric tensor left invariant by crystallographic transformations. It is shown how the present approach leads to general laws, which (if restricted to the Euclidean case) throw a new light on the structure of usual crystallography. The concept of natural lattice is introduced because it allows a fairly simple illustration of the properties mentioned above. In this way one directly arrives at arithmetic functions Pk(n) and Apk(n) having integral values for integral k and n. By means of these functions all crystallographic transformations can be expressed and parametrized by the integer n for all n∊Z. The Euclidean case is obtained for |n] < 2, the Galilean for |n| = 2, and the Minkowskian (relativistic) one for |n| > 2. As illustration, the geometric classes of two-dimensional crystallographic point groups are tabulated and the correspondence with the international notation in the Euclidean case is given. Furthermore, the symmetries of natural lattices are briefly discussed and completely set up. Actually the Galilean case (n = 2), even if included, needs still further investigation. The aim of the paper is to give a first synthetic view of the basic principles underlying crystallography. © 1969, Walter de Gruyter. All rights reserved.