Elements of a multimetrical crystallography

Abstract

Different metrics can be assigned to a given lattice of symmetry translations in such a way that the lattice is left invariant by circular and by hyperbolic rotations, respectively. Therefore, a crystallographic point group can he defined having as generators elements of orthogonal groups of same dimension and different signature. This explains the term multimetrical crystallography. Combining lattice translations with those point groups one obtains multimetrical space groups, which can be interpreted as symmetry groups of point-like crystal structures. The hexagonal close-packed structure and the Wurtzite crystal structure are discussed from this point of view and their multimetrical symmetry derived on the basis of the theory of integral binary and ternary quadratic forms. The connection with the group of units of quadratic fields is briefly explained.