A matrix approach to symmetry

In sharp contrast to other methods which focus on the consequences of symmetry (such as dot products, d spacings etc.), the matrix approach deals with symmetry in its most abstract form-represented as matrices. The basis of the matrix approach is to generate the matrices that transform the lattice into itself. The resulting group of matrices defines the holohedry of the lattice. These matrices may be used both theoretically and practically to analyze symmetry from any cell defining the lattice. The mathematics and algorithms used to analyze symmetry become extremely simple since they are based on manipulating integers and simple rational numbers using elementary linear algebra. The matrix approach provides the conceptual and practical framework required to perform experimental procedures in a logical and general manner. In practice, the symmetry matrices may be used to define the metric symmetry, the directions of the symmetry axes, the Laue symmetry, group-subgroup relationships, and conventional or standard cells. Because of its fundamental nature, the matrix approach should provide the basis for further experimental and theoretical advances in symmetry and symmetry-related topics in crystallography as well as in chemistry, physics and mathematics.