Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Classification of crystal lattices

R. L. E. Schwarzenbergera1

a1 University of Warwick

There are two natural ways in which to attempt a classification of the lattices which occur in crystallography or, more generally, of n-dimensional lattices. The first method is algebraic and proceeds by classifying the symmetry groups of the lattices. Thus an n-dimensional lattice is a discrete subgroup

S0305004100047162_eqnU001

of Rn and determines a symmetry group

S0305004100047162_eqnU002

which is a subgroup of the orthogonal group On. Two lattices T1 and T2 are said to determine the same crystal system if the symmetry groups G(T1) and G(T2) are conjugate: that is there exists a linear isomorphism ø: RnRn such that

S0305004100047162_eqnU003

(Received September 09 1971)