Lattices and reduced cells as points in 6-space and selection of Bravais lattice type by projections

A characterization of crystallographic unit cells as vectors in a Euclidean six-dimensional space (E⁶ in the usual mathematical notation; here termed G⁶) is introduced, in which the non-triclinic Bravais lattice types form one-, two-, three- and four-dimensional linear subspaces. This formalism makes the determination of the 'best' Bravais lattice (or lattices) for a particular experimentally determined cell a process of determining Euclidean distances in G⁶ from the cell to its projections into the subspaces of the lattice types. The elements of vectors in the space are drawn from the Niggli matrix with the unsymmetrical elements doubled. A cell is first reduced and all its nearly Buerger-reduced cells are used in the distance determinations. Thus the smallest distance provides information about both the propriety of the lattice type selection and the instability of the cell reduction.