The use of a redundant axis in defining the basis of a lattice

The description of n-dimensional space by a basis of (n + 1) vectors, a_i (i = 1, . . . , n + 1), is discussed, with particular reference to the Miller–Bravais system of indexing hexagonal crystals. It is shown that if a hyperplane with normal h makes intercepts h_i⁻¹ on the a_i and a vector u has components u_i relative to the a_i, then h. u = Σh_iu_i without any further restrictions on the h_i or u_i. Furthermore, it is possible to find a basis a^†_i(i = 1, . . . , n + 1) for reciprocal space such that h = Σh_ia^†_i is always true and indeed there are n degrees of freedom available for choosing such a basis. Criteria which may lead to a unique choice of a^†_i are discussed.