Miller-Bravais indices
/ Bravais: bruh-VAY /
Hexagonal crystals have an awkward feature: in their flat basal plane there are three axes that all look identical, spaced 120 degrees apart, not two. If you force the usual three-index Miller scheme onto them, symmetry-equivalent faces end up with indices that look nothing alike, which is confusing. Miller-Bravais indices fix this by using four numbers, (hkil), one for each of the three in-plane axes (h, k, i) plus one for the vertical c axis (l).
The third index is not free — it is fixed by the other two through the rule i = -(h + k). It carries no new information at all; it exists purely so that the three equivalent prism faces of a hexagonal crystal show off matching index patterns. For instance the three side faces of a hexagonal prism become (1 0 -1 0), (0 1 -1 0) and (-1 1 0 0): permutations of one another, obviously the same family {1 0 -1 0}. In the plain three-index scheme those same faces would read (100), (010) and (-110), whose kinship is hidden. Directions use the same four-index style, [uvtw], again with t = -(u + v), and converting from a three-index [UVW] takes a small formula: u = (2U - V)/3, v = (2V - U)/3, t = -(u + v), w = W.
So the redundant index is a piece of deliberate bookkeeping that makes hexagonal symmetry visible at a glance. You will meet it constantly in the structure of metals like magnesium, titanium, zinc and cobalt, and it is why the close-packed basal plane of such a metal is written (0001) rather than (001).
The three prism faces of a hexagonal metal read (1 0 -1 0), (0 1 -1 0), (-1 1 0 0) in the four-index scheme — clearly one family, since each is a permutation of the others. The basal (close-packed) plane on top is (0001).
The extra index, always i = -(h+k), makes equivalent hexagonal planes share an obvious pattern.
The i index is redundant, not extra information — it is completely determined by i = -(h + k). Its only job is to make hexagonal symmetry legible; it is sometimes shown as a dot, as in (h k . l).